What are the limits of Mathematics?

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What are the limits of mathematics?

Oskar John Hollinsworth

October 24, 2014

Abstract

In this dissertation I will outline informally1 the arguments for key limitative results in meta-mathematics,

accompanied by the history of philosophical thought surrounding these developments. I will focus on the work

of Godel, Tarski, Lob and Turing in eliminating any hope of a fulfilment of Hilberts program. These results will

be shown to demonstrate that the consistency, soundness and truthfulness of mathematics cannot be trusted

in the way that mathematicians had previously always assumed. Then I will critically evaluate the claims of

mathematical philosophers with regard to the impact of these results on the limits of artificial intelligence and

the human mind.

Contents

1 Introduction 2

2 What is Mathematics? 2

3 Naive Set Theory and Russells Paradox 4

4 Godels First Incompleteness Theorem 5

5 How limiting are these theorems? 7

6 Minds and Machines 7

7 Computers and Decidability 9

8 Conclusion 10

Appendices 11

A Formal Systems as the Foundation of Mathematics 11

A.1 First-order Logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11A.2 Peano Arithmetic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16A.3 Naive Set Theory and Russells Paradox . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

B Godels First Incompleteness Theorem 19

B.1 Godel Numbering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19B.1.1 Primitive Recursive Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19B.1.2 Coding Theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

B.2 The Formalised Liars Paradox . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24B.3 Extending the First Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

C Godels Second Incompleteness Theorem 27

C.1 An Informal Introduction to Surpassing the First Theorem . . . . . . . . . . . . . . . . . . . . . . . 27C.2 The Hilbert-Bernays-Lob Derivability Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29C.3 The Formalised First Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

1See the Appendix for a much more formal treatment of these results

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C.4 The Formalised Second Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31C.5 Extending the Second Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

D Turing Machines 34

D.1 An Informal Introduction to the Church-Turing Thesis . . . . . . . . . . . . . . . . . . . . . . . . . . 34D.2 -recursive functions and Turing computation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34D.3 Decidability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37D.4 Halting Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

Glossary 38

1 Introduction

I chose to research the proofs and implications of these modern results in mathematical logic primarily because of theelegant ideas they describe which are readily accessible to non-mathematicians. They have profound philosophicalimplications on the fundamental limits of minds and machines which lead to interesting active debate even in thepresent day. As a result, I wanted my project title to reflect the fact that I wished to come to my own conclusionon the extent of the impact of these theorems. The proofs of these results also rely only on elementary numbertheory which does not exceed the difficulty of multiplication and division, although complete constructions are

highly complex. Ultimately it was the intriguing claim by Godel that we can know that there are some things whichwe can never know which made me decide that this would be an interesting yet very accessible field to researchin depth. My progress towards understanding the arguments behind these results was impeded by sub-optimalsources which wasted my time as I tried to decipher the information they contained. I was finally able to reallythrive once I had developed enough skill at using these sources efficiently to study the concise academic literaturein the field.

2 What is Mathematics?

Mathematicians are famous for their stubborn demand for rigorous proof of their conjectures. Fermats LastTheorem was doubted by mathematicians for 357 years before any possible doubt was allayed by a general, absolute

and irrefutable proof by Andrew Wiles published in 1995.2

Mathematicians take real pleasure not in their results,but the justification for their results3. These proofs consist of a deductive series of logical steps starting fromcertain assumptions or axioms. If you accept the axioms then you must necessarily accept the proven statements(or theorems) derived from them. For mathematicians to maximise their certainty that these proofs are sound, theirultimate dream is always to reduce the reasoning process into such unambiguous steps that the rules of inference canbe formalised to typographical rules for the transformation of strings of symbols that need not be understood4. Oncea demonstration of a mathematical statement is fully understood it should be possible to create an unequivocalalgorithm which a computer could follow to produce the same theorems from the given axioms, even though itmight unthinkingly go through every other possible theorem before reaching the desired result. The combinationof these axioms and laws of transformation which can be used to prove theorems is known as a formal system ofmathematics and is the central object of study in meta-mathematics.

Human reason may seem to transcend simple computation, but the basic rules are easy to formulate. This is knownas First-order Logic5 and formulating it mechanically has surprising implications. Theorems such as ifP is truethenPis true may come as no surprise (Lemma 3 p.12), but the Principle of Explosion is a more surprising result(Theorem 1 p.14). This theorem states that if a single inconsistency is proven, then any well-formed formula isprovable. In other words, a single contradiction is contagious and will explode into the whole theory in which it iscreated.

2Andrew Wiles. Modular elliptic curves and Fermats Last Theorem. In: Annals of Mathematics 142 (1995), pp. 443551.3G.H Hardy once famously said to Bertrand Russell: If I could prove by logic that you would die in five minutes, I should be sorry

you were going to die, but my sorrow would be very much mitigated by pleasure in the proof.4David Hilbert described it so: Mathematics is a game played according to certain rules with meaningless marks on paper.5Appendix A.1,p.11

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Figure 1: In Hyperbolic Geometry there are many different curved lines which are parallel to another line and gothrough the same point because they can curve away from each other.

Figure 2: In Elliptical Geometry all lines curve round the sphere so they have to meet at some point, meaning thatparallel lines do not exist (see previous page).

This formalisation of Logic was so elegant that it led many such as the famous mathematician and philosopherBertrand Russell to conclude that all mathematical knowledge ought to be derivable from Logic with extendingdefinitions6. Hilbert became interested in these mathematical foundations from studying Non-Euclidean Geometries.These controversial theories rejected Euclids Parallel Postulate, leading to triangles in which the sum of the anglesis never 180. The parallel postulate states that there is only one unique line which goes through a given pointand is parallel to another given line. This is clearly true for a flat surface like a table, but not for a curved surface

where the lines can circle back round to where they started. In hyperbolic geometry, this axiom is replaced with theassumption that there are infinitely many such lines, leading to triangles with interior angles which sum to strictlyless than 180, which describes the surface of a saddle. By instead assuming that there are no parallel lines, themathematics then describes elliptical geometry, where the sum of the interior angles in a triangle is strictly morethan 180, such as on the surface of a sphere.

Hilbert initially struggled to have his controversial ideas recognised. Immanuel Kant believed that space was flat bydefinition9 and supposing otherwise was self-contradictory. In the face of disbelief that Euclids Parallel Postulatecould be violated, Hilbert proved10 the consistency of these non-Euclidean geometries by showing that if EuclideanGeometry was consistent then non-Euclidean Geometry must also be consistent. Mathematicians take great pleasurein proving philosophers wrong and Hilbert was particularly clearly vindicated when Albert Einstein showed11 that

6In his own words: all mathematics deals exclusively with concepts definable in terms of a very small number of logical concepts,and . . . all its propositions are deducible from a very small number of fundamental logical principles.

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Figure 1: Coral Jasmin. Crazy hyperbole, crazier crocheting and the craziest me. url: https://www.google.co.uk/search?q=hyperbolic+geometry&source=lnms&tbm=isch&sa=X&ei=Cx4KVJO3IZKf7Aa_woGADA&ved=0CAgQ_AUoAQ&biw=1920&bih=955#tbm=

isch&q=hyperbolic+geometry+parallel+lines&facrc=_&imgdii=_&imgrc=fSM61UxxkI2aCM%253A%3BBNK0-qtNyJzS-M%3Bhttps%253A%

252F%252Fcoraljasmin.files.wordpress.com%252F2012%252F05%252Fhyperbolic-space.jpg%3Bhttp%253A%252F%252Fcoraljasmin.

wordpress.com%252Fcategory%252Fscience%252F%3B3283%3B1843; Wikimedia Commons. Hyperbolic triangle. url: http://commons.wikimedia.org/wiki/File:Hyperbolic_triangle.svg#mediaviewer/File:Hyperbolic_triangle.svg; Annenberg Learner. Sphericaland Hyperbolic Geometery. url: http://www.learner.org/courses/mathilluminated/units/8/textbook/04.php

8Figure 2: Ravi Bhoraskar. Non Euclidean Geometries. url: http: // blogbloggityblog. wordpress. com/ 2011/09/09/ non-euclidean-geometries/; Wikimedia Commons. Triangle trirectangle. url: http://commons.wikimedia.org/wiki/File:Triangle_trirectangle.png

9Space is . . . but a pure intuition(Immanuel Kant. Critik der reinen Vernunft. Penguin Classics, 1781)10David Hilbert. Grundlagen der geometrie. Leipzig, B.G. Teubner, 1903.11Albert Einstein and Marcel Grossman. Entwurf einer verallgemeinerten Relativitatstheorie und eine Theorie der Gravitation. In:

Zeitschrift fur Mathematik und Physik 62 (1913), pp. 225261.

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Hilberts Geometries actually describe our universe better than Euclids. Similarly, Cartesian geometry suggeststhat Euclidean Geometry is built up just from algebra. In this way, a hierarchy of mathematics seemed to besuggested, in which the consistency of all of mathematics depended on strong logical foundations 12. Hilbert listedthe task of proving the consistency of basic arithmetic as the second of his famous 23 great unsolved problems at thestart of the twentieth century.13 Hilberts beliefs about how research in foundational mathematics should proceed(that is, primarily focused on this reductionist formal justification for complex mathematics) is commonly knownas Hilberts program.

3 Naive Set Theory and Russells Paradox

Continually justifying the premises of ones previous proofs with more proofs quickly leads to an infinite regress.Hilbert was interested in a finitistic proof of consistency, which did not require appealing to infinitely many stepsin the proof. The hope was that a very small finite set of axioms could be chosen from which all of mathematicswas derived. At the turn of the twentieth century, naive set theory was created as an attempt to provide exactlythat. Numbers could simply be described as sizes of sets so in that sense set theory is even more fundamental tomathematics than arithmetic. Frege was attempting to derive all of fundamental mathematics from a few rulesabout sets in his great work: The Basic Laws of Arithmetic.14

Unfortunately, there was a a serious problem; Bertrand Russell was able to easily infer a paradox from the axioms

of naive set theory.15

This may seem like a minor misdemeanour, but recall that a single inconsistency could beused to infer any proposition in a sufficiently complete formal system of mathematics (specifically one includingfirst-order logic). Therefore we can see that this system, although capable of deriving that 1 + 1 = 2, was alsocapable of deriving Moon=Cheese.

The paradox at the heart of Russells proof is known as the Barbers Paradox. It is usually presented as follows:There is a town with only one barber. This barbers job description is to shave exactly those who do not shavethemselves. Who shaves the barber? If he does shave himself, then by the barbers job description, the Barbershould not shave him, therefore he should not shave himself. But if he does not shave himself, then by the barbers

job description he must shave him, therefore he does in fact shave himself. This strange circle of argument in whichyou successively have to believe that the barber does and does not shave himself is described as a strange loop byDouglas Hofstadter.16 Russell formulated17 the strange loop in naive set theory as follows. Lets call a set normalif it is not a member of itself and lets call the set of all normal sets Russells Set. Is Russells Set itself normal? If

it is indeed normal, then by definition it must be a member of Russells Set, so it is a member of itself and thus itmust not be normal. If it is not normal though, it must contain itself (by the definition of normal) and thus as amember of Russells Set it must be normal.

This paradox was very important because it demonstrated that fatal flaws can indeed go unnoticed in formalsystems, such that all of Freges work was essentially meaningless. A theorem of naive set theory has no bearingon truth. This is a particularly frightening result, as it demonstrated once and for all that it is not enough tosimply assume that axioms which seem reasonable must be consistent. Frege was happily deriving theorems in hisconception of naive set theory,18 completely unaware that he could just as easily have proven anything, includingthe negation of all his theorems. The prospect that mathematicians today could be living under the same illusionwithout even knowing it is what makes Russells Paradox so interesting.

As a result of Russells contribution, more restrictive systems were designed to ensure that paradoxes were trulyforbidden. This may sound like a simple task but recall that these symbols are meaningless, so to them xand notx

(the very definition of a contradiction) is just another string to be mechanically derived. Formal systems are purelytypographical, semantics has no bearing on their results whatsoever, except as far as the rules of the system are

12Marcus Du Sautoy describes it in The Music of the Primes like so: Hilbert had been pushed further and further back and was nowhaving to question the very basis on which mathematics had been founded.

13David Hilbert. Mathematical Problems. In: Bulletin of the American Mathematical Society 8 (1902), pp. 437479.14Gottlob Frege, Philip A. Ebert, and Marcus Rossberg. Basic Laws of Arithmetic. OUP Oxford (31 Oct 2013), 1893 and 1903.15Bertrand Russell. The Principles of Mathematics. Michigan University Press,digitised by Google, 1903. url: https://archive.

org/details/principlesmathe00russgoog .16Douglas R. Hofstadter. Godel, Escher, Bach: An Eternal Golden Braid. New York: Basic Books, 1999. url: http://www.

questiaschool.com/read/10035000 .17Russell, The Principles of Mathematics.18Frege, Ebert, and Rossberg, Basic Laws of Arithmetic.

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Figure 3: Prime factor decomposition of the first 1000 numbers

designed to preserve the truth of the axioms. An improved theory arose19 in 1908 as Zermelo-Fraenkel Choice SetTheory (ZFC) was developed, truly the crowning achievement of mathematics. To this day, no fatal inconsistencieshave been found and ZFC is the basis for all fundamental mathematics. As we know though, mathematicians arenot satisfied by over a hundred years of checking and failing to find a contradiction, they still demand a proof ofits consistency. In that vein, Godel originally set out to prove the consistency of Principia Mathematica20 whereBertrand Russell had attempted to build up all of mathematics from First-Order Logic and Peano Arithmetic(PA) used in conjunction with definitions for introducing new concepts. Ultimately though, he was clever enoughto realise and then prove that it could not be done.

4 Godels First Incompleteness Theorem

PA just describes the natural numbers or counting numbers (1 , 2, 3 . . .). This is surely about as simple as mathe-matical study can be. Notwithstanding, Kurt Godels work21 showed incredibly fundamental limitations that mustexist in any system closely related to Peano Arithmetic. His first incompleteness theorem states that such a systemcould never be both consistent and complete, meaning that either we can prove anything even if it is untrue or thereare true statements which we cannot prove - both terrifying potential flaws. His second incompleteness theorem

states that if a similar family of systems contains a statement about its own consistency then it must necessarilybe inconsistent, preventing its consistency from ever being proven or even being somehow incorporated into theaxioms.

As is common in mathematics, the proof of his results is even more interesting. Godels First IncompletenessTheorem centres around constructing a formalised Liars Paradox within Peano Arithmetic. The Liars Paradoxis simply the claim This sentence is a lie. Suppose that were true, then it would be true when it says that it isa lie. But if its a lie then what the sentence says is correct, so it must be true. Once again, this is a paradox ofself-reference. Surely then basic arithmetic is not capable of enough self-reference to fall victim to such a paradox.However, this is not the case, due to Godels most brilliant innovation. He proposed a beautifully simple and elegantidea: what if we numbered the theorems of arithmetic? Then we could essentially encode information about thetheorems of arithmetic within arithmetic itself. He proposed what is now known as a Godel Numbering system23.These systems are usually built up from prime numbers and give rise to very large codes24. This is an effective

19Ernst Zermelo. Untersuchungen uber die Grundlagen der Mengenlehre. I.. In: Mathematische Annalen 65 (1908), pp. 261281.20Russell, The Principles of Mathematics.21Kurt Godel. Uber formal unentscheidbare Satze der Principia Mathematica und verwandter Systeme, I.. In: Monatshefte fur

Mathematik 38 (1931), pp. 17398.22Figure 3: Math Medics. The Prime Factorization of the First 1000 Integers. url: http://www.sosmath.com/tables/factor/

factor.html23Appendix B.1, p.19

24A simple two line proof of the Law of the Excluded Middle gives the number 10 10102.062518063586396

which is larger than a Googolplex.According to the astronomer Carl Sagan, just writing out this number in decimal would require more space than exists in the entirevisible universe (Carl Sagan. Cosmos: A Personal Voyage, Episode 9: The Lives of the Stars. 1980). The mathematician JohnLittlewood estimated that the odds of a mouse surviving for a week on the surface of the Sun due to random quantum fluctuations arebetter than one in a googolplex (Daniel Tammet. Thinking in Numbers: How Maths Illuminates our Lives. Hodder and Stoughton,

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strategy because prime numbers are to number theory as the elements are to chemistry. The Fundamental Theoremof Arithmetic25 states that all numbers are either prime or the product of primes. As a result, just as the elementsin a compound can be used to predict its properties, the prime factor decomposition of a number can be used toclassify it into a group. This ensures that no two theorems are assigned the same Godel number because theyare made up of different elements (i.e. prime factors). Moreover, it further ensures that the Godel number of atheorem can be used to determine its constituent symbols by observing the elements that make up the compound(i.e. the prime factors in the Godel number), thus leading to different types of numbers being assigned to differenttypes of theorems and to a natural arithmetical relationship between the Godel numbers of derivations and theircorresponding theorems. This allows a function to be expressed within arithmetic to describe the special relationbetween the numbers of theorems and their proofs. The relation is actually very simple in style, it is just built upthrough many levels of recursion from the most primitive arithmetical relations. As a result, it is known that theseprimitive recursive relations are captured by Peano Arithmetic, meaning that we can be sure that PA can provethat these provability relationships exist.

I will now explain a faulty method to complete G odels proof to highlight a subtlety in the argument. Lets say thata number n is Richardian26 if and only if it does not have the nth property in some numbered list of properties.We can imagine such a list as follows:

1. Odd

2. Even

3. Square4. Richardian

5 . . . .

In this example, the number 3 is Richardian because it is not a square number so does not possess the thirdproperty. Numbers 1 and 2 are not Richardian because 1 is indeed an odd number and 2 is obviously an evennumber, so they actually do possess the first and second properties respectively. More interestingly, is the number4 Richardian given this listing of properties? Suppose it is, then it does not have the fourth property, so it is notRichardian. But if its not Richardian, then it does have the fourth property so it must be Richardian. Surely thenthis derives a contradiction demonstrating the inconsistency of arithmetic? This argument is fallacious because ittakes a semantic, metamathematical concept (Richardianism) and inserts it into a rule-based mechanical theorem-prover for arithmetic. This is a misapplication of terms and the argument does not hold. The key to Godels successwas in his construction of a purely arithmetical statement which could be interpreted with another meaning fromoutside of the system such that as outsiders we know that it cannot be proven without creating a paradox. ProfessorRebecca Goldstein27 makes a good analogy to that of a narcissist. No matter what narcissists are seemingly talkingabout, there is always some subtle way in which they are managing to still just talk about themselves. In thesame way, the sentence that Godel constructs seems to be an inert question of basic arithmetic, but its subliminalmessage about itself is what leads ultimately to self-referential paradox.

The formalised Liars Paradox in Godels First Incompleteness Theorem involves constructing the sentence G whichsays that there is no number which codes for a proof which satisfies the numerical relationship with the G odelnumber ofG such that it could be a proof ofG. Put more simply, G expresses an arithmetical formula which istrue if and only if the metamathematical statement G is not provable is true. Suppose thatG were a theoremof the system, then what it says should be true, so it should be true that G is not provable and thus G is not atheorem. This implies that no consistent theory of arithmetic can decideGeither way28 under pain of such paradox.Consequently, G is not provable and thus we can conclude that G must be true, as it says precisely that: Gis not

provable. This is an incredible result as it seems that the potential reinterpretation of these meaningless strings ofsymbols (designed just to describe arithmetic) leads to constraints on what the system can prove. Additionally, wecould then add this formula (GPA) to the axioms of the theory in order to strengthen it (resulting in PA+GPA).However, this would simply create a new system with a new Godel numbering scheme and a new arithmetical relation

2012).25Theorem 8 p.2326Jules Richard. Les Principes des Mathematiques et le Probleme des Ensembles. Revue Generale des Sciences Pures et Appliquees,

1905.27Professor Rebecca Goldstein. Godels Incompletness Theorems in the Context of Philosophy. New College of the Humanities.

March 18, 2013. url: https://www.youtube.com/watch?v=HHMKnwCT_Dk&index=4&list=WL.28By this we mean that G cannot be proven and neither can its negation: not G, which states that G is provable.

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for recognising the proof of a theorem based on their codes. The same process would lead to a new Liar sentencefor the new theory (GPA+GPA). We could always repeat this process ad infinitum to produce an infinite number oftrue but unprovable formulae: GPA, GPA+GPA, GPA+GPA+GPA+GPA . . . This shows that arithmetic is incompletable,there will always be an infinite number of elementary questions in arithmetic which we cannot answer with ourcurrent theory.

5 How limiting are these theorems?

There are in fact a variety of ways to circumvent Godels Incompleteness Theorems by avoiding the constraints thatthey assume about the theory. One possibility is to fundamentally abandon the idea of typographical formalism,then you could trivially create a consistent and complete system by adding all true arithmetical statements to theaxioms. This is known as True Arithmetic and is disliked by mathematicians for its lack of focus on rigorous proof.Another option is to not be sufficiently expressive about arithmetic, so formalisations of limited geometry can bepotentially both consistent and complete. A proof of consistency can also come from outside of the system suchas from ZFC set theory, but Godels Theorems would still apply to the consistency and incompleteness of ZFCitself. A more interesting proof of consistency (due to Gentzen29) comes from modified Peano Arithmetic to onlyallow induction over the ordinals up to0. The fact that these transfinite numbers cannot be expressed in normalPeano Arithmetic was the key to its success, but it also makes the proof highly questionable in justifying weakassumptions with stronger assumptions, which is the exact opposite of what Hilbert had hoped for. However, it isalso a weaker theory in that it cannot express normal mathematical induction. Overall, I would argue that this stilldoes not really capture Hilberts original program as it just transfers the question to another equally questionableformal system and does not build up naturally from logical, fundamental truths.

Tarskis Theorem on the undefinability of truth30 can be avoided in several ways. One is to use a meta-language31

which consists of an arbitrary number of languages, each successive language can then contain the truth predicatefor the language before it. However, the problem would remain that the truth-predicate for the meta-langaugewould be undefinable. It is also not finitistic as after any finite number of steps the highest level language wouldnot have a definable truth-predicate. A different solution is offered by Kleene who described logics with a thirdvalue of a truth predicate which can stand for both true and false or neither true nor false. Thus we could simplydescribe these more problematic sentences with this new value for the truth predicate and hence avoid paradox.This solution is also rather unsatisfying as we cannot know how many sentences have this undefined truth value orif a particular sentence is undefined. A more philosophical approach is to simply decide that mathematical truth is

subjective, there is no ultimate truth and the concept of omniscience is logically impossible. I would interpret thisvery strongly as saying that the intuitive idea we have of ultimate truth is an illusion. We can never achieve thisideal, but rather we must strive to constantly expand our view of truth.

6 Minds and Machines

A common philosophical claim made based on Godels First Incompleteness Theorem is that the human mind musttranscend any machine. As we have already argued, any mechanical system of mathematics encounters a Liarssentence G which it cannot prove, but as humans we can deduce that G is true. Therefore, we will always knowthings that machines cannot. However, this argument is subtly fallacious, as it neglects the possibility that it is notthe machine which is limited, but the mind of the mathematician who cannot dream up a smart enough machine

to formalise their understanding of mathematics. As Godel32

himself noted, the mathematical conclusion of thisargument consists of separate options which can be chosen based on ones philosophical inclinations. Rationaliststend to argue this proves that humans are limited in that we can never formalise our own understanding of mathe-matics so Hilberts Program will always be incomplete. Others such as Kurt Godel himself believed that the answerto this dilemma indeed was that humans would always surpass machines.

29Gerhard Gentzen. Neue Fassung des Widerspruchsfreiheitsbeweises fur die reine Zahlentheorie. In: Forschungen zur Logik undzur Grundlegung der exakten Wissenschaften 4 (1936), pp. 1944.

30This essentially says that a formal system of arithmetic cannot define the concept or predicate of arithmetical truth. See AppendixC.1 p.27 for a more detailed but still informal look at the result.

31Paul Christiano et al. Definability of Truth in Probabilistic Logic. In: Machine Intelligence Research Institute (2013). url:http://intelligence.org/files/DefinabilityTruthDraft.pdf .

32Either mathematics is too big for the human mind or the human mind is more than a machine.

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Figure 4: Futurists and Science-fiction authors have long considered the possibility of an Intelligence Explosion

One can also start from the reverse assumption of naturalism to reach interesting conclusions. In this view wemay suppose that mathematicians too are just rule-following theorem-proving machines and therefore we can applyGodelian incompleteness to them in order to show that humans beings will never be able to prove certain thingsabout arithmetic and we can never know the consistency of our own beliefs. There are certain truths about ourown reasoning process that we cannot ever know. It would require an alien mind surpassing our own just to noticethe existence of such a statement though. We will never be able to point to such a fundamentally undecidablestatement, although one may well be known but unrecognised. For example, perhaps we will never manage to provethe Riemann Hypothesis because we fundamentally cannot, regardless of how hard or long we try, but we couldnever know that this was the case. There are ways to avoid this conclusion naturally. One could argue that aswe sometimes notice our own inconsistencies, perhaps we are inconsistent formal systems. Moreover, inconsistencyis to be expected from a poorly evolved mammalian species which shares most of its DNA with the chimpanzee.However, this would put into question the presumption that mathematicians can come to know absolute, immutabletruths of all possible universes. Personally, I think this shows that we will always have an incomplete understandingof mathematical truth.

If we live in a materialistic, reductionist universe then it may seem obvious that either free will and consciousnessexist at all scales (i.e. in the human mind but also in rocks or the Higgs Boson) or it is simply an illusion. However,Douglas Hofstadter argues33 that holism may still be necessary to explain human behaviour in this view. He arguesas follows. Supposing that the brain is built up from simpler entities, there must be some level at which the braincan be viewed as a system which Godels First Incompleteness Theorem applies to and yet there is certainly a levelwhere we know thatG is clearly true. Therefore, just as Hilberts program is quashed by Godelian Incompleteness,reductionist attempts to understand free will or consciousness may too be futile. In this view, we have to imaginethe brain as a formal system of incredible complexity but built up from primitive computations at its lowest levels,

such that it is another mechanical system which must face the limits of Godelian Incompleteness. This is certainlya plausible argument given a naturalistic world-view, but with how little we know about the brain I do not acceptthat it can be established with great certainty.

Lobs Theorem35 has also been argued by Eliezer Yudkowsky36 to present a major obstacle to designing an ArtificialIntelligence (AI). I.J. Good argued37 that AIs should one day learn to improve themselves leading to an intelligenceexplosion38. There is a so-called Lobian obstacle for such self-modifying artificial intelligences, which is that arational agent seemingly cannot choose to rewrite its own source code. This AI would want to be able to trustitself after making an alteration to its source code, thus it would expect a proof of the soundness of its new sourcecode. However, Lobs Theorem states that a theory cannot prove its own soundness, so it certainly cannot provethe soundness of a stronger theory. Therefore, if an AI changes its own source code it will have to make it weakerin its mathematical power, which does not really seem like an improvement at all. On the other hand, if this AIalways chooses to keep its original source code then it is incredibly limited in its ability to improve itself.

This points to fundamental limits in self-reference and reflective rationality in Artificial Intelligence. Perhaps we

33Hofstadter, Godel, Escher, Bach: An Eternal Golden Braid.34Figure 4: Daniel Dewey. Intelligence Explosion and the Long-Term Future of Artificial Intelligence. url: http://www.danieldewey.

net/tedxvienna.html ; James Cameron. Terminator 2. url: http://geektyrant.com/news/2013/2/4/new-terminator-2-vfx-tests-show-cutting-edge-tech-for-the-ti.html

35This essentially says that a formal system cannot prove its own soundness, i.e. if it proves something then it is true. See AppendixC.1 p.27 for a more detailed but still informal look at the result.

36Eliezer Yudkowsky and Marcello Herreshoff. Tiling Agents for Self-Modifying AI, and the Lobian Obstacle. In: MachineIntelligence Research Institute (2013). url: http://intelligence.org/files/TilingAgentsDraft.pdf.

37Irving John Good. Speculations Concerning the First Ultraintelligent Machine. In: Advances in Computers 6 (1965), pp. 3188.38This refers to the positive feedback cycle when machines create new machines which will be even better at making new machines.

This kind of change builds on itself so could grow exponentially or even faster.

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Figure 5: The hypothetical machine which solves the Halting Problem halts if and only if it does not halt (giventhe addition of the looper program and that the input is its own name)

have to accept some weaker result than the soundness of a new theory before we choose to embrace it. One simplesolution to the problem is to simply imagine a list of theories in which successive theories can prove the soundnessof the previous theory and as the AI self-modifies it could simply descend down this list of theories as it rewritesitself in order to be able to prove the soundness of its new code. However, this would eventually hit some kind

of bottom (potentially at ZFC) as it approaches the weakest mathematical system, creating an upper limit on thenumber of rewrites that an AI could perform in its lifetime, which is distinctly unsatisfying. There are some limitedpossibilities suggested for reversing this process such that it has no last theory rather than no first theory, but it isnot a problem with any accepted solution.39

A related problem is known as the procrastination paradox. Imagine that an AI has to choose between acting now(lets say pressing a button) or modifying its source code so that it acts later. Assume the AI wants to press thebutton but is under no time pressure to do so quickly. In that case the AI will conclude that if it does not pressthe button now, its next decision will be to either press the button or to modify its source code such that it willpress the button eventually. Therefore the AI must conclude that the button will eventually be pressed so it can

just modify its source code rather than pressing the button now. This argument is independent of how many timesthe same decision has previously been made therefore the AI will never actually press the button. This seems tobe intimately related in some way to the L obian obstacle as the advantage of the obstacle is that it would preventsuch paradoxes (which come from an AI being able to rewrite its own source code).

7 Computers and Decidability

The great mathematician Alan Turing set to work on solving the Halting Problem: can we find an effective procedurefor determining whether a computer program will finish and halt normally or instead become lost in an infiniteloop and crash? Turing was able to invent an elegant and simple argument that this problem could not be solved.Suppose that it was solved, so we could instruct a machine H to determine whether a given program will haltsuccessfully. Turing proposed a couple of simple modifications that we could make to the machine to lead inevitablyto paradox. Add a looping device to the end of the program which if given the input Yes will go into an infiniteloop and if given the input No will simply halt immediately. Now let us give this new machine H+ its own nameas an input. Will it halt? Suppose it does, thenHwill return the output Yes and given the input Yes the looping

program will loop forever, so it will not halt. Suppose it does not halt, then H will return the output No andthe looper given the input No will simply halt. Either supposition leads to contradiction and so we are forced toconclude that this paradoxical machine Hdoes not exist and thus the Halting Problem is not solvable accordingto the Church-Turing Thesis.41 Even without the assumption of a Windows operating system, Turing was able toconclude that we cannot trust our computers not to crash.

39Benja Fallenstein and Nates Soares. Problems of self-reference in self-improving space-time embedded intelligence. In: MachineIntelligence Research Institute (2013). url: http://intelligence.org/files/ProblemsSelfReference.pdf.

40Figure 5: Mohit Dhingra. Problem on NFA construction. url: http:/ /functionspace.org/topic/ 264/ Problem- on- NFA-construction

41See Appendix D.1 p.34 for an informal introduction to Turings ideas.

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Now we can apply all of the results so far to questions of the nature of science, in particular, the search for a GrandUnified Theory of Physics. Laplace put this hypothesis the most transparently in his famous thought-experimentwhich has become known as Laplaces Demon42. Imagine that there was a machine of incredible computationalpower which knew all the fundamental laws of physics and all the information about the starting conditions of theuniverse. Classical physics states that any future state of the universe would be hence mathematically determined,with nothing left to chance. As a result, this machine could predict any event in the universe that ever happenedincluding what I am about to write next, what you are thinking right now and what you will do when you finishreading this. However, this extreme statement of determinism clearly contradicts Turings result as we could usesuch a machine to solve the Halting Problem by predicting whether a program would halt based on a physicaldescription of the situation. Therefore, Laplaces Demon is actually not possible. In light of this interesting result,we can actually now examine the full impact of Godelian Incompleteness on the universe itself, as if we assumethat the universe is deterministic then it is essentially a mechanical system in which the theorems being proven arethe states of the universe being exhibited over time. The universe is also axiomatised according to the essentialphysical assumption that there are underlying immutable laws43. It is sufficiently strong to capture arithmeticas every number theorist who has ever existed worked within our universe and so activities in the universe aresufficiently complex to model arithmetic. It is clearly consistent as in classical physics we cannot have two thingshappening at once. Consequently we must conclude that there are fundamental truths about our universe whichcannot be known within the universe. We will never be able to create a complete predictive model of the universe.If44 we were to one day be woken up to reality, only to find out that our universe was actually a simulation, thenperhaps we could suddenly understand fundamental underlying truths as we look down into our universe from the

outside. On the other hand, there are modern developments in physics which could prevent such a bizarre scenario.There are non-deterministic interpretations of quantum mechanics which could introduce fundamental randomnessto the physical laws, making them not axiomatisable or reducible to simple computation. The possible existence ofa multiverse of infinite universes could also complicate matters. As a result, I would argue that the implications ofGodelian Incompleteness cannot be fully understood until fundamental disagreements over the nature of physicallaw are finally settled.

8 Conclusion

In summary, what can we say about Peano Arithmetic and related systems? The formalised theorems of Godeldemonstrate not just that they can never prove certain true statements, such as their own consistency, but they

know that they never could prove them in the sense that a computer could derive G odels Incompleteness Theoremsfrom Peano Arithmetic given enough time (although obviously the computer would not recognise their incrediblesemantic significance). Lob showed45 that PA also knows that it can prove what it can prove, but it knows nothingabout what it cannot prove. We also know that PA cannot even discuss questions of its own truthfulness withoutbecoming inconsistent. Accordingly, Hilberts Second Problem46 has been settled in the negative. The nature offoundational mathematics is not as it seems. Complex mathematics does not arise from logic and arithmetic, butin fact assumptions about infinity are necessary to justify the truth of arithmetic. Consequently I must concludethat the nature of mathematics is not yet fully understood and perhaps there is a line which mathematical proofcannot pass. Some mathematical problems will always be intractable unless we adopt a far more sophisticatedunderstanding of what mathematics actually is.

42Laplace originally spoke of something usually translated as an intellect, he said: for such an intellect nothing would be uncertainand the future just like the past would be present before its eyes.

43Laplace: all the effects of nature are only mathematical results of a small number of immutable laws44Wachowski and Wachowski. The Matrix. March 31, 1999.45See Appendix C.1 p.27 for a more detailed but still informal look at the results.46Hilbert, Mathematical Problems.

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Appendices

A Formal Systems as the Foundation of Mathematics

A.1 First-order Logic

Symbols: {(, ),P,Q,R, , , , , }

Well-formed variables are of the form {P,Q,R}or any of those three letters followed by any number of primes (e.g.P orR)

Strings of any well-formed variables (xand y) are themselves well-formed if they are made up of the following fourforms:

1. x

2. (x y)

3. (x y)

4. (x y)

Laws of Transformation:Rule 1 (Conjunction Introduction).xy(x y)

11Rule 2 (Conjunction Elimination).(x y)xyRule 3 (Double Negation). x xRule 4 (Conditional Introduction).

(+ x) y (x y) is used here to represent known theorems of the system. The process of adding x to the system temporarily isknown as the Push and removing it again is known as the Pop47.The string represented by x here is known asthe Premise. The fact that +x strictly includes x implies that theorems can be carried down through a pushbut not back up after the pop; this principle is known as the Carry-down Rule. 48

Rule 5 (Conditional Elimination or Modus ponens).x(x y)yRule 6 (Contrapositive Rule). (x y) (y x)Rule 7 (Junction Conversion). (x y) (x y)Rule 8 (Disjunctive Syllogism). (x y) (x y)

I will add the following symbol, which is unnecessary, but will be used to make theorems easier to read.Definition 1 (Biconditional Introduction and Elimination).

((x y) (y x)) (x y)

This will be a useful shorthand when x and y are very large strings.Lemma 1 (Disjunction Commutativity). ((P Q) (Q P))

47Note that push and pop are not part of the system.48Hofstadter, Godel, Escher, Bach: An Eternal Golden Braid.

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Proof.

Push

(P Q) Premise

(P Q) Switcheroo

(Q P) Contrapositive

(Q P) Double-negative

(Q P) Switcheroo

Pop

((P Q) (Q P)) Hypothetical Rule

But P and Q were randomly chosen variables, so we could simply perform the same proof again with the roles ofP andQ reversed to obtain the following theorem.

((Q P) (P Q)) Repeat process

((P Q) (Q P)) Biconditional Introduction (p.11)

Lemma 2 (Conjunction Commutativity). ((P Q) (Q P))

Proof.

Push

(P Q) Premise

P Conjunction Elimination

Q Conjunction Elimination

(Q P) Conjunction Introduction

Pop

((P Q) (Q P)) Conditional Introduction

((Q P) (P Q)) Repeat process

((P Q) (Q P)) Biconditional Introduction

Lemma 3 (Tautology). (P P)

Proof.

Push

P Premise

Pop

(PP) Conditional Introduction

(PP) (P P) Conjunction Introduction

(P P) Biconditional Introduction

Lemma 4 (The law of the excluded middle). (P P)

Proof.

(P P) Tautology

(P P) Disjunctive Syllogism

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Lemma 5 (The law of non-contradiction). (P P)

Proof.

(P P) The Law of the Excluded Middle

(P P) Double Negation

(P P) Junction Conversion

(P P) Double Negation(P P) Conjunction Commutativity

As a short hand for using this result we will introduce an extra definition for convenience.Definition 2 (Contradiction). (P P) for any formula taking the place of the free variable P. Thereforethe law of non-contradiction translates to .Lemma 6 (Proof by Contradiction49). (Q ) Q

Proof. First the forward implication.

PushQ Premise

Q Contrapositive Rule

The law of non-contradiction

Q Conditional Elimination

Pop

(Q ) Q Conditional Introduction

Now the reverse implication:

Push

Q Premise Push

Q Premise

Q Carry-down Rule

(Q Q) Conjunction Introduction

Definition of contradiction

Pop

Q Conditional Introduction

Pop

Q (Q ) Conditional Introduction

Combining:

(Q ) Q Forward Implication

Q (Q ) Reverse Implication

Q (Q ) Biconditional Introduction

49Also known as Reductio ad absurdum or Modus tollens

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Lemma 7 (Case analysis). ((((P R) (Q R)) (P Q)) R)

Proof.

Push

(((P R) (Q R)) (P Q)) Premise

((P R) (Q R)) Conjunction Elimination

(P R) Conjunction Elimination(Q R) Conjunction Elimination

(P Q) Conjunction Elimination

(R P) Contrapositive Rule

(R Q) Contrapositive Rule

Push

R Premise

(R P) Carry-down Rule

(R Q) Carry-down Rule

P Conditional Elimination

Q Conditional Elimination(P Q) Conjunction Introduction

Pop

(R (P Q)) Conditional Introduction

(R (P Q)) Junction Conversion

((P Q) R) Contrapositive Rule

((P Q) R) Double Negation

R Conditional Elimination

Pop

((((P R) (Q R)) (P Q)) R) Conditional Introduction

Theorem 1. The Principle of Explosion50: ((P P) Q)

Proof.

Push

(P P) Premise



Push

Q Premise

P Carry-down RuleP Double Negation

Pop

(Q P) Conditional Introduction

(PQ) Contrapositive Rule and Double Negation

Q Conditional Elimination

Pop

((P P) Q) Conditional Introduction

50Also known as ex contradictione quodlibet (from a contradiction, anything follows)

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Lemma 8 (Nested Conditionals). P (Q R) (P Q) R

Proof. First the forward implication:

Push

P (Q R) Premise

Push

P Q Premise

P (Q R) Carry-down Rule


(Q R) Conditional Elimination

Q Conjunction Elimination


Pop

(P Q) R Conditional Introduction

Pop

((P(Q R)) ((P Q) R)) Conditional Introduction

Now the backwards implication:

Push

(P Q) R Premise

Push

P Premise

Push

Q Premise

P Carry-down Rule

P Q Conjunction Introduction(P Q) R Carry-down Rule


Pop

Q R Conditional Introduction

Pop

P (Q R) Conditional Introduction

Pop

(((P Q) R) (P(Q R))) Conditional Introduction

Combining:

((P (Q R)) ((P Q) R)) Forwards implication

(((P Q) R) (P (Q R))) Reverse Implication

P (Q R) (P Q) R Biconditional Introduction

Lemma 9 (Conditional Syllogism). ((P Q) (Q R)) (P R)

Proof.

Push

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(P Q) (Q R) Premise

P Q Conjunction Elimination

Q R Conjunction Elimination

Push

P Premise

P Q Carry-down Rule

Q Conditional EliminationQ R Carry-down Rule


Pop

P R Conditional Introduction

Pop

((PQ) (Q R)) (P R) Conditional Introduction

A.2 Peano Arithmetic

We will now outline what must be added to first-order logic to create a sufficiently strong theory of arithmetic.

Language for arithmetic ={0, S, +,p,q,r, , =, , , :}Definition 3. 0 is a number and ifx is a number then S x is a number.

Ifx and y are numbers or numerical variables then x = y is a well formed formula. Ifx is a variable which occurs in(x) then x: ((x)) and x: ((x)) are well-formed formulae. All well-formed formulae can be substituted intofree propositional variables51. p,q,r are numerical variables ifx is a numerical variable then x is also numericalvariable.

Ifx and y are numbers or numerical variables then they obey the following arithmetical axioms:Axiom 1 (Equality Introduction). x: (x= x)Axiom 2 (Zero is not a Natural Number). x: (0=Sx)

Axiom 3 (Succession is an Injective Function). x: y: (Sx = Sy x = y)Axiom 4 (Basis of Addition). x: (x + 0 =x)Axiom 5 (Induction of Addition). x: y: (x + Sy = S(x + y))Axiom 6 (Basis of Multiplication). x: (x 0 = 0)Axiom 7 (Induction of Multiplication). x: y: (x Sy = (x y) + x)Axiom 8 (Equality Elimination). ((x= y) (x)) (y)Axiom 9 (Induction Schema). (((0) x : ((x) (Sx)) x : (x)) where (x) is an open well-formedformula with at least the variable x free.

The following are true for any variables x, y and any open formula :Axiom 10(Universal Elimination or Universal Instantiation). (x: ((x))) (y) assuming that there is no clashof variables.Axiom 11 (Universal Introduction or Universal Generalisation).

(x) (x : ((x))) assuming that the variable x was not previously used as a free variable in the derivation of(x), meaning thatx could have been any other variable.Axiom 12 (Existential Introduction). (y) x: ((x)) wherex is a free variable replacing a constant termy in. The term y is known as a witness to(x).Axiom 13 (Existential Elimination). (x: ((x)) ((x) ) ) assuming that x does not appear again as afree variable in .Lemma 10 (Quantification Conversion). x: ((x)) x: ((x))

51the kind of variables used in first-order logic

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Proof. First the forward implication:

Push

x: ((x)) Premise

(x) Universal Elimination (p.16)

Push

x: ((x)) Premise

Push

(x) Premise

(x) Carry-down Rule

(x) (x) Conjunction Introduction

Definition 2 (p.13)

Pop

(x) Conditional Introduction

Existential Elimination

Pop

x: ((x)) Conditional Introduction

x: ((x)) Proof by contradiction (p.13)

Pop

x: ((x)) (x: ((x))) Conditional Introduction

Then the reverse implication:

Push

x: ((x)) Premise

Push

x: ((x)) Premise

(x) (x: ((x))) Universal Introduction(x) (x: ((x))) Substitution

x: ((x)) (x) Contrapositive Rule

x: ((x)) (x) Double Negation

(x) Conditional Elimination

x: ((x)) Existential Introduction

x: ((x)) Carry-down Rule

(x: ((x))) (x: ((x))) Conjunction Introduction

Definition 2

Pop

(x: ((x))) Conditional Introduction

x: ((x)) Proof by Contradiction

Pop

(x: ((x))) x: ((x)) Conditional Introduction

x: ((x)) x: ((x)) Biconditional Introduction (p.11)

We can build up Peano Arithmetic in Set theory, demonstrating it is truly at the most fundamental level ofmathematical foundations.Definition 4 (Number). is a number and ifis a number then {} is a number.

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This defines an equivalent to the successor function, where a set is added to its own elements. Thus the set ofnumbers consists of:

0 =

1 = {}

2 = {, {}}

3 = {, {} , {, {}}}

4 = {, {} , {, {}} , {, {} , {, {}}}}

. . .

This allows arithmetic to be modelled by set theory. Therefore, if arithmetic is capable of enough self-reference toface fundamental limitations in order to maintain consistency, then the absolute foundations of mathematics suchas set theory must face the same limitations.

A.3 Naive Set Theory and Russells Paradox

The fatal principle is that sets of the type {x|P(x)}are always well-formed and are defined by definition 5.Definition 5 (Membership Introduction and Elimination).((x {x|P(x)}) P(x))

Definition 6 (Membership Negation). x /y (x y)Definition 7 (Russells Set). R= {x|(x /x)}Theorem 2 (Russells Paradox52).

((R R) (R /R))

Proof. Starting from the critical axiom listed above, let P = (x /x). Therefore, Russells Set is well formed.

((x R) (x /x)) Rule of Assimilation

((R R) (R /R)) Substitution x R

Theorem 3.

Moon = Cheese

Proof.

((R R) (R /R)) Russells Paradox

((R R) (R R)) Definition 6

((R R) (R R)) Biconditional Elimination

((R R) (R R)) Biconditional Elimination

((R R) (R R)) The Law of the Excluded Middle (p.12

((((P R) (Q R)) (P Q)) R) Case Analysis (p.14)

(((R R) ) ((R R) ) ((R R) (R R))) Substitution

The complicated last theorem is simply the result of substituting values into the Case Analysis Lemma from thethree preceding theorems for values of the variables P, Q, and R. Now by Conditional Elimination we have:

((R R) ((R R)))

Now we need only combine this result with the Principle of Explosion to complete the proof. Theorem 1 states:

((P P) Q

52Russell, The Principles of Mathematics.

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Recall that P and Q are merely variables. Let P = (R R) and Q = (Moon = Cheese).

(((R R) (R R)) (Moon = Cheese) The Principle of Explosion

((R R) ((R R))) Russells Paradox

Moon = Cheese Conditional Introduction

The end of that proof was not technically accurate because the letters required to write Moon = Cheese do notactually exist in the language of Logic or Set Theory, but this demonstrates the claim that anything expressible innaive set theory is provable.

B Godels First Incompleteness Theorem

B.1 Godel Numbering

Theorem 4 (Chinese Remainder Theorem). For any relatively prime sequence q0, q1 . . . q n1, qn, then for eachi inthe range from 0 to (q0 q1 . . . qn1 qn) 1 there is a unique sequence:

i(mod q0), i(mod q1) . . . i(mod qn1), i(mod qn). There are onlyq0 q1 . . . qn1 qnpossible sequences of remain-ders, so these values ofi will produce every possible unique sequence of remainders.

Proof. Suppose that the sequences were not all unique. Then there would be two values ofi, say i1 and i2 where0 i1 < i2 < q0 q1 . . . qn1 qnwhich give identical sequences. Let i= i2 i1. Ifi1 = i2(mod qk) then clearlyi must be a multiple ofqk. This is true for all qk in the sequence (q0, q1 . . . q n1, qn), so i must be a multipleof every term in the sequence. As the terms in the sequence are relatively prime, i must therefore be a multipleof the product q0 q1 . . . qn1 qn. However, i is the difference between two values ofi, which both must bestrictly less than the product, so i should have been less than the product. Contradiction.

There are qk unique numbers in a modulo qk53 for each term in the sequence. As a result, if we multiply to get

the total possible combinations of remainders we get q0 q1 . . . qn1 qn. In summary, there are no repeatedsequences created and there are as many sequences formed as there are possible sequences, so these values ofi mustproduce precisely every combination of remainders.

B.1.1 Primitive Recursive Functions

Definition 8. Primitive recursive functions ultimately consist of just three so-called primitive functions:

1. The Successor Functiony = S(x) = x + 1

2. The Zero Functiony = Z(x) = 0

3. The Identity Functiony = Ini (x1, x2, x3 . . . xn1, xn) = xi

Complex functions are built up from these using the following rules of recursion:

1. Iff andg are primitive recursive functions, then f(g(x)) is a primitive recursive function

2. Let g and h be primitive recursive functions. The function f, defined by the basis fc(0) = g(c) and theinductive rulefc(Sx) = hc(x, fc(x)), is also primitive recursive54.

This class of functions is useful to define because they can clearly be captured by a sufficiently strong system ofarithmetic, which we will now prove.Theorem 5(Expressibility of primitive recursive functions). All primitive recursive functions can be expressed bythe language of arithmetic.

53Recall that in a modulo qk counting system, by convention we write all numbers as one of{0, 1, 2 . . . qk 2, qk 1} which has sizeqk.

54c here stands for any number of values which are required in calculating the function. For example, 2( x+ 1) 3 would require thenumbers{1, 2, 3} in memory to know how much to add, subtract and multiply

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Proof.

1. LA can express the three primitive functions.

1 The Successor Functiony = S(x) =x + 1 can be represented by the well-formed formula of arithmetic(y = Sx) using the model S0 = 1, SS0 = 2. . . making the mechanical successor function satisfy thesemantic successor function.

2 The Zero Function y = Z(x) = 0 is expressed by the well-formed formula (x= x y = 0).

3 The Identity Function y = Ini (x1, x2, x3 . . . xn1, xn) = xi is expressed by the well formed formula((x1 = x1) (x2 = x2) (x3 = x3) . . . (y= xi) . . . (xn1 = xn1) (xn= xn)) .

2. Expressibility inLA is stable under the rules of primitive recursion.

(a) The first rule is trivially expressed by (y= f(t)) (t= g(x)).

(b) The second rule is much more complicated. This can be proven using Godels Function Lemma toexpress a recursively defined list through a coding system similar to Godel numbering.

Suppose we had a sequence of natural numbers k0, k1k2 . . . kn1, kn then we could encode all of the

information of this list into the number N=k00 k11

k22 . . .

kn1n1

knn . We can then decode it

using a primitive recursive function exp(N, i) = x Nwhich satisfies the formula (xi |N (x+1i |N)).

The problem is that Peano Arithmetic does not include exponentiation. This can be fixed by instead

using two coded numbersaandb, then settinga,b(i) to be the remainder whenais divided byb(i+1)+1.Formally, a,b(i) = y if and only ifa = S(b Si) f+ y wheref is its greatest possible value55 less thana. The Chinese Remainder Theorem implies that there will always be values ofa and b to satisfy this-function, as we will now demonstrate.

Let s be the largest number of the set {n, k0, k1k2 . . . kn1, kn}. Let b = s!. Now imagine the sequenceb(i+ 1) + 1 for all values of i between 0 and n (inclusive): b+ 1, 2b+ 1, 3b+ 1 . . . b(n+ 1) + 1. Wenow prove that this must give a relatively prime sequence of numbers. Suppose, they were not relativelyprime, then there would have to be two numbers in the sequence bj + 1 and bk + 1, or explictly s!j+ 1and s!k+ 1, which had a prime factor in common, say p. The first s numbers cannot divide s!j+ 1 asthey all leave remainder 1, so the common divisor p must not be any of those numbers, i.e. p > s(n).

This common factor p must also be a factor of their difference b(k j). Also, as a factor ofbj+ 1, pcannot be a factor of b. Therefore, by process of elimination, it must have been a factor in k j , so

p (k j). As k and j index two ofn + 1 numbers, their difference must be at mostn. If we put thosetwo facts together then p n. Contradiction.

Consequently, the sequenceb(i + 1) + 1 for all values ofi between 0 and n (inclusive) is relatively prime.The Chinese Remainder Theorem implies that there must therefore be some value ofa for every possiblesequencea(mod b + 1), a(mod 2b + 1) . . . a(mod bn + 1), a(mod b(n + 1) + 1). As a result, there must besome value ofa for which this procedure produces the series k0, k1k2 . . . kn1, kn exactly. Thus we havedemonstrated the existence of a function.

These two coded numbers a and b can then be further encoded to one number, for example usingN = 2a 3b, which could be decoded by factorisation and counting, so we can stick to just one codenumber to simplify our-function. Generalising, for any sequence of natural numbersk0, k1k2 . . . kn1, knthere is a coded number Nwith a function such that N(i) = ki for all i n.

Now we are ready to express the function fc(x) = y recursively such that fc(0) = g(c) and fc(Sx) =hc(x, fc(x)). It defines a set of numbers leading up to y for which we can specify a function:k0, k1k2 . . . kx1, kx where k0 = g(c) = N(0), kx = y = N(x) and for all i < x: ki+1 = hc(i, N(i)) =N(Si). Therefore, we define fc(x) = y if and only if y obeys the following formalised formula inarithmetic:

N : ((y= N(x)) (g(c) = N(0)) (i < x : (hc(i, N(i)) = N(Si))))

wherei < x: is short fori: (v: ((v= 0) (v+ i= x))).

55Formally this is ((f < a) g: f < g )

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Theorem 6. Primitive Recursive functions are those which can be expressed using for-loops in pseudo-code

Proof. Pseudo-code is a high-level language which is ultimately reducible to machine code, which is ultimatelyreducible to basic computations on binary numbers. Binary numbers can be mapped to the natural numbers, sothese basic functions must be primitive recursive. Now we just need to show that recursively defined functions canbe expressed using only for-loops. The classic style of a recursive definition is:

1. definef(x):

2. if x = 0 then:

3. return g(c)

4. else:

5. return h(c, x 1, f(x 1))

This can easily be defined non-recursively using just for-loops:

1. definef(x):

2. fList=create(EmptyList)

3. fList.add(g(c))4. for i in range i = 0 to i = x 1:

5. fList.add(h(c,i,f(x)))

6. return fList[final]

This creates a list, initialised to [g(c)], then uses an iterative loop to gradually add values until the end result is alist whose final value is the answer to f(x): [g(c), h(c, 0, g(c)), h(c, 1, h(c, 0, g(c))) . . . f (x)].In summary, we have proven that primitive recursive functions can be defined using only for-loops and thatprograms using only for-loops must be primitive recursive. Therefore, functions are primitive recursive if and onlyif they can be written out in a program using only for-loops (no while-loops).

Definition 9 (Express). The open formula (x) expresses a property P if and only if the following two conditionshold:ifx has property P, then (x) is true inLA,IA.ifx does not have property P, then(x) is true in LA,IA.Definition 10(Capture). The open formula(x) captures a property Pif and only if the following two conditionshold:ifx has property P, then A (x).ifx does not have property P, thenA (x).Theorem 7. A captures all primitive recursive functions

Proof. Firstly, the three primitive functions are trivially captured. For succession simply add anS, for zero simplywrite 0 and for identity simply do nothing to the number in question. Thus all three reduce to the ability to provethat two identical numbers are equal, which Axioms 1 and 10 (p.16) ensure easily. Moreover, under recursion of thefirst kind, primitive recursive functions are still clearly captured. Assuming that f(t) and g (x) are both captured,

then Axiom 1 ensures that (y = f(t)) (t = g(x))) captures f(g(x)). Recursion of the second kind on capturedfunctions must also give a captured function as its result because we can reduce this recursion to for-loops aswe have seen. This means that there will be no open-ended searches and thus all quantification will be bounded.Therefore, we just have to prove that bounded quantification of captured formulae results in more complicatedrelations which can also be captured. There are obviously two types of bounded quantification:

x c : ((x)) (1)

x c : ((x)) (2)

where (x) is a captured formula. The first type will be proven if true due to axiom 12, as it is a computablemechanical procedure to check all of the numbers from 0 to c to see if they satisfy (x). Then if and only if

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such a value is found will axiom 12 derive the formula. The second type needs to be written out explicitly asx : (x c) ((x)) and for a specific value of c could be written out more explicitly as (x = 0 x = 1 x =2 . . . x= c) (x). Now we can see that it will be proven if true because assuming that it is correct then we canprove one of(0), (1), (2) . . . (c). Then using lemma 7 (p.14) we can prove it to be correct.

We can now show that the formulae will also be proven false if they are not true using the relation from lemma 10:x: ((x)) x: ((x)). This shows that proving either type false is equivalent to proving a formula of theother type true. Therefore, all primitive recursive functions can be reduced to captured arithmetical relations.

B.1.2 Coding Theorems

We now define a Godel numbering system and primitive recursive operations on those numbers. First we assignAtomic Godel Numbers to all the symbols in the language of arithmetic. A simple trick of modular arithmetic allowsus to assign numbers to infinite numerical and propositional variables. The numerical variables can be assignedAtomic Godel Numbers which are equal to 0 in modulo 3:

p q r p q r p q r . . .3 6 9 12 15 18 21 24 27 . . .

The propositional variables can be assigned Atomic Godel Numbers which are equal to 1 in modulo 3:

P Q R P Q R P Q R . . .

1 4 7 10 13 16 19 22 25 . . .

The remaining symbols can then be assigned the lowest numbers left, which are equal to 2 in modulo 3:

0 S + = ( ) :2 5 8 11 14 17 20 23 26 29 32 35 38 41

Godel numbers are then assigned to expressions based on the Atomic G odel Numbers of their constituent symbols.Suppose an expression is made up of the symbols s0s1s2 . . . sn2, sn1, sn which have corresponding Atomic GodelNumbers {g0, g1, g2 . . . gn2, gn1, gn}, then the expression will be assigned the Godel number

g00

g11

g22

. . . gn2n2

gn1n1

gnn . Then a sequence of expressions can be assigned a Super Godel Number based on the

Godel numbers of their constituent expressions in an identical fashion.

For example, let us deduce the Super Godel Number of the proof of the Law of the Excluded Middle (lemma 4p.12):

(P P) Tautology

(P P) Disjunctive Syllogism (p.11)

First we look up on our table the Atomic G odel Number of the symbols used.

(P P) 35, 23, 1, 26, 23, 1, 38

(P P) 35, 11, 32, 23, 1, 38

Now we assign each formula its Godel number.

(P P) 235 323 51 726 1123 131 1738

(P P) 2

35

3

11

5

32

7

23

11

1

13

38

Now we give the entire proof (a sequence of formulae) a Super Godel Number Nby iterating the process on theGodel Numbers of the formulae.

(P P)(P P) N= 2(2353235172611231311738) 3(2

353115327231111338)

In order to comprehend the magnitude of this number, let us write it in powers of 10:

N= 1010102.062518063586396

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This means that N is larger than a Googolplex. According to the astronomer Carl Sagan, just writing out thisnumber in decimal would require more space than exists in the entire visible universe.56 The mathematician JohnLittlewood estimated that the odds of a mouse surviving for a week on the surface of the Sun due to randomquantum fluctuations are better than one in a googolplex.57 There may be much more efficient Godel numberingsystems for this theory but this is the easiest and most natural to describe mathematically whilst ensuring notwo objects in the system at the same level (symbol, expression, sequence) have the same Godel number. This isguaranteed by the fundamental theorem of arithmetic.Theorem 8 (The Fundamental Theorem of Arithmetic). Every natural number greater than 1 can be expresseduniquely as a product of primes.

Proof. First we show that every natural number greater than 1 can be expressed as a product of prime numbers inat least one way. Let(x) be the statement that x can be expressed as a product of primes. We will proceed usingproof by strong mathematical induction, which consists of two steps:

1. Basis Step: prove(2)

2. Inductive Step: prove

ni=2

((i)) =n+1i=2

((i))

Then we will have proven (x) forx 2 as required. The Basis Step is trivial as 2 is prime. For the Inductive Stepwe must first assume the antecedent, then we have (2) (3) (4) . . . (n 1) (n). Now we just have toshow (n + 1), that is, that n + 1 has a prime factorisation. Ifn + 1 is prime then this is trivially true, otherwise,n+ 1 = ab for some factors a and b which must both be greater than 1 and less than n+ 1, so by assumption58

have prime factorisations. Let us say that a = p1 p2 . . . pk1 pk andb = q1 q2 . . . ql1 ql. Thereforewe can deduce that n + 1 = ab = p1 p2 . . . pk1 pk q1 q2 . . . ql1 ql. This is a prime factorisationofn + 1 and thus we have proven by strong induction that we can express every natural number greater than 1 asa product of prime numbers in at least one way.

Now we must prove that such expressions are unique. We will proceed using proof by contradiction. Let us supposethat there is some integer n with two different expressions as a product of primes and derive a contradiction. Bysupposition,n= p1p2. . .pk1pk = q1q2. . .ql1qlwhere the two lists of primes are written in ascendingorder and are different lists. Now we can cancel any common prime factors, isolating the terms which differ between

the lists. By assumption, this will leave us with a few completely unique prime factors remaining from the originallists which distinguish the lists from each other. Therefore r1 r2 . . . ra1 ra= s1 s2 . . . sb1 sb. Itis clear that r1 is a factor ofr1 r2 . . . ra1 ra and thus is also a factor ofs1 s2 . . . sb1 sb. As r1is prime, it must divide one of the factors ofs1 s2 . . . sb1 sb in order for it to be a factor of the product.Therefore,r1 is a factor ofsj for some value ofj . As r1 and sj are both prime, this can only be the case ifr1 = sj ,which contradicts our assumption that the prime factorisations were different. In summary, every number has aprime factorisation (induction) and it must be unique (contradiction).

Now we will define functions on Godel numbers. It should be intuitive from the example that there is somearithmetical relation, albeit it incredibly complicated, between the Godel number of a theorem and the SuperGodel number of a valid proof of that theorem.Definition 11. Prf(x, z) stands for the arithmetical formula in A which is true if and only ifx is the Super Godelnumber of a proof of the theorem with Godel number z . By proof we mean a sequence of well-formed formulaewhich start with the axioms, end with the desired theorem and in which each successive formula follows by a ruleof inference.

Now we can express statements about provability encoded as arithmetical relations between G odel numbers. Wewill also need to define a relation which will allow formulae to reference themselves, or more specifically, their ownGodel numbers.

56Sagan, Cosmos: A Personal Voyage, Episode 9: The Lives of the Stars.57Tammet, Thinking in Numbers: How Maths Illuminates our Lives.58We assumed (2)(3)(4) . . .(a) . . .(b) . . .(n1)(n)

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Definition 12. diag((x)) = ((x)). In words: given the Godel number n of an open formula (x),substitute the numeral n into the formula in exchange for the free variable x and then return the Godel number ofthe resulting formula. This resulting formula is known as the diagonalisation of.

Now we have to convince ourselves that these arithmetical formulae can be captured in A.Theorem 9. Prf(m, n) and diag(n) are captured in A

Proof. We will prove this by showing that they are primitive recursive and thus captured in A as we proved in

Theorem 7 (p.21). Unsurprisingly, the most rigorous way to prove this is constructively, building up definitions inA which refer to only primitive recursive functions. This was achieved by Kurt Godel in his original proof.

59 Itwas incredibly tedious but Godel wanted to make sure that no-one could question his conclusions. Here we will becontent to give an informal argument that it should be theoretically possible to do what Godel did.In order to prove that these are primitive recursive functions we will exploit theorem 6 and simply outline roughlya computer algorithm to express these functions.

1. define diag(n):

2. decoden to get the open well-formed formula (x)

3. replace the free variable x with the numeral n

4. return the Godel number of the resulting string

Coding in a Godel numbering system simply involves prime factorisation, which can be done with bounded searchesonly. Therefore, diag(n) is captured.

1. define Prf(m, n):

2. decode the Super Godel number m into a sequence of formulae

3. condition1 = is this a sequence of well-formed formulae?

4. condition2 = is each formula derivable from the previous one?

5. condition3 = is the last formula the same as decode(n)?

6. if condition1 and condition2 and condition3 then:

7. return Prf(m, n) is a theorem

8. else:

9. return Prf(m, n) is a theorem

This only involves bounded searches as there are only a finite number of types of well-formed formula (xy, xy, xy . . .) and only a finite number of axioms or rules of inference. Therefore, Prf(m, n) is captured.

B.2 The Formalised Liars Paradox

Theorem 10 (Godels First Incompleteness Theorem). LetA be a formal system sufficiently powerful to captureelementary arithmetical statements. IfA is consistent then there is a well-formed arithmetical formula G which istrue but not provable so A is incomplete. Even ifG is incorporated into the axioms or other axioms added so Gis provable, then there will always be a new similar unprovable but true formula which can be constructed, so A

is essentially and necessarily incompletable.

Proof. We first introduce some arithmetical statements which allow us to model statements about provability. Wedefine the following formula to have Godel number n.

x: Prf(x, diag(y))

59Godel, Uber formal unentscheidbare Satze der Principia Mathematica und verwandter Systeme, I..

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We will now diagonalise, that is, we will substitute the numeral for the Godel number of this open formula (n) intothe open variable (y). Thus by definition60 the resulting formula will have Godel number diag(n). Now we willwrite out this formula for which we have already determined the Godel number.

x: Prf(x, diag(n))

We will now simplify this formula in order to interpret its significance by defining G = diag(n). We now have thefollowing theorem:

G x: Prf(x, G)

If we consider the meta-mathematical meaning of this arithmetical statement, we are forced to conclude that Gis true if and only if it is not provable. In other words, G is a well-formed arithmetical formula which states anarithmetical claim which is true if and only if the meta-mathematical statement Gis not provable is true. Thereforewe can conclude that formula G is true if and only if it is not provable.

G G is not provable

Assuming that A is sound, there is an easy semantic argument to complete the proof. SupposeG were provable,then what it says would be true, that is, it would be true that G is not provable. Suppose G were provable,then what G says would be true, so G would be provable. Contradiction either way. Therefore, this strongerassumption implies thatG is formally undecidable.

Now we argue just from the assumption of consistency. Suppose thatG is a theorem ofA, that is, G is provablein A. Consequently, there exists a set of theorems from whichG is derived. Let k equal the Super Godel numberof the derivation ofG, then (by definition of Prf and the fact that Prf is captured) the following would also have tobe a theorem:

Prf(k, G)

We can now recall what the theorem G actually says, then apply Quantification Conversion (p.16) and UniversalElimination (p.16):

x: Prf(x, G)

x: Prf(x, G)

Prf(k, G)

This clearly contradicts our previous result soG must not be a theorem. There is a more complicated proof of thereverse implication just assuming consistency. We will make the stronger assumption of consistency in order toprove it. Suppose G is a theorem, so it is provable. This theorem states the following:

x: (Prf(x, G))

Assuming consistency, then G cannot be a theorem and therefore Prf(k, G) cannot be a theorem for any k byExistential Introduction (p.16). In summary, x: (Prf(x, G)) is a theorem and for every value ofk, Prf(k, G) isa theorem (as Prf is captured (p.24)), making the system inconsistent, contradicting our initial assumption andso G must not be a theorem. Thus we conclude once again that G must be formally undecidable so G is notprovable, but that is exactly what G says, so it must be true.

We have found an arithmetical formula G which is true but unprovable in arithmetic. Consequently we must

conclude that arithmetic is incomplete. Moreover, we can always add an undecidable formulae like G to the axiomsof the system. This will still be consistent as the old Prf(x, z) will not have included this new axiom G in its code, soit will be effectively stating thatGwasnot provable. As a result, if we addedGto the axioms of the system, it wouldcreate a new system with a new Godel numbering system, a new Prf function and a new undecidable statement.Applying the same process would always produce a similar true but unprovable statement in the stronger systemof arithmetic. This process could be repeated ad infinitum, demonstrating that there are an infinite number of truebut unprovable statements and arithmetic is essentially incompletable.

Let us put that more precisely. In order to strengthen arithmetic and thus decide previously undecidable formulae,we will be forced to add axioms to our system. A simple way of doing that would be to simply add the traditional

60See Definition 12 p.24

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Godel sentence for a system to the axioms of that system. This would create a series of progressively strongersystems:

PA, PA + GPA, PA + GPA+ GPA+GPA . . .

If these new undecidable statements are not provable in the stronger system, then they are definitely not provablein any of the weaker systems. Thus we can construct the following array of undecidable statements in PeanoArithmetic:

GPA, GPA+GPA, GPA+GPA+GPA+GPA . . .

Thus there is a countably infinite set of undecidable statements in any consistent theory of arithmetic.

B.3 Extending the First Theorem

The clever trick of diagonalisation that we used to form G obviously generalises to all circumstances where we wantto create a statement that references a property of its own Godel number.Theorem 11 (Diagonalisation Lemma). For every open formula (), there is a formula such that () is a theorem.

Proof. Let the diagonalisation function ((y)) = ((y)). Let ((((n)))). Applying thedefinition of:

((((n))))

(((((n)))))

()

Theorem 12 (Rossers Trick). There is an undecidable proposition R in A.

Proof. Diagonalisation allows us to create the sentence:

R p: (Prf(p, R)) q p : (Prf(q, R))

Suppose R is a theorem, then it is provable so there is some p for which Prf(p, R) is a theorem. By ExistentialIntroduction (p.16), p: (Prf(p, R)) must also be a theorem. Assuming the consistency ofA, Ris unprovableso A Prf(q, R) for any q, so certainly ifq p. By making q p a premise (i.e. (q= 0) (q= 1) (q=2) . . . (q= p 1) (q= p)), then in each of the possible cases we would derive Prf(q, R) so using the CaseAnalysis Lemma (p.14) in the general disjunctive case it will still prove Prf(q, R). Therefore, by ConditionalIntroduction, (qp) Prf(q, R) so by Universal Introduction q p : ((Prf(q, R))) and thenapplying Quantification Conversion (p.16) A (q p : ((Prf(q, R))).

On the other hand, by the rule of Conditional Elimination on the Rosser Sentence, (p.11), q p : (Prf(q, R)must be a theorem as the antecedent inR is provable andR is a theorem. Contradiction. ThusR is not a theorem.

Suppose R was a theorem. Then for some q, A Prf(q, R) so by Existential Introduction A q :(Prf(q, R)). Assuming consistency then A Prf(p, R) for all p. If we write R explicitly then we can useDisjunctive Syllogism (p.11) and Junction Conversion (p.11) to rewrite (x y) in the form (x y).

R (p: (Prf(p, R)) q p : (Prf(q, R)))

p: (Prf(p, R)) (q p : (Prf(q, R)))

By Conjunction Elimination we can derive just the second half: (q p : (Prf(q, R))). Applying QuantificationConversion (p.16),A (q p : (Prf(q, R))). Thus the value ofqwhich provesRmust be greater than p toavoid contradiction. Thereforep qmeaning we can derive p q: (Prf(p, R)). We now have a bound for thewitness to the existence ofp, so we can just check this finite number of cases in order to contradict it. Assuming theconsistency ofA, then we will derive Prf(0, R) Prf(1, R) . . . Prf(q 1, R) Prf(q, R). Then as wesaw in the first case we can derive p q: (Prf(p, R))) so by Quantification Conversionp q: (Prf(p, R)).Contradiction.

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Theorem 13 (Provability cannot be captured). is not captured within A.

Proof. First we apply the Diagonalisation Lemma to the property of provability:

A

Supposing that provability was captured, we would have the following theorem for all :

A

Combining these two formulae obviously leads to contradiction:

A

Therefore, provability cannot be captured.

The intuitive reason for this result is that while Prf(x, z) is a bounded arithmetical relation, x:(Prf(x, z)) could bewitnessed by any value ofx. Thus the task of deciding this formula is not primitive recursive and is unsurprisinglynot always possible.It is also important to note where this limitation lies. Provability is clearly semantically expressed by x:(Prf(x, z)),but the transformation laws of the system are not always capable of deciding it. This weakness in the proof system

is necessary to prevent the system from proving too much and thus becoming inconsistent.Theorem 14 (Tarskis Undefinability Theorem). TrueA() cannot be expressed within A.

Proof. Suppose we had some arithmetical relation TrueA() which expressed truth. We could imagine that thispredicate held true if and only if input with the Godel number of a true formula.

First we apply the Diagonalisation Lemma to the property of truth:

A TrueA()

Assuming thatA is sound, then:

TrueA()

Now if we suppose that is true, then it is not true, in which case it is true. Therefore, A has not just failed toprove a true statement about the property of truth, but TrueA() cannot actually be correct in its assignment oftruth values to formulae. Consequently, truth is not captured or even expressed by A.

There is a nice corollary of this result.Lemma 11 (Proofs always need to be longer than you think) . For every function f(), there is a formula which has no proof smaller than f().

Proof. Suppose that this lemma was not correct, so there is some function f() which is essentially an upperlimit on all proofs. We could then capture withx f() : (Prf(x, )), as this involves only bounded searchesso is primitive recursive and thus captured by A. This contradicts Theorem 13 (p.27), so there must be somewell-formed formula which has no proof smaller than f(), such that w

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