The absolute zero vector. - ardix.be · The absolute zero vector. By Karel Van de Rostyne Abstract Set theory [1] and theory of vector spaces [2] are considered as mature theories

By Karel Van de Rostyne

[email protected] 1 / 13

The absolute zero vector.


Abstract Set theory [1] and theory of vector spaces [2] are considered as mature theories where it is safe to build on.

However, when one uses those theories in reality they still contain a contradiction that make’s set theory an

inconsistent system. We will give a series of simple examples in different domains of mathematics and physics to

demonstrate this. We will pinpoint the contradiction and show that applying some elementary logic removes the

contradiction that is the cause of the inconsistency. On top of that is shown that removing this inconsistency offers

the opportunity to extend set theory so that it becomes a more powerful theory.

Introduction The essence of the article can be demonstrated by a simple example from reality to show where it goes wrong.

Example 1. Consider an ordered basis B with basis vectors: { 𝟏𝐅𝐱 , 𝟏𝐅𝐲

, 𝟏𝐅𝐳 } so that force vectors can be represented

[3]. Consider also a point A where maximum 1 force act on so that the force can be described by formula 𝑓 (𝑡).

Suppose that at time t = 𝑡0 no force act on point A. Then “the set of forces that act on point A when t = 𝑡0” is {} or Ø.

When someone uses the formula to calculate “the set of forces that act on point A when t = 𝑡0” the answer for 𝑓 (𝑡0)

will now be { (0, 0, 0) } = {0 }. Both answers for the question “the set of forces that act on point A when t = 𝑡0” are

considered correct. {0 } is the answer when a formula is used, while Ø is the answer when no formula is used. It is

obvious that 2 different answers (element and no element) that represent exactly the same situation presented in

the same basis is not possible. With such system one can prove whatever one wants. (E.g. Equal and different at the

same time, 1 = 0 etc.)

The contradiction The contradiction that is shown in example 1 is easy to explain. A force is a kind of vector where one can consider a

situation without forces. And for such kind of vectors it is evident to choose the origin of the basis so that 0

correspond with the situation without this vector Ø (no force from example 1). In the further text we will only

consider basis where �� correspond with Ø.

So if there are 2 mutual excluding ways to represent the same situation (no force from example 1):

0 that is an element

Ø that is no element

then one gets an inconsistent system.

Example 2. A similar inconsequence as in example 1 is seen for continuous sets versus discrete elements.

Consider a basis to represent a force with one basis vector: { 𝐹𝑋 }

Then A ={q . 𝐹𝑋 | q ∊ [-10, 10]} will be considered as a continuous interval, and no one will doubt that 0 𝐹𝑋

= 0 is an

element of set A. When we consider just the situation without forces (= 0 ∊ A), then this situation is usually

described as Ø. Because 0 correspond with Ø one can as well write A = {q . 𝐹𝑋 | q ∊ {[-10, 0 [ U ] 0, 10]}} which is no

continuous interval.

mailto:[email protected]



Example 3. Consider the concept of the empty set.

Choose a basis where 0 correspond with Ø.

0 correspond with Ø (1)

It is obvious that following expression is correct.

0 ϵ { 0 } (2)

Substitute (1) in (2) => 0 ϵ {Ø} => 0 ϵ Ø

This is a contradiction because the empty set cannot contain elements.

The list of examples and problems is too long to elaborate on it. It are internal contradictions, conflicts with all

possible domains of mathematics (logic, vector spaces, continuity,...), conflicts with the daily reality, physics, etc. No

matter how one looks at it, the conclusion is always that the demonstrated contradiction leads to an inconsistent

system. And such a system has of course no probative force and may not be used to build on it.

The absolute zero vector

The solution for the inconsistency that is shown is simply accept that when 0 correspond with Ø, one should not

consider 0 as an element. So when 0 or (0, 0, …, 0) correspond with Ø, then 0 is just a coordinate for Ø, but it

represents Ø and is therefore no element.

Definition

We will call from now on 0 that correspond with Ø the “absolute zero vector” ( noted as 0′ ).

So per definition: 𝟎′ = Ø

Remarks:

𝟎′ is no element because it correspond with Ø.

Ø has all characteristics of the absolute zero vector. E.g. 𝑣 - 𝑣 = 0′ = Ø

𝟎′ is unique because Ø is unique [4].

Note that 0′ is unique while 0 is only unique per vector space. When a vector space is considered with 0′ as zero

vector, then of course only components (of 0′ ) are considered that are required to define elements of the vector

space.

When you read the text again with the insight that 0′ = Ø you will notice that there are no inconsistencies anymore.

Example 4.

{q . 𝐹𝑋 | q ∊ [-10, 10]} = {q . 𝐹𝑋

| q ∊ {[-10, 0 [ U ] 0, 10]} U 0′ } {q . 𝐹𝑋

| q ∊ {[-10, 0 [ U ] 0, 10]} U Ø} {q . 𝐹𝑋

| q ∊ {[-10, 0 [ U ] 0, 10]}}




Similarity between set theory and theory of vector spaces Set’s and vectors are both ways to describe quantities. Therefore it is not surprisingly that sets and vectors are

interchangeable under certain circumstances. Without the concept of the absolute zero vector this is impossible

because then one bumps on inconsistencies like will be shown. The absolute zero vector offers however the

opportunity to make this interchangeability possible and extend set theory (for sets that apply to 3 additional

criteria) with typical vector operations that makes it a powerful theory that can be used in daily life by almost

everyone. The key ideas to do this will be briefly explained in the further text.

E.g. consider a common daily action with groups like the merging of two fruit baskets. To know the result of this

action, one cannot use set theory.

Summing sets makes the inconsistency very visible and is therefore not accepted (e.g. {𝑣 } + {−𝑣 } = Ø is not accepted

because 𝑣 + (−𝑣 ) = 0 what, without the concept of 0′ , conflicts with Ø). Merging fruit baskets is also not a union of

sets. {apple, 2 pears} U {apple, 3 pear} = {apple, 2 pears, 3 pears} while {2 apples, 5 pears} would be the answer that

one expect when the fruit baskets are merged. Note that this are all valid sets and that the concept of multisets is

completely irrelevant in the context of this text. (See addendum for more info.)

Also other common daily actions like the multiplication of a set with a scalar are impossible with the current set

theory (e.g. 0 {𝑣 } = Ø is not accepted because 0 𝑣 = 0 what, without the concept of 0′ , conflicts with Ø).

The insight that 0′ = Ø offers the opportunity to extend set theory so that it allows those actions.

The idea is that sets, that apply to the further mentioned criteria, can be treated as vectors so that following

operations are possible:

(+) Addition of sets. E.g. 𝐴 +�� with A and B sets

( . ) Multiplication of a set with a scalar. E.g. 5.𝐴 with A a set.

The criteria so that these operations are allowed are:

It must be possible to consider each element of the set as a vector wherefore the addition and the scalar

multiplication are defined. This can be verified with the axioms for a vector space [2].

When considering a one dimensional basis [3] for each individual element of the set, then the zero vector should

always correspond with the absolute zero vector.

All elements of the set should be linear independent [5]. E.g. {5 𝑣 } is ok, but {5 𝑣 , 9𝑣 } does not apply to this

criterion.

Note that daily life objects (e.g. apples) can be treated as vectors. With 0′ as zero vector you can check that all

axioms of vector spaces are satisfied [2]. On first sight we don’t know a physical inverse element for most physical

objects. But nowhere in the axioms is specified that the inverse element should be a physical element that you can

find in daily life. Inverse elements can as well be virtual, like the removing of an object etc. So it is up to users to

verify that objects apply to all axioms of a vector space in the way and range they will use it. Mathematicians on the

other hand should offer a theoretical framework that is consistent and powerful enough to solve problems so that

conclusions are correct in the domain the user has defined, whatever the nature of the elements is.

In the next example most insights of this text are used.




Example 5.

Example with sets. Equivalent example with vectors.

Choose a basis with following basis vectors:

{ 1Fx , 1 apple, 1 kg sand, 1 Euro }

Set 𝐴 = { 5 𝐹𝑋 , 2 apples, 500 g sand }

Set �� = {-5 𝐹𝑋 , 4 apples, 2 kg sand, 15 Euro}

𝑎 = (5, 2, 0.5, 0)

�� = (-5, 4, 2, 15)

0 . (𝐴 +�� ) = 0 . ( { 5 𝐹𝑋 , 2 apples, 500 g sand } +

{-5 𝐹𝑋 , 4 apples, 2 kg sand, 15 Euro} )

= 0 . { 0 𝐹𝑋 , 6 apples, 2.5 kg sand, 15 Euro }

= 0 . { 0′ = Ø, 6 apples, 2.5 kg sand, 15 Euro } = 0 . { 6 apples, 2.5 kg sand, 15 Euro } = { 0 apples, 0 kg sand, 0 Euro } = { Ø, Ø, Ø} = Ø

0 . (𝑎 +�� ) = 0 . ( (5, 2, 0.5, 0) + (-5, 4, 2, 15) ) = 0 . (0, 6, 2.5, 15) = (0, 0, 0, 0)

= 0′ = Ø

Notice the similarities between sets that are used as vectors and the classic way to represent vectors.

Sets that can be treated as vectors. Vector spaces with ordered basis

where �� correspond with 𝟎′ (= Ø).

Elements have a name and can be scaled. E.g. 7𝑣 , apple, 2 Euro

Basis elements have a position, and only scale factors are used to represent an element.

Empty set: Ø Absolute zero vector: 0′

Some operations: #, U, ∩, +, -, . Some operations: +, -, .

All Axioms of a vector space are applicable. All Axioms of a vector space are applicable.

There is in fact another general criterion that is required to ensure that theories that make use of elements become

more consistent and robust theories. This criterion states that elements may not contain contradictions. So a “thing”

that is defined as a “circular square” should be excluded as element because it cannot exist. The element 0

corresponding with Ø (= no element) is also an example of an contradictory element. But this contradiction is

removed by introducing 0′ that is no element. The criterion that an element should not contain contradictions is

definitely much more than a theoretical detail, but that is going beyond the scope of this article.




References

[1] R. C. Freiwald, „An Introduction to Set Theory and Topology,” Washington University in St. Louis, 2014. [Online].

Available: http://dx.doi.org/10.7936/K7D798QH.

[2] E. Weisstein, "Vector Space," A Wolfram Web Resource., [Online]. Available:

http://mathworld.wolfram.com/VectorSpace.html.

[3] E. Weisstein, "Vector Space Basis," A Wolfram Web Resource., [Online]. Available:

http://mathworld.wolfram.com/VectorSpaceBasis.html.

[4] A. Kanamori, "ZFC," Encyclopedia of Mathematics, [Online]. Available:

http://www.encyclopediaofmath.org/index.php?title=ZFC&oldid=19298.

[5] E. Weisstein, „Linearly Independent,” A Wolfram Web Resource., [Online]. Available:

http://mathworld.wolfram.com/LinearlyIndependent.html.




Universes

Abstract The concept of the absolute zero vector has consequences on multiple domains of mathematics, physics and daily

life. The most important consequence is however that there will be shown that the theory of the absolute zero

vector leads directly to the conclusion that there exist an infinite number of sets that have the following two

characteristics:

The elements of the set physically exist.

The elements of the set obey to rules that are completely comparable with the laws of physics that we know

from our universe.

Because such sets have the full potential to explain all aspects of our universe, it make sense that sets that obey to

those 2 criteria are called universes. With a worked out example we will describe in detail a complete mini universe

so that the explained principles become clear.

The absolute zero vector (𝟎′ ) exists.

The reasoning starts with the fact that the absolute zero vector exists. One can consider this trivial, because Ø = 0′

and Ø of course exists. But there are many ways to prove this. Who likes can see a few of them in Appendix A.

Sets of vectors that exist.

For the absolute zero vector is proven that it exists. But then all that is equal to 0′ of course also exist.

(A exists, and A = B ⇒ B exists. If B would not exist, then A ≠ B, what conflicts with the initial assumption.)

We can thus write:

∑𝐯𝒊

𝑛

𝑖=1

= 𝟎′ ⇒ { 𝐯𝟏 , 𝐯𝟐 , … , 𝐯𝐧 } exists (1)

Each of the vectors v1 , v2 , … , vn should of course be possible. “possible” should be interpreted in the strict logical

sense as “not impossible” what is equivalent with “not contradictory”. We will explain in the next paragraph that the

condition that a vector should be possible is a severe condition and the reason that laws of nature exist.

Note that the vectors v1 , v2 , … , vn from expression 1 can of course also exist in the dimensions that we know from

our universe. (e.g. speed, acceleration, electric fields etc.)

Laws of nature. The last step in the reasoning is to show that there are sets of vectors that have as property that they exist

(expression 1: sum equals the absolute zero vector), and where vectors obey to all kind of rules comparable with the

laws of nature that we know from our universe [6]. It is obvious that we will use examples of our universe to explain

this mechanism.




Let’s consider the physical quantity “speed”. As we know, speed can be considered as a vector, but speed is also by

definition a displacement per unit of time or mathematically formulated v = dx/dt.

This definition has several consequences:

All required dimensions [7] should exist.

A displacement dx when there is no space, or dt when there is no time doesn’t make sense and is therefore of

course impossible. Besides that, a displacement is always a property of an object. A displacement without object

(= displacement of nothing) does not make sense.

For those reasons, talking about a speed vector implies a multi-dimensional vector space with at least basis

vectors for speed but also for space, time and basis vectors required to define objects. These conditions are of

course satisfied for our universe.

Continuity [8].

The differentials [9] dx and dt cannot exist for discontinuous elements. Those differentials thus impose

conditions on the continuity of the vectors in the set that is considered. Note that the space and time we live in

is continuous in the same way. There are no discontinuous places and times where the universe stops.

Dimensions are related to each other with formulas.

"v" is "dx/dt". Therefore the condition v = dx/dt should of course always be satisfied for each vector (of an

object). But this condition automatically implies other formula’s. E.g. by integrating [10] dx = v dt, when v is

constant, we get ∆x = v ∆t. But that is exactly the formula we’ve all learned in the physics lessons for constant

speeds. Because (for vectors we know from our universe) we get exactly the same formula’s we know from

physics (laws of nature), it is obvious that we can do calculations in precisely the same way we’ve learned from

physics lessons.

Now it should be clear that we cannot just use expression (1) for a speed v (≠ 0′ ) and write

“�� − �� = 𝟎′ ⇒ { �� , −�� } 𝒆𝒙𝒊𝒔𝒕𝒔“. As expanded above, a speed v implies a multi-dimensional vector space with at

least basis vectors for speed, space, time and basis vectors required to define objects. On top of that there should

also exist (an infinite number of) other elements so that conditions about continuity and the formulas from the

definitions (v = dx/dt etc.) are respected. In fact there should exist a set of vectors where laws are applicable that are

completely comparable with the laws of nature in our universe.

In our universe (that also has as property that it exists), we can see that all vectorial physical quantities are

connected to each other by relationships. We’ve mentioned already speed, but there is also acceleration that is a

change of speed per unit of time (a = dv/dt), force is mass multiplied with acceleration (F = m ∙ a), electric field is a

force per unit of charge (E = F/q) etc. [6]. If there was a vectorial physical quantity that was not connected with

others by a relationship then it could not influence other vectorial physical quantities in any way, and there would be

no possibility to observe it.

The principle proposed here explains that there are sets of vectors that are elements of vector spaces where exist all

kind of rules comparable with laws of nature. Those vector spaces contain a basis for physical quantities that are

functions of other physical quantities so that there exist relations comparable with laws of nature. Note that “being

function of” is a relative concept because if a = f(b) then b = f−1(a). Vectors that trespass the laws imposed by the

vector space are a contradiction and are therefore obviously impossible. We can extend this principle to an

unlimited number of rules. (theory of relativity, quantum mechanics, gravity, …) [6] It is then a matter of

mathematics to write all the known laws of physics in the form of a basis for a vector space. Determine the set of

vectors corresponding with our universe as accurate as possible is then a task for astronomers and physicists.




