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. — MoK

Question: taking one - the number - as a point on the number line, do you mean that there are more than one ones that are not one on the number line that are at a distance of zero from the point one - that are not one but some other number called one and understood to be one - or not one? Can you please make sense of this in terms a high-school mathematician can understand?

Perhaps this post help you. What I showed in that post is that the distance between consecutive means tends to zero for large $i$. The last step of my argument seems to be problematic because of the way the division is defined between cardinal numbers. You don't need to worry about the division of the cardinal numbers though since I don't need that step. All I need is to show that the distance between consecutive means is $d_i=\frac{(b-a)}{2^i}$. This distance is zero when $i$ is strictly infinite. I am currently trying to make sense of this post though and I have to say it is very technical.

Perhaps this post help you. — MoK

Um, no. Especially when you qualify it yourself as being hard for you to make sense of and very technical. Perhaps re-read my question - which I imagine you understand - and try answering in the same spirit and language.

That is, two points on the number line, zero distance between them: how can they not be the same point, or number, and if not the same, how can there be zero distance between them?

That is, two points on the number line, zero distance between them: how can they not be the same point, or number, and if not the same, how can there be zero distance between them? — tim wood

You are correct in your observation. If the distance between two points is zero then we are dealing with the same point. This means that there cannot be a point between points since they are the same point. This is against the argument that there is always a point between two points on the real number. What I showed is that the distance between consecutive "means" (by "mean" I mean the point between two points) tends to zero. I hope things are clear now. If not please let me know so I would elaborate.

tends to zero. — MoK

Sure, gets smaller. Which in the very expression of which says that there is again a smaller - always. I suppose you can introduce a rule or limit that says at that limit the distance is zero, but then the point is identical with itself. I don't see how you get out of or around this.
t

I looked at all your posts and didn't find the proof that no non-zero real number is an infinitesimal. — MoK

Clearly, you're not reading.

I gave you a proof in the very post to which you have now replied:

The proof that no non-zero real number is an infinitesimal is immediate from the fact that for every [non-negative] real number x there is a positive real number y such that y < |x|. — TonesInDeepFreeze

You need it spelled out for you again?

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

Theorem: No non-negative real number is an infinitesimal.
Proof: Suppose x is a non-negative real number. Since x is a non-negative real number, |x|/2 is a positive real number and |x|/2| < |x|. So it is not the case that |x| < |x|/2|. So it is not the case that for every positive real number y, |x| < y. So x is not an infinitesimal. So no non-negative real number is an infinitesimal.

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. — MoK

You skip that I addressed that. There is no operation of infinite division in the real number system. There are infinite sequences of divided results, and the one you have in mind converges to 0. It is very crank (confused, ignorant and fallacious) to conflate the limit of a sequence with an out-of-thin-air claim of an operation of infinite division. And the fact that the limit of the sequence is 0 does not refute that between any two distinct real numbers there is real number strictly between them. Indeed, the convergence to 0 itself depends on the fact that between any two distinct real numbers there is a real number strictly between them.

No recognition that you were just blowing smoke when you referenced a source that you didn't even read to see that it says the OPPOSITE of your claim?:

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. — TonesInDeepFreeze

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. — MoK

This is incoherent at best, wrong at worst. I explained this to you at length. But look. You are trying to prove there are two points without a third between them, by claiming there are two points at a distance of zero. Can you see the circularity of your argument?

I explained this to you at length in a post you didn't bother to engage with.

I am a retired physicist — MoK

I believe you. Physicists attempting to do math are often a source of humor and/or horror to mathematicians. But I'm sure you know that :-)

Honestly. I explained this to you at length. What you've worked out for yourself is the equivalence of the two definitions of a dense set: (1) That there is third point between any two; and (2) That there is a sequence of distinct points approaching any given point as a limit. You proved that with your bisection idea. Your idea is essentially correct. Your intuition about what it means is mathematically wrong.

It is very crank (confused, ignorant and fallacious) to conflate the limit of a sequence with an out-of-thin-air claim of an operation of infinite division. — TonesInDeepFreeze

What Tones said.

I am a retired physicist and my knowledge of mathematics is very rusty due to my age — MoK

Don't feel bad. I"m a very old retired mathematician and have had to look up filter trying to understand @sime 's comments.

So if no real number is an infinitesimal, numbers are then what is relation to geometry. Is 2 then 2 points, or are all numbers a point?

According to Wki both Cauchy (in Cours d'Analyse) and Edwars Nelson also compared infinite points to the numberline. Long before hyperreals i believe. The great writer and philosopher George Berkeley rejected infinitesimals on both mathematical and philosophical grounds

What about imaginary numbers, however? Stephen Hawking, in his attempt at find the wave function of the universe, proposed his (yep) No Boundary Proposal in 1983. I like to apply this "theorem" to consciousness. Hawking uses imaginary numbers to describe time as it goes backwards, behind the Big Bang. How are we to understand mathematically a state not having any boundaries? There is always a "here" and "there" in our experience. That is, except in consciousness wherein we can go deeper and deeper and we find no edge. The "limit" seems to be death, but in our experience we are infinite. Hence we can think about infinities..

Note: if the world is a hologram, then it is proven there is a "thing-in-itself".

The kalam cosmological argument gives a great example of infinities embedded in another. The argument fails in its purpose because eternity, an infinity, contains all steps of infinity. There can be that infinity if there is the eternity. QED?

Just some philosophy and context for this forum

If i cut a cake horizontally starting from the halfway point upwards with each slice being half the size of the one immediately below, what would the top of the cake look like? Isn't it indefinite? But you can definitely look at the cake, from all angles, and see that it has definite position in relation to its parts. So how do we reconcile the indefinite with the definite? I think this is what must be asked about the continuum. Hawking would say that four dimensional Euclidean space, with a time dimension that both 1) acts as space, and 2) is described by imaginary numbers, gives an answer to this question. That is to say, the universe as a whole gives the answer to the continuum. But how do imaginary numbers relate to geometry?

Sure, gets smaller. Which in the very expression of which says that there is again a smaller - always. I suppose you can introduce a rule or limit that says at that limit the distance is zero, but then the point is identical with itself. I don't see how you get out of or around this. — tim wood

I consider the number of the operation (by operation I mean dividing by two) to be strictly infinite. @TonesInDeepFreeze however claims that such an operation does not exist in the real number system: "There is no operation of infinite division in the real number system. There are infinite sequences of divided results, and the one you have in mind converges to 0."

I am however confused by this statement. How could you have an infinite sequence of divided results without infinite division!?

I have already qualified myself as a high-school "mathematician" - that being why I try to make sense in English. Let's try this: it seems to me you are confusing ideas of number with limit. For example, the decimal expansion of the square root of two goes on forever, is infinite, so what is the last digit? You want to divide something a "strictly" infinite number of times: great, how many times is that?

You need it spelled out for you again?

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

Theorem: No non-negative real number is an infinitesimal.
Proof: Suppose x is a non-negative real number. Since x is a non-negative real number, |x|/2 is a positive real number and |x|/2| < |x|. So it is not the case that |x| < |x|/2|. So it is not the case that for every positive real number y, |x| < y. So x is not an infinitesimal. So no non-negative real number is an infinitesimal. — TonesInDeepFreeze

You didn't provide this argument before. Did you? You just defined infinitesimal!

You skip that I addressed that. There is no operation of infinite division in the real number system. There are infinite sequences of divided results, and the one you have in mind converges to 0. — TonesInDeepFreeze

How could you have an infinite sequence of divided results without infinite division operations?

It is very crank (confused, ignorant and fallacious) to conflate the limit of a sequence with an out-of-thin-air claim of an operation of infinite division. — TonesInDeepFreeze

I was not confusing these. In fact, I mentioned the number of division operations to be strictly infinite.

And the fact that the limit of the sequence is 0 does not refute that between any two distinct real numbers there is real number strictly between them. Indeed, the convergence to 0 itself depends on the fact that between any two distinct real numbers there is a real number strictly between them. — TonesInDeepFreeze

That is understandable but I was not arguing against that. I just argued that if the number of divisions is strictly infinite then you cannot get anything new by dividing the result further since the result is zero. I now know that the number of division operations cannot be infinite in the real number system. I don't know why!

This is incoherent at best, wrong at worst. I explained this to you at length. But look. You are trying to prove there are two points without a third between them, by claiming there are two points at a distance of zero. Can you see the circularity of your argument?

I explained this to you at length in a post you didn't bother to engage with. — fishfry

It seems that there is no operation of infinite division in the real number system. That was something I didn't know.

I believe you. Physicists attempting to do math are often a source of humor and/or horror to mathematicians. But I'm sure you know that :-) — fishfry

Oh yeah, I can guess that. We, physicists, work with the infinities all the time. Of course, mathematicians do not agree with how we deal with infinities but strangely physics works. :)

What Tones said. — fishfry

I am not confusing the two.

Don't feel bad. I"m a very old retired mathematician and have had to look up filter trying to understand sime 's comments. — jgill

I am not feeling bad. At worst I am wrong and learn a new thing. At best I am right so others learn a new thing. Thank you for your support anyway. :)

So if no real number is an infinitesimal, numbers are then what is relation to geometry. Is 2 then 2 points, or are all numbers a point? — Gregory

Yes, there is no infinitesimal in the real number system. I don't understand the rest of your comment.

According to Wki both Cauchy (in Cours d'Analyse) and Edwars Nelson also compared infinite points to the numberline. Long before hyperreals i believe. The great writer and philosopher George Berkeley rejected infinitesimals on both mathematical and philosophical grounds — Gregory

I didn't read Berkeley at all so I don't know what he is arguing about. Is he arguing that there is no infinitesimal in the real number system or he is arguing that there is no infinitesimal in any mathematical system? According to @TonesInDeepFreeze there are mathematical systems with infinitesimal.

What about imaginary numbers, however? Stephen Hawking, in his attempt at find the wave function of the universe, proposed his (yep) No Boundary Proposal in 1983. I like to apply this "theorem" to consciousness. Hawking uses imaginary numbers to describe time as it goes backwards, behind the Big Bang. How are we to understand mathematically a state not having any boundaries? There is always a "here" and "there" in our experience. That is, except in consciousness wherein we can go deeper and deeper and we find no edge. The "limit" seems to be death, but in our experience we are infinite. Hence we can think about infinities.. — Gregory

I don't understand what you are trying to argue here.

The kalam cosmological argument gives a great example of infinities embedded in another. The argument fails in its purpose because eternity, an infinity, contains all steps of infinity. There can be that infinity if there is the eternity. QED? — Gregory

The Kalam cosmological argument states that there cannot be an infinite number of past events therefore there is a beginning. The rest of the argument is about proof of the existence of God which I don't agree with.

If i cut a cake horizontally starting from the halfway point upwards with each slice being half the size of the one immediately below, what would the top of the cake look like? — Gregory

The top of the cake looks like the top of the cake no matter how many times you divide the cake horizontally.

Isn't it indefinite? — Gregory

It is not.

But you can definitely look at the cake, from all angles, and see that it has definite position in relation to its parts. So how do we reconcile the indefinite with the definite? — Gregory

I don't understand what you are talking about.

I think this is what must be asked about the continuum. Hawking would say that four dimensional Euclidean space, with a time dimension that both 1) acts as space, and 2) is described by imaginary numbers, gives an answer to this question. That is to say, the universe as a whole gives the answer to the continuum. But how do imaginary numbers relate to geometry? — Gregory

We are not talking about space and time here. Whether space and time are continuous or not is the subject of other threads.

I have already qualified myself as a high-school "mathematician" - that being why I try to make sense in English. — tim wood

Ok, I try my best to answer your questions. Perhaps, others (mathematicians @TonesInDeepFreeze, @fishery, and @jgill) would participate and answer your questions in a simple manner.

Let's try this: it seems to me you are confusing ideas of number with limit. — tim wood

No, I am not confusing the ideas of the number and the limit.

For example, the decimal expansion of the square root of two goes on forever, is infinite, so what is the last digit? — tim wood

There is no last digit. The square root of two is an irrational number. The Irrational set, the set of all irrational numbers, is a subset of the real number set. Almost most of the real numbers do not have the last digit (the number of digits is infinite).

You want to divide something a "strictly" infinite number of times: great, how many times is that? — tim wood

Bigger than any countable number.

There is no last digit.... Bigger than any countable number. — MoK

Just so, so when you talk about something that "strictly" can be, but which cannot itself be, then you tell me what the sense is.

I dont think that a real number can't be divided infinitely. The area of a circle is pi-r-squared wherein pi represents an aspect of space (the area). Each decimal would be a tiny and tinier slice of space and this goes on forever. So the space represents the number as we visualize it and the the number represents the space. Infinity is in both.

As for Hawking, physical explanations shed light on mathematical concepts just as the reverse is true.

"But nobody in that century or the next could adequately explain what an infinitesimal was. Newton had called them 'evanescent divisible quantities,' whatever that meant. Leibniz called them 'vanishingly small,' but that was just as vague.. Pierre Bayle's 1696 article on Zeno drew the skeptical conclusion that, for the reasons given by Zeno, the concept of space is contradictory."
Internet Encyclopedia of Philosophy

If there are no infinitesimals, than an infinity of zeros can equal anything. Does this mean that 0×infinity=everything? But an argument against infinitesimals and discreteness is that space by definition is that which is divisible. How can there be something in-between space and a point? Where do we even begin with a continuum? (At least Banach-Tarski's paradox makes more sense in this context) Geometric objects seem to be in themselves the opposite of Gabriel's horn. Instead of an infinite surface area for a finite volume we seem to have in the continuum an infinity of space bounded by finite (beginning and end) space

More latter..

Just so, so when you talk about something that "strictly" can be, but which cannot itself be, then you tell me what the sense is. — tim wood

I don't understand your objection to my comments. I used "strictly" when I wanted to see what is the result of an interval divided infinite number of times by two. Mathematicians think that there is no operation of infinite division in the real number system. I don't know why and I asked @TonesInDeepFreeze for an explanation. Perhaps he can answer my question in a simple term so you can understand as well.

what is the result of an interval divided infinite number of times — MoK

That's what I understand as a limit (process) - that can be approached but never reached, but if it could be reached, would yield whatever - the process of whatevering being nothing in itself, but useful as a tool.

How could you have an infinite sequence of divided results without infinite division!? — MoK

An operation symbol takes only finitely many arguments.

For example, the operation of division (x/y) takes only two arguments (x and y), and is defined accordingly:

Df. If y not= 0, then x/y = the unique z such z*y = x.

Every operation symbol, whether, primitive or defined, takes only finitely many arguments. The reason is that every formula is finite in length, so every term is finite in length, so no operation symbol can take infinitely many arguments.

So we have division, which is a binary operation. But we also prove that for every positive real number r, there exists a function f whose domain is the set of positive natural numbers and such that, for every positive natural number n:

f(1) = r
f(n+1) = f(n)/2

f is a function from the set of positive natural numbers into the set of real numbers.

Notice that it is trivial to prove that for no n is it the case that f(n) = 0.

Then we prove:

There exists a unique real number x such that for every positive real number y, there exists a positive natural number n such that |f(n) - x| < y.

Then we write:

lim[n = 1 to inf] f(n)

'lim' is a variable binding operator, but it can be reduced to a regular operation symbol:

Df. If g is a function from the set of positive natural numbers into the set of real numbers, and there exists a unique real number x such that for every positive real number y, there exists a positive natural number n such that |g(n) - x| < y, then Lg = the unique real number x such that for every positive real number y, there exists a positive natural number n such that |g(n) - x| < y

You see that 'L' is an operation symbol that takes only finitely many arguments - in this case, one argument.

The argument itself is an infinite set (an infinite sequence in this case), which is okay, because the operation symbol takes only finitely many arguments - in this case, one argument.

And we prove, regarding the function f we previously defined:

Lf = 0

In everyday parlance, "the limit of f is 0" and for no n is it the case that f(n) = 0.

There's no operation of infinite division (no operation takes infinitely many arguments), but there are infinite sequences that converge to 0, defined by, for example, halving each previous entry in the sequence. But if we call taking the limit of such a sequence f "infinite division" then that's okay as long as we are clear that that is what we mean and not some other undefined notion and that we recognize that for no n is it the case that f(n) = 0.

Moreover, the point is sustained that the fact that f converges to 0 does not refute that no non-zero real number is an infinitesimal.

You didn't provide this argument before. Did you? You just defined infinitesimal! — MoK

That is blatantly false and with an exclamation mark that is a cherry on top of falsehood. I wrote:

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. — TonesInDeepFreeze

This latest time I just made that proof even more explicit. That proof essentially had been given you at least a few times by other posters, but you still didn't understand (chose not to understand?). So I gave it again for you. And then you still skipped it and asked me to give a proof even though I already had. And so I gave it to you even more explicitly. And that explicitness is itself more than you would need, as the matter is so extremely simple to begin with:

No non-zero real is an infinitesimal, since, unlike an infinitesimal, every non-zero real is such that there is real between it and 0.

/

And still curious whether you understand now that "aleph_1/(2^aleph_1)" is nonsense.

According to TonesInDeepFreeze there are mathematical systems with infinitesimal. — MoK

Don't take my word for it. Look up 'non-standard analysis', 'hyperreal', 'infinitesimal', