## Continuum does not exist

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Here is the argument:

D) By continuum I mean a set of distinct points without an abrupt change or gap between points
A) Assume that continuum exists (assume that D is true)
P1) There is however either a gap between all pairs of points of the continuum or there is no gap
P2) We are dealing with the same point of the continuum if there is no gap between a pair of points
C1) Therefore there is a gap between all pairs of distinct points of the continuum (from P1 and P2)
C2) Therefore, the continuum does not exist (from A and C1)

To challenge the argument (basically to challenge P1) people claim that continuum is the real number. They argue that any finite interval on the real number line can be infinitely divided into smaller intervals. The real number, however, is constructed from two parts, an integer part and a decimal part. The number of digits of the decimal part however is the number of points of the natural number so-called the cardinality of the natural number (Aleph_0). It however can be shown that there is the smallest interval on the real number so-called infinitesimal. Infinitesimal can be constructed as follows: 0.0...01 by "..." I mean Aleph_0 0. Therefore, there is the smallest interval on the real number so there is a gap between each consecutive points of the real number.
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By continuum I mean a set of distinct points without an abrupt change or gap between pointsMoK

A point is an abstract mathematical entity which doesn't correspond with any phenomenon in the world of our everyday existence. The same is true of a continuum.
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A point is an abstract mathematical entity which doesn't correspond with any phenomenon in the world of our everyday existence.
The center of mass of your body is a point. The center of mass of your computer is a point as well. There is a distance between these two points. The question is whether this distance is discrete or continuous.

The same is true of a continuum.
Well, that is the subject of discussion.
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The center of mass of your body is a point. The center of mass of your computer is a point as well. There is a distance between these two points. The question is whether this distance is discrete or continuous.MoK

A point does not exist in the everyday world. It is an abstraction, and idealization - imaginary. It has no size. It 's zero dimensional. It does not take up space. A center of gravity is a point and, as such, is also an abstraction, imaginary. And, as I noted, a continuum is also a mathematical idealization. It doesn't exist. It's imaginary.

A continuum exists in the same manner that a point does.
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A point does not exist in the everyday world. It is an abstraction, and idealization - imaginary. It has no size. It 's zero dimensional. It does not take up space. A center of gravity is a point and, as such, is also an abstraction, imaginary.
Are you claiming that something which is an abstraction cannot exist?
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Hypostatisation. Another case of folk mistaking a way of talking for a thing.
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Are you claiming that something which is an abstraction cannot exist?MoK

It exists in your mind, your imagination, but not in the physical world. I can imagine a point. I can also imagine a line, which is continuous.

I don't think we're getting anywhere. I'm going to leave it at that.
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Hypostatisation. Another case of folk mistaking a way of talking for a thing.
Ok!
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It exists in your mind, your imagination, but not in the physical world. I can imagine a point. I can also imagine a line, which is continuous.

I don't think we're getting anywhere. I'm going to leave it at that.
As you wish.
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C1) Therefore there is a gap between all pairs of distinct points of the continuum (from P1 and P2)
C2) Therefore, the continuum does not exist (from A and C1)
MoK
C2 doesn't follow at all. In the real numbers, there being a gap between 4 and 13 does not imply that the real numbers (or even the rationals) is not a continuum. You need to demonstrate that there is nothing between some pairs of points that are not the same point. Then you've falsified the continuum premise.

A point is an abstract mathematical entity which doesn't correspond with any phenomenon in the world of our everyday existence
I disagree. Yes, a point can be an abstraction, but can also correspond to a location in space say.
It indeed does not 'take up space', by definition. You seem to imply that something real must take up space.
Perhaps it's all a linguistic quibble, but it isn't one that takes apart the OP argument.

The center of mass of your body is a point.MoK
Only in classical physics, and our universe isn't classical. But I accept your refutation of the rebuttal to the OP. Do you accept my rebuttal?
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A point is an abstract mathematical entity which doesn't correspond with any phenomenon in the world of our everyday existence
— T Clark
I disagree.

Even if we disagree, the OP still doesn't make sense. Whatever a point is, a line is the same kind of thing and a line is continuous by definition. A line is expressed as f(x) = mx + b, which means it is defined for any real number "x."
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C2 doesn't follow at all. In the real numbers, there being a gap between 4 and 13 does not imply that the real numbers (or even the rationals) is not a continuum.
C1 states that there is a gap between all pairs of distinct points of the continuum.

You need to demonstrate that there is nothing between some pairs of points that are not the same point. Then you've falsified the continuum premise.
Are you challenging (P1)? If yes, I already illustrated that given the definition of the real number one can construct the smallest number or the smallest interval so-called infinitesimal.