In general, physical vectors are considered as passive elements that obey to laws of nature. Why there are laws of

nature and the mechanism behind how they work is not known. Here we show that vectors themselves impose the

laws of nature because of what they are (their definition).

The mechanism described in the previous section shows that there is a reason why some sets of vectors should exist

(equals the absolute zero vector). In the next example will be shown in more detail how the principles explained

result in the existence of sets of vectors where rules apply that match with the laws of nature that we know.

This is a short introduction that is only meant to show that sets of elements comparable with our universe should

exist. For scientists and mathematicians that prefer correct and critical thinking above all kind of dogmatic

inconsistent ideologies to explain the reality, this base text can open an almost unlimited source of knowledge that is

able to explain all facets of our reality in an thorough way.

Example: A mini universe Preliminary note: All the mathematics in this example serve only 2 purposes:

The sum of the vectors in the resulting set equals the absolute zero vector.

The resulting set of vectors in the considered vector space should contain no contradictions. This is in fact 100%

equivalent with using the well-known rules of physics.

Description of the example.

In this example we will consider a simple mini universe with an electric field where charged particles move precisely

like in our universe.

To avoid that everything should be written in 3 spatial dimensions, only one spatial dimension will be used. The

relations between the physical quantities that will be considered are [6]:

E = F/q = m ∙ a/q = (m/q) a = c ∙ a; (with c = m/q)

a = dv/dt

v = dx/dt

With

E: electric field (N/C)

F: force (N)

a: acceleration (m/s²) of a particle with c ≠ 0 kg/C

v: speed (m/s) of a particle with c ≠ 0 kg/C

x: position on an x-axis (m)

t: time (s)

q: charge (C: Coulomb)

m: mass (kg)

c = m/q (kg/C)

The speed component of a vector indicates, per definition, where the c value of the vector is “moved” to. This is

comparable with the speed that describes where matter or mass is “moved” to. Because the c component of the

vector is all the time moved in the direction of the speed, the c component is constant for the vectors on the path in

space and time that exists as a result of the speed.

The formula’s in this example, and the way they work are chosen so that they correspond with the laws of nature

that we know in our universe. There are of course an infinite number of other possibilities with different kind of

rules.




It is also an option to work out the example without using the c physical quantity. In that case we have to use mass

m and charge q, what leads to more calculations. As shown before, vectorial physical quantities as electric fields

imply several things like the existence of all the required physical quantities, conditions related to the continuity of

the set of vectors, and the results should of course be in accordance with the given formulas. In addition to those

requirements the sum of the vectors of the resulting set should equal the zero-vector. As shown before, is this the

reason why a set of vectors exists.

In order to make it easy we will work with a particle that corresponds to a single vector. A moving particle then

corresponds with a set of vectors with c ≠ 0 kg/C on a continuous path in space and time. The “space” around the

particle corresponds with vectors with c = 0 kg/C. To get a movement that is identical to the movement of an

electron in an electric field in our universe we choose for

c = mass electron/charge electron

= 9.109534 ∙ 10−31 kg/1.6022 ∙ 10−19 C

= 5.69 ∙ 10−12 kg/C

The example.

Consider now an ordered basis B with basis vectors: { 𝟏𝐄 , 𝟏𝐚

, 𝟏𝐯 , 𝟏𝐱

, 𝟏𝐭 , 𝟏𝐜

} These are the physical units for

respectively the following physical quantities E, a, v, x, t, c.

We first define a set of vectors V that forms the space and time of the mini universe and that contains an electric

field.

V = { vectors in the electric field} U {vectors outside the electric field}

= { (E, 0, 0, x, t, 0) | E = (10 - x)/106; x ∊ [5, 15]; t ∊ [−tmax, tmax]} U

{ (0, 0, 0, x, t, 0) | x ∊ { [−xmax,5[ U ]15, xmax] }; t ∊ [−tmax, tmax]}

Notice that the sum of the vectors in set V is the zero-vector. As explained before this is the condition so that the set

exists.

We consider now a particle located in the electric field from which we specify one vector.

t0 = 0 s; v0 = 1258 m/s; x0 = 10 m; c0 = 5.69 ∙ 10−12kg/C

Based on the definitions, we can find the other components.

E0 = (10 - 10)/106 = 0 N/C; a0 = E0/c0 = 0/5.69 ∙ 10−12 = 0 m/s²

A physical quantity c ( ≠ 0 kg/C) in an electric field correspond with an acceleration (of the particle) that is given by

following equation. (see relations between the physical quantities given in the beginning of this example)

d²x/dt² = a = E/c = (10-x)/( 106 ∙ 5.69 ∙ 10−12) = 175,747 (10-x)

This is a differential equation for which we give immediately the complete solution:

{ (E, a, v, x, t, 5.69 ∙ 10−12) | E = (10 - x)/106; a = -527,241 sin(419.2219 t); v = 1258 cos(419.2219 t); x =10 + 3

sin(419.2219 t); t ∊ [t1, t2] }

Notice that these vectors replace vectors in set V (original situation without particle) that are located on the same

path in space and time (x, t), so that each location in space and time occur only once (not twice) in set V.

In order that set V (with the moving particle) exists, the sum of the vectors in this set should equal the zero-vector.

We will check this per physical quantity:

E, x, t: was initially ok (sum = 0), and the particle changes nothing to this.

a, v: as long as multiples of full periods (t2 – t1 = k . 2π/419.2219 s with k an integer) are considered is this ok (sum =

0), and there is nothing that should be compensated. It doesn’t matter where the period starts.

c: here is a problem (sum > 0). This should be compensated in one way or another, so that the sum for this physical

quantity equals zero. One of the many solutions to compensate this, is introducing a second particle that exists as

long as the first particle, and wherefore the charge is opposite (c = -5.69 ∙ 10−12kg/C). The introduction of multiple




particles, with different c values, that exist longer or shorter is of course also possible. The place of the particles

doesn’t matter, as long as it is within the space and time formed by the vectors of set V. In other words, the new

(compensating) particle replaces vectors from set V, and there are no new vectors created. Here is an example of a

compensating particle:

{ (0, 0, 0, -7, t, -5.69 ∙ 10−12) | t ∊ [t1, t2] }

Figure 1 | Mini universe. Space, electric field and particles.

In general there are lots of compensation mechanisms possible, so that the sum of the vectors of a set equals the

zero-vector. Here are some examples:

Particles with opposite charge. Like negative electrons and positive protons.

Wave motion. Like an electric field that becomes alternating positive and negative.

Particles that turn around each other. In case there are multiples of complete periods considered, the resultant of

the speed and acceleration of the particles both equal zero.

In this mini universe there is still a problem related to the begin and endpoints of the “particles”. A speed defines the

path that a particle follows in space and time. Creating and disappearing of particles goes against the definition of

speed of a particle (c ≠ 0 kg/C), and are therefore no valid options. This mini universe is however a simple example to

illustrate the basic concepts explained in this text. In more advanced examples particles will not consist of one vector

and there will be mechanisms for the beginning and ending comparable with the situation in our universe, but this

goes beyond the objectives of this example.




Appendix A. Proves that the absolute zero vector exists.

Prove 1. Based on a property of the empty set, that Ø is unique [4].

Ø = 0′ def. (1)

Ø is unique (2)

Substitution of (1) in (2) => 0′ is unique. In other words, there exists precisely 1 absolute zero vector. (If 0′ would not

exist, then this would be equivalent with the existence of 0 absolute zero vectors.)

Prove 2. Logic

We start with two conditional expressions that both lead to the same conclusion: Ø exists.

Logically at least one of these two conditional expressions is true. => Ø exists (unconditionally)

Expression 1: “No elements” exists (1)

“No elements” is equivalent with Ø (2)

Substitution of (2) in (1) => Ø exists (3)

Expression 2: There are elements that exist.

Choose an existing element a

{a} exists (4)

{a} = {a} U Ø (5)

Substitution of (5) in (4) => ({a} U Ø) exists

=> Ø exists (6)

At least one of conditional conclusions (3) and (6) is true

=> Ø exists (unconditionally) (7)

Ø = 0′ (8)

Substitution of (8) in (7) => 0′ exist

Prove 3. Reality (per construction)

Consider a one dimensional basis for forces (0 correspond with Ø).

We can now consider an existing situation with “no forces”. (1)

“No forces” (Ø) correspond with 0′ (2)

Substitution of (2) in (1) => existing situation with 0′ => 0′ exist

Add another basis vector to this basis wherefore 0 correspond with Ø. (e.g. electric field). We can now consider an

existing situation without electric field and without force. => 0′ for this two dimensional basis exists. Repeat this

operation for all basis vectors where (0 correspond with Ø). Note that the basis vectors can be added in an arbitrary

order. => 0′ exists (for all basis where 0 correspond with Ø)




References

[4] A. Kanamori, "ZFC," Encyclopedia of Mathematics, [Online]. Available:

http://www.encyclopediaofmath.org/index.php?title=ZFC&oldid=19298.

[6] H. &. M. J. Ohanian, Physics for Engineers and Scientists, 1 red., New York: W. W. Norton & Company, 2007.

[7] E. Weisstein, „Dimension,” A Wolfram Web Resource., [Online]. Available:

http://mathworld.wolfram.com/Dimension.html.

[8] E. Weisstein, „Continuity,” A Wolfram Web Resource., [Online]. Available:

http://mathworld.wolfram.com/Continuity.html.

[9] E. Weisstein, „Derivative,” A Wolfram Web Resource., [Online]. Available:

http://mathworld.wolfram.com/Derivative.html.

[10] E. Weisstein, „Integral,” A Wolfram Web Resource., [Online]. Available:

http://mathworld.wolfram.com/Integral.html.

[11] P. A. Kosso, A Summary of Scientific Method, 1 red., Dordrecht Heidelberg London New york: Springer, 2011.




Addendum Multisets

Some people think that this text has one way or another something to do with multisets. E.g. that {5 apples} is no

set, but a multiset. Here are just a few reasons why this idea is not to the point.

This counter argument goes against the definitions of set and multiset.

https://en.wikipedia.org/wiki/Multiset

https://en.wikipedia.org/wiki/Set_(mathematics)

o {7 Euro, 2 pears} is a set (and because a multiset is a generalization of sets, of course also a multiset)

Reason: Elements of a set can be anything and the elements of the set are different.

o {7 Euro, pear, pear} is a multiset and no set.

Reason: It is not allowed for a set that elements appear more than once.

This thesis goes against the generally accepted use of sets.

{2 𝑣 } is everywhere accepted as a valid set.

This thesis conflicts in many ways with logic and common sense.

o {2 𝑣 } should be no set, and {�� } should be a set, but what if �� = 2 𝑣 ?

o What do we do with 1000 kg sand?

This is for some people no valid set. But if we call it 1 ton sand, is it then a valid element of a set?

o Maybe we have to consider the sand grains. How hard do we have to push the sand so that it can be

considered as one block? In which axiom’s is this kind of stuff written?

o How are we going to write 1000 stones?

{1000 stones} is not allowed as element of a set, and with the syntax of multisets we have a lot of writing

work {stone, stone, …, stone}

o …

Also { 5 apples, 2 apples } is a valid set.

o Prove:

As shown { 5 apples } and { 2 apples } are both valid sets.

When applying the definition of a union on both sets one get:

{ 5 apples } U { 2 apples } = { 5 apples, 2 apples }

The union of sets is a set, and nowhere is written that it is not allowed to do a union of the sets above.

o Or just stick to the definition of sets.


Multisets are used for a complete different purpose than what users of this argument think. It has nothing to do

with the topic of this text. I will not elaborate on it, but multisets are for example indispensable to note things

like results …


https://en.wikipedia.org/wiki/Multiset



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