Only in classical physics, and our universe isn't classical. But I accept your refutation of the rebuttal to the OP. Do you accept my rebuttal?
You can define it in quantum physics as well. Of course, you cannot measure it.
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Even if we disagree, the OP still doesn't make sense
I don't disagree with that

C1 states that there is a gap between all pairs of distinct points of the continuum.MoK
What do you mean by 'a gap'? If you mean that the two distinct points are not the same point, then yes, by definition. There's a gap between 4 and 13.

If you mean by 'a gap' that there is nothing between these two distinct points, then C1 is false. For instance, 10 is between 4 and 13. 'The gap' is not empty, and C2 seems to rely on any such gap being empty.

Are you challenging (P1)?
Without a definition of a gap, P1 is ambiguous. It states that either G or ~G, which is tautologically true, making P1 empty. The word 'distinct' is not part of P1.
My challenge was C2 following from C1, or any of what preceded C2.

It however can be shown that there is the smallest interval on the real number so-called infinitesimal.MoK
Show it then. What about the number that is halfway between this smallest positive number and zero? You've shown that it doesn't exist?
Funny that the smallest number happens to be a perfect power of 0.1
What are the odds of that?

You can define [the center of mass of a body] it in quantum physics as well. Of course, you cannot measure it.
Define it then, without making classical assumptions (like a particle having a location, or some counterfactual property.
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What do you mean by 'a gap'? If you mean that the two distinct points are not the same point, then yes, by definition. There's a gap between 4 and 13.
By a gap, I mean an interval.

Without a definition of a gap, P1 is ambiguous. It states that either G or ~G, which is tautologically true, making P1 empty. The word 'distinct' is not part of P1.
It shouldn't be.

Show it then. What about the number that is halfway between this smallest positive number and zero? You've shown that it doesn't exist?
Well, I construct the infinitesimal in this way: 0.0...01. By "..." I mean Aleph_0 zero. The next number is then 0.0...02 therefore there is a gap, 0.0...01 between these two numbers. One could say how about 0.0...011? It can be shown that 0.0...011 is 0.0...02 by simple math. 0.0...011=0.0...01+0.0....01. By "...." (where dots appears four times) I mean Aleph_0+1. But Aleph_0+1=Aleph_0 therefore 0.0...01+0.0....01=0.0...01+0.0...01=0.0...02.

Define it then, without making classical assumptions (like a particle having a location, or some counterfactual property.
This is off-topic but I give it a try. Consider a hydrogen atom for example. R is the center of mass position operator of the atom that is related to the position operator of the nucleus (r_n) and the position operator of the electron (r_e). The relation is R=(m_n*r_n+m_e*r_e)/(m_n+m_e) where m_n and m_e are the mass of the nucleus and electron respectively. The center of mass therefore can be calculated as <Psi(R,t)|R|Psi(R,t)>.
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In ordinary mathematics, from axioms we define 'is a real number' and we prove that there is a set whose members are all and only the real numbers. And we prove that there is a total ordering on the set of real numbers such that every non-empty subset with an upper bound has a least upper bound (equivalently, every non-empty set with a lower bound has a greatest lower bound). We call that 'the standard ordering'. Then we define 'is a complete ordered field'. And we prove the existence of operations on the set of real numbers such that, along with the standard ordering, we have a complete ordered field. Then we prove that all complete ordered fields are isomorphic with the system of real numbers.

There are different notions of 'the continuum' and 'a continuum'. An ordinary mathematical notion is that the continuum is the set of real numbers along with the standard ordering of the real numbers; then a continuum is any set and ordering on that set that is isomorphic with the continuum.

But you say that you use the terminology to mean that "a continuum is a set of distinct points with no abrupt change of gap between the points".

So what are the definitions of "abrupt change" and "gap between"? But even more basically, what are your axioms?

You do say:

By a gap, I mean an interval.MoK

There is however either a gap between all pairs of points of the continuum or there is no gapMoK

Between any two different real numbers there is an interval.

But the ordinary definition of continuum is not "there are no intervals between points". It is only by you personally redefining 'continuum' that you infer that there is no continuum.
One could as easily define 'is a mammal' by 'is an omnipotent animal' to then conclude that there are no mammals.

We are dealing with the same point of the continuum if there is no gap between a pair of pointsMoK

[x x] is the interval whose only member is x.

(x x) is the empty interval.

Therefore there is a gap between all pairs of distinct points of the continuumMoK

Of course, between any two different real numbers there is a non-empty interval.

Therefore, the continuum does not existMoK

We have:

If x < y, then (x y), (x y], [x y) and [x y] are non-empty intervals.

[x x] is a singleton interval.

From that it follows that there is an interval between any real numbers.

But it is only your ersatz definition of 'continuum' that leads you to infer that there is not a continuum from the fact that between any two points there is an interval. We might as well define 'is a computer' by 'can make infinitely many calculations in finite time' to conclude that there are no computers.

people claim that continuum is the real number.MoK

There are uncountably many real numbers. There is not just one real number that is to be called "the real number".

Maybe you mean that we say that the continuum is the set of real numbers. I would rather be more exact in saying that the continuum is the set of real numbers along with the standard ordering of the set of real numbers.

The real number, however, is constructed from two parts, an integer part and a decimal part.MoK

Again, there is not a real number that is called "the real number".

The usual constructions of real numbers are as either a Dedekind cut or an equivalence class of Cauchy sequences. Then we prove that every real number is representable as an integer along with a denumerable sequence (binary, decimal or whatever, as suits). And we prove that every integer along with such a sequence corresponds with a unique real number.

Infinitesimal can be constructed as follows: 0.0...01 by "..." I mean Aleph_0 0MoK

I guess you mean that there are denumerably many 0s followed by 1. There are rigorous treatments of nonstandard analysis in which there are infinitesimals. And there is the sequence {<x y> | x in w+1 & (x in w -> y = 0) & (x = w -> y = 1)}. But you are merely handwaving. It doesn't follow that such a sequence is an infinitesimal or represents an infinitesimal. Rather, you have to define 'is an infinitesimal' then prove that such sequences are or represent infinitesimals. Then define operations and orderings that include infinitesimals. To see how that is actually done, look up 'nonstandard analysis'. Moreover, that there are systems with infinitesimals doesn't contradict that there are other systems without infinitesimals.
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Let's start the discussion by defining the different gaps. Given a set of points, points of real number for example, I define G1 as an interval between two distinct points. I define G2 as an interval between two immediate points with no point between them (what I call an abrupt change in OP). I am interested in understanding whether there are gaps of type G2 in the set of real number given the definition of the set of real number which can be found here. The set of real can be defined as a set of numbers each number has an integer and decimal fraction. The decimal fraction contains an infinite number of digits, to be precise Aleph_0, where Aleph_0 is the cardinality of the natural number. To show that G2 exists in the set of real number I simply construct an infinitesimal_1 as follows: infinitesimal_1 = 0.0...01 where by "..." I mean an infinite number of zero to be more precise by the infinity in here I mean Aleph_0. I can show the second smallest number is 0.0...02 with no number between. Think of a number between, 0.0...011. This number can be divided into two parts 0.0...01 and 0.0...001. 0.0...001 however is 0.0...01 given the property of Aleph_0 in which Aleph_0+1=Aleph_0 therefore 0.0...011 is equal to 0.0...02.

I am sure that mathematicians define things in a more rigorous way when it comes to infinitesimal. I am interested in whether you can show that there is an infinitesimal in the set of real number given your definition.
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I define G2 as an interval between two immediate points with no point between them (what I call an abrupt change in OP). I am interested in understanding whether there are gaps of type G2 in the set of real number given the definition of the set of real number which can be found here.MoK

There are provably no such G2 gaps in the real numbers. Between any two distinct real numbers, there is always another one strictly between them.

The real numbers are dense. That means that between any two distinct real numbers, there is a third one between them. For example between 5 and 7 we find 6. Between .001 and .002 we find .0015. And so forth.

In fact there's a formula to find the exact midpoint between two distinct reals. If $x$ and $y$ are distinct reals, then $\frac{x+y}{2}$ is halfway between them.

Also note that even the hyperreal numbers of the nonstandard reals, which have actual infinitesimals, are also dense. Between any two hyperreals there's another one distinct from those two.

There are no "adjacent points" in the real numbers. When you think of the real numbers, don't think of a string of bowling balls. Think of infinitely stretchy maple syrup or taffy. You can stretch and stretch but there are always more points.

infinitesimal_1 = 0.0...01MoK

This notation is meaningless. In decimal notation, each digit position to the right of the decimal point corresponds to a negative natural number power of 10. That is, starting from the decimal point and going right, the position values are 1/10, 1/100, 1/1000, and so forth.

There is not an "infinitieth" position. There is just no such thing in the notation. And that's not how the hyperreals work.

To emphasize this point, note that in even in the hyperreals, you can always divide by 2. So if .000...1 was a hyperreal infinitesimal, what is .000...1/2? You have no notation for that. In fact, .000...1 is meaningless in the reals and meaningless in the hyperreals.

And what if we divide by 10? You'd need an extra 0 in there somewhere ... but you can't add another 0 if you already have Aleph-0 of them. Make sense?

Forget .000...1. No such notation. Meaningless in every context, real or hyperreal.
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My context here, unless mentioned otherwise, is ordinary mathematics, which is classical mathematics as found in calculus for the sciences, which is axiomatically formalized in set theory. I don't opine that that is the only context we should consider; only that when you talk about "real numbers", without qualifying that you don't mean other than the ordinary context, I surmise that you are talking about the ordinary context. And when I say "an object exists with property P" I mean that from the axioms we prove the theorem that we render in English as "There exists an x such that x has property P". I don't opine in this immediate context as to the philosophical aspects of such existence theorems.

We provide (with definitions from the primitives for membership (e) and identity (=), and from the axioms) such basics as set abstraction, subset, union, singleton, ordinal, function, sequence, finite, infinite, natural number, the set of all and only the natural numbers (w [read as 'omega']), denumerable, ordering and operations on the natural numbers, recursion, transfinite recursion, cardinal, aleph, irrational number, the set of all and only the rational numbers, ordering and operations on the rational numbers, Dedekind cut, equivalence class, and Cauchy sequence.

Most pertinently here, we define the property "is a real number", and we prove the existence of the set of all and only the real numbers (R), a particular ordering on that set (<), two particular elements (0 and 1), and two particular operations (+ and *). We also define "is a completed ordered field" and we prove that the system <R < 0 1 + *> (I have to use the symbol '<' for both less-than and the opening bracket) is a complete ordered field and that all complete ordered fields are isomorphic with one another. Thus we have the number system of real numbers that is unique within isomorphism. (Then we also define subtraction (-), negative, positive, non-negative, non-positive, division (/), absolute value (| |) and intervals (( ), [ ), ( ], [ ]).

And we define "the continuum" as R along with <. That is the ordered pair with R and <. So, the continuum is the set of real numbers considered vis-a-vis the standard ordering of the set of real numbers. Since we proved the existence of the set of all and only the real numbers and we proved the existence of the standard ordering on that set, we proved the existence of the ordered pair, and thus are entitled to name it, as we name it "the continuum". The continuum exists.

Usually "is a real number" is not rigorously defined by "is an integer followed by a decimal sequence". Rather, an integer followed by a decimal sequence is taken to be merely a representation of a real number.

The two most common rigorous definitions of "is a real number" are:

Df. r is a real number if and only if r is an equivalence class of Cauchy sequences

or

Df. r is a real number if and only if r is a Dedekind cut.

But we can define "is a real number" as a certain kind of sequence as long as we eschew infinitely many consecutive 9s:

Df. r is a real number
if and only if
r is a sequence whose domain is w and
r(0) is an integer and
for n>0, r(n) is in {0 1 2 3 4 5 6 7 8 9} and
for all n in w, there is an m in w such that n<m and f(m) not= 9

So, r(0) is the integer part, then for each n>0, r(n) is the nth digit in the infinite expansion, and there are not infinitely many consecutive 9s.

They key point is that all three - the equivalence class of Cauchy sequences definition, the Dedekind cut definition, and the decimal sequence definition - all provide a complete ordered field, and all complete ordered fields are isomorphic with one another.

But what about infinitesimals? A possible definition:

i is an infinitesimal if and only if 0 < |i| and there is no real number strictly between 0 and i.

or

i is an infinitesimal if and only if 0 not= i and there is no real number strictly between 0 and i.

Then it is easy to prove that no real number is an infinitesimal.

But the fact that no real number is an infinitesimal doesn't prevent us from formulating another system in which there do exist infinitesimals.

Perhaps the three main two approaches to having a number system with infinitesimals are:

(1) In set theory, we use models. Thus we have two different number systems to use: The real number system and a different number system that has infinitesimals.

(2) In set theory, we use ultrafilters. Thus we have two different number systems to use: The real number system and a different number system that has infinitesimals.

(3) To set theory we add a primitive unary predicate "is standard" and additional axioms. In that theory we also have the real number system and a different number system with infinitesimals.

In any case, having a different number system doesn't disprove the existence of the other number system and doesn't disprove the existence of the continuum.

/

Your terminology about gaps conflates the predicate "is a gap" with the nouns "G1" and "G2".

Let's fix that with clear definitions of predicates:

Df. g is a G1-gap if and only if g is an interval

Df. g is a G2-gap if and only if g is an interval between two different points q and r such that there is no point strictly between q and r.

We prove that in the real number system there are no G2-gaps.

You propose to have 0.0...01 serve as an infinitesimal.

We could as easily write that as 0.0...1.

But two things:

(1) We need to be clear what 0.0...1 is.

(2) You need to define an ordering that has your supposed infinitesimal in the field of the ordering and to prove the needed theorems.

Regarding (1):

First:

A sequence is a function whose domain is an ordinal.

Every cardinal is an ordinal, but not every ordinal is a cardinal.

Ordinal addition is different from cardinal addition. But we use the same symbol '+' for ordinal addition and cardinal addition. So, we consider context to see whether '+' is being used for ordinal addition or cardinal addition. (And '+' for addition of real numbers is different also.)

So I'll use '+' for addition of real numbers, '#' for ordinal addition, and '+' [in bold] for cardinal addition.

The least infinite ordinal is w. w is also a cardinal, referred to as 'aleph_0' when we emphasize it as a cardinal.

{n | n is a natural number} = N = w = aleph_0 = card({n | n is a natural number}).

(aleph_0)+1 = aleph_0 = w.

w#1 not= w.

w#1 not= aleph_0.

Second:

Lets look at three things you've mentioned:

(a) 0.0...1

(b) 0.0...2

(c) 0.0...11

(a) and (b) represent sequences whose domain is w#1.

(c) represents a sequence whose domain is w#2.

Regarding (2):

You've not defined any such ordering.

Moreover, though you are free to have such things as (a), (b) and (c) in a number system that you may formulate, they do not represent real numbers as we have represented real numbers. And if they represent infinitesimals, then they are not in any number system isomorphic with the set of real numbers. So, your argument is profoundly ill-conceived. Even if you went on to fulfill your own formulation of a number system with infinitesimals, then that would not be the real number system nor isomorphic with the real number systems, thus it is ludicrous to say, as you do, that there is an interval in the reals between two different points q and r such there is no real strictly between q and r.

Moreover, your profoundly ill-conceived argument does not in any way support your titular assertion that the continuum does not exist.
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Between any two distinct real numbers, there is always another one strictly between them.
That is not correct: Consider two numbers on the real number such as $a$ and $b$. Let's define the mean as $c_1=\frac{a+b}{2}$. We can determine the next mean as either $c_2=\frac{a+c_1}{2}$ or $c_2=\frac{c_1+b}{2}$ in which in the first case we approach to $a$ from the right and in the second case we approach to $b$ from the left. Let's work with the first approach: $c_2=\frac{a+c_1}{2}$. The next mean can then be determined by $c_3=\frac{a+c_2}{2}$. We can write $c_{i+1}=\frac{a+c_i}{2}$. The distance between $c_{i+1}$ and $a$ is $d_{i+1}=c_{i+1}-a=c_i-c_{i+1}=\frac{(2^i-1)*a+b}{2^i}-\frac{(2^{i+1}-1)*a+b}{2^{i+1}}$. Therefore, we have $d_{i+1}=\frac{2*((2^i-1)*a+b)-((2^{i+1}-1)*a+b)}{2^{i+1}}=\frac{b-a}{2^{i+1}}$. So, $d_{\text{Aleph}_1}=\frac{b-a}{2^{\text{Aleph}_1}}\lt\frac{\text{Aleph}_1}{2^{\text{Aleph}_1}}=0$. Therefore, your statement does not follow.
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It appears all you have shown is the distance between consecutive means tends to zero. The last sentence is a little weird. The previous sentence says it all if one takes a limit.

I haven't kept up with this nonsense. Kudos to those knowledgeable who have.
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Therefore, we haveMoK

$\frac{\aleph_1}{2^{\aleph_1}}$?

The burden is on you to justify this notation. Perhaps you can begin with your theory of cardinal division. You'll have a hard time making sense of it. Trust me, if there were any mathematical theory that justifies this notation, I'd have heard of it. I haven't and there isn't.

BUT! Your underlying idea of continually subdividing an interval is correct. It just doesn't show what you think it does.

If as @jgill notes you are simply proving that you can find a sequences approaching $a$ and $b$ as limits, that just amounts to a restatement of the fact that the real numbers are dense: that between any two, there's a third. An equivalent condition is that we can find a sequence approaching as a limit any given real number.

And the equivalence is shown by continually taking the midpoint, as you did. But your notation is fanciful and undefined. Your basic idea is correct, but it does not show that there are two reals without a third between them. On the contrary, it shows that if between any two reals there's a third between them, then we can find a sequence approaching as a limit any given real.

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

ps -- How did you get to $\aleph_1$ as the limit of 1, 2, 3, 4, ...? Did you mean perhaps $\aleph_0$?

In summary, your idea is correct, even if your notation isn't. But your idea only shows that the two definitions of a dense set coincide. If you can always find the midpoint between two points, then you can always find sequences converging to each of the two points.
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It appears all you have shown is the distance between consecutive means tends to zero.
Well, I showed that the distance between consecutive means is zero if the number of divisions is $\aleph_1$.

The last sentence is a little weird.
Well, that is the division of two cardinal numbers. I googled about the division of cardinal numbers and I found two references here.
The previous sentence says it all if one takes a limit.
Ok, I see what you mean.
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I googled and I found two references about the division of cardinal numbers. You can find the references here.
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Thank you very much for your post. It made many things clear to me after I read it a few times. I am still not convinced that infinitesimal does not exist though. Can you prove it? I however can show that the distance between consecutive means tends to zero. You can find my argument here. Please let me know what do you think.
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For a mathematics for the sciences, ordinarily we use a complete ordered field. That requires having a non-empty set, a 2-place relation (<) on the set and two 2-place operations (+ *) on the set such that for all x, y and z:

ORDERED FIELD
x+(y+z) = (x+y)+z (associativity of addition)
x+y = y+x (commutativity of addition)
EyAx x+y = x (additive identity element)
Theorem: E!yAx x+y = x
Definition: 0 = the unique y such Ax x+y = x
Ey x+y = 0 (additive inverse)
EyAx x*y = x (multiplicative identity element)
Theorem: E!yAx x*y = x
Definition: 1 = the unique y such that Ax x*y = x
0 not= 1
x*y = y*x (commutativity of multiplication)
x*(y*z) = (x*y)*z (associativity of multiplication)
x*(y+z) = (x*y)+(x*z) (distributivity)
x not= 0 -> Ey x*y = 1 (multiplicative inverse)
(x < y & y < z) -> x < z (transitivity)
exactly one: x < y, y < x, x = y (trichotomy)
x < y -> x+z < y+z (monotonicity of addition)
(0 < z & x < y) -> x*z < y*z (monotonicity of multiplication)

COMPLETE ORDERED FIELD
In set theory, we prove that there is a carrier set (called 'R') for such a system and such that, for any upper bounded non-empty subset of S of R, S has a least upper bound. With that and the rest of the set theory axioms we can do the mathematics of derivatives and integrals for the sciences.

/

An alternative is to have a system with infinitesimals. But still, ordinarily, we need to define <, + and * and to prove whatever theorems are needed for the machinery of mathematics.

To just wave a hand and say "Voila, this is my infinitesimal" does not provide the needed definitions of < + and * with infinitesimals nor the needed proofs.

So how do we go about proving the existence of a system with infinitesimals? For your purposes, it would help to first define 'is an infinitesimal'. I provided a definition previously, but I notice that many authors include 0 as an infinitesimal. So perhaps use this definition:

x is an infinitesimal if and only if, for every positive real number y, |x| < y.

It has been proven for you that for every real number x there is a positive real number y such that y < |x|.

So no non-zero real number is an infinitesimal.

One more time: No non-zero real number is an infinitesimal. The proof that no non-zero real number is an infinitesimal is immediate from the fact that for every real number x there is a positive real number y such that y < |x|. We don't need to keep going over this over and over.

I mentioned previously different ways of proving the existence of systems with infinitesimals.

I am still not convinced that infinitesimal does not exist though. Can you prove it?MoK

I can't prove it, since it is not provable. I never said that there are not infinitesimals. I explicitly said that there are systems with infinitesimals and I even mentioned different ways of developing them. But no non-zero real number is an infinitesimal, which has been proven to you over and over. That you are still asking about this suggests that you are not paying attention to the posts.

(I mentioned previously using the compactness theorem for having a system with infinitesimals. But an interesting thing about that is that we prove the existence of such a system but (unlike with the reals) we cannot define a particular such system: The compactness theorem for uncountable languages relies on Zorn's lemma. Same problem with the ultrafilter method.)

I googled and I found two references about the division of cardinal numbers. You can find the references here.MoK

That source itself points out that when the numerator is less than the denominator, there is no definition of numerator/denominator. The very source you point to disputes your claim that (aleph_1)/(2^aleph_1) is properly defined. And you should have proven that for yourself when you first thought of it [here 'X' stands for the Cartesian product]:

K*L = card(K X L). And we have the theorem that if L <= K and K is infinite and L is non-zero, then K*L = K.

The definition of x/y:

x/y = the unique z such that z*y = x. If there is no such unique z, then x/y is not properly defined.

Suppose here that L <= K and K is infinite and L is non-zero:

If L = K, then there are Z such that Z*L = K, but there is no unique such Z. So L/K is not properly defined.

If L < K, then the unique Z such that Z*K = L is L. So L/K = L.

If K/L were properly defined, then K/L would be the unique Z such that Z*L = K. But there is no such Z at all, let alone a unique one. So K/L is not properly defined.

You did an Internet search but didn't even bother to read what you found, instead recommending that other people read it even though, unknown to you since you didn't even read it, it says the exact opposite of your claim that you made up out of thin air.

"(aleph_1)/(2^aleph_1)" is pure bunk.

the distance between consecutive means tends to zeroMoK

The sequence of half distances converges to 0. So what? That doesn't prove that it's not the case that between any two different real numbers there is another different real number.

Well, I showed that the distance between consecutive means is zero if the number of divisions is aleph_1.MoK

"number of divisions is aleph_1" is undefined.

I recommend that you learn how axiomatic mathematics and definitions work
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Formally, the classical continuum "exists" in the sense that that it is possible to axiomatically define connected and compact sets of dimensionless points that possesses a model that is unique up to isomorphism thanks to the categoricity of second order logic.

But the definition isn't constructive and is extensionally unintelligible for some of the reasons you pointed out in the OP. Notably, Dedekind didn't believe in the reality of cuts of the continuum at irrational numbers and only in the completeness of the uninterpreted formal definition of a cut. Furthermore, Weyl, Brouwer, Poincare and Peirce all objected to discrete conceptions of the continuum that attempted to derive continuity from discreteness. For those mathematicians and philosophers, the meaning of "continuum" cannot be represented by the modern definition that is in terms of connected and compact sets of dimensionless points. E.g, Peirce thought that there shouldn't be an upper bound on the number of points that a continuum can be said to divide into, whereas for Brouwer the continuum referred not to a set of ideal points, but to a linearly ordered set of potentially infinite but empirically meaningful choice sequences that can never be finished.

The classical continuum is unredeemable, in that weakening the definition of the reals to allow infinitesimals by removing the second-order least-upper bound principle, does not help if the underlying first-order logic remains classical, since it leads to the same paradoxes of continuity appearing at the level of infinitesimals, resulting in the need for infinitesimal infinitesimals and so on, ad infinitum.... whatever model of the axioms is chosen.
Alternatively, allowing points to have positions that are undecidable, resolves, or rather dissolves, the problem of 'gaps' existing between dimensionless points, in that it is no longer generally the case that points are either separated or not separated, meaning that most of the constructively valid cuts of the continuum occur at imprecise locations for which meta-mathematical extensional antimonies cannot be derived.
Nevertheless this constructively valid subset of the classical continuum remains extensionally uninterpretable, for when cut at any location with a decidable value, we still end up with a standard Dedekind Cut such as (-Inf,0) | [0,Inf) , in which all and only the real numbers less than 0 belong to the left fragment, and with all and only the real numbers equal or greater than 0 belonging to the right fragment, which illustrates that a decidable cut isn't located at any real valued position on the continuum. Ultimately it is this inability of the classical continuum to represent the location of a decidable cut, that is referred to when saying that the volume of a point has "Lebesgue measure zero". And so it is tempting to introduce infinitesimals so that points can have infinitesimal non-zero volume, with their associated cuts located infinitesimally close to the location of a real number.

The cheapest way to allow new locations for cuts is to axiomatize a new infinitesimal directly, that is defined to be non-zero but smaller in magnitude than every real number and whose square equals 0, as is done in smooth infinitesimal analysis, whose resulting continuum behaves much nicer than the classical continuum for purposes of analysis, even if the infinitesimal isn't extensionally meaningful. The resulting smooth continuum at least enforces that every function and its derivatives at every order is continuous, meaning that the continuum is geometrically much better behaved than the classical continuum that allows pathological functions on its domain that are discontinuous, as well as being geometrically better behaved than Brouwer's intuitionistic continuum that in any case is only supposed to be a model of temporal intuition rather than of spatial intuition, which only enforces functions to have uniform continuity.

The most straightforward way of getting an extensionally meaningful continuum such as a one dimensional line, is to define it directly in terms of a point-free topology, in an analogous manner to Dedekind's approach, but without demanding that it has enough cuts to be a model of the classical continuum. E.g, one can simply define a "line" as referring to a filter, so as to ensure that a line can never be divided an absolutely infinite number of times into lines of zero length, and conversely, one can define a collection of "points" as referring to an ideal, so as to ensure that a union of points can never be grown for an absolutely infinite amount of time into having a volume equaling that of the smallest line. This way, lines and points can be kept apart without either being definable in terms of the other, so that one never arrives at the antimonies you raised above.
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E.g, one can simply define a "line" as referring to a filter,sime

Hmmm. Care to explain? (I recall having difficulty with filters, ultra filters, etc. in grad school a half century ago. I only encountered them in passing - not in my specialty area)
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But the definition isn't constructive and is extensionally unintelligible for some of the reasons you pointed out in the OP.sime

In set theory, we prove the existence of a particular complete ordered field and that it is unique within isomorphism. I didn't adduce any reasons that that is not constructive nor even say that it is not constructive.

Dedekind didn't believe in the reality of cuts of the continuum at irrational numberssime

What passage in what paper by Dedekind are you referring to?

so that one never arrives at the antimonies you raised above.sime

I didn't mention any antinomies.
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Hmmm. Care to explain? (I recall having difficulty with filters, ultra filters, etc. in grad school a half century ago. I only encountered them in passing - not in my specialty area)

I'm not sure what @sime meant by that statement either. But ultrafilters are just a set theory gadget that lets you rigorously construct the hyperreals of nonstandard analysis.
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I am a retired physicist and my knowledge of mathematics is very rusty due to my age. I however understand what you are saying. Thanks for helping me to learn new things and refresh what I learned a long time ago.

For a mathematics for the sciences, ordinarily we use a complete ordered field. That requires having a non-empty set, a 2-place relation (<) on the set and two 2-place operations (+ *) on the set such that for all x, y and z:

ORDERED FIELD
x+(y+z) = (x+y)+z (associativity of addition)
x+y = y+x (commutativity of addition)
EyAx x+y = x (additive identity element)
Theorem: E!yAx x+y = x
Definition: 0 = the unique y such Ax x+y = x
Ey x+y = 0 (additive inverse)
EyAx x*y = x (multiplicative identity element)
Theorem: E!yAx x*y = x
Definition: 1 = the unique y such that Ax x*y = x
0 not= 1
x*y = y*x (commutativity of multiplication)
x*(y*z) = (x*y)*z (associativity of multiplication)
x*(y+z) = (x*y)+(x*z) (distributivity)
x not= 0 -> Ey x*y = 1 (multiplicative inverse)
(x < y & y < z) -> x < z (transitivity)
exactly one: x < y, y < x, x = y (trichotomy)
x < y -> x+z < y+z (monotonicity of addition)
(0 < z & x < y) -> x*z < y*z (monotonicity of multiplication)

COMPLETE ORDERED FIELD
In set theory, we prove that there is a carrier set (called 'R') for such a system and such that, for any upper bounded non-empty subset of S of R, S has a least upper bound. With that and the rest of the set theory axioms we can do the mathematics of derivatives and integrals for the sciences.
Ok, thanks for the elaboration. I got that.

An alternative is to have a system with infinitesimals. But still, ordinarily, we need to define <, + and * and to prove whatever theorems are needed for the machinery of mathematics.

To just wave a hand and say "Voila, this is my infinitesimal" does not provide the needed definitions of < + and * with infinitesimals nor the needed proofs.

So how do we go about proving the existence of a system with infinitesimals? For your purposes, it would help to first define 'is an infinitesimal'. I provided a definition previously, but I notice that many authors include 0 as an infinitesimal. So perhaps use this definition:

x is an infinitesimal if and only if, for every positive real number y, |x| < y.

It has been proven for you that for every real number x there is a positive real number y such that y < |x|.

So no non-zero real number is an infinitesimal.

One more time: No non-zero real number is an infinitesimal. The proof that no non-zero real number is an infinitesimal is immediate from the fact that for every real number x there is a positive real number y such that y < |x|. We don't need to keep going over this over and over.
I looked at all your posts and didn't find the proof that no non-zero real number is an infinitesimal. Could you please provide the proof?

The sequence of half distances converges to 0. So what? That doesn't prove that it's not the case that between any two different real numbers there is another different real number.

Between any two distinct real numbers, there is always another one strictly between them.
This was a reply to the above comment from @fishfry who claimed between any two distinct real numbers, there is always another one strictly between. The distance between two points is zero if the number of divisions is strictly infinite so there cannot be a point between two points in this case.
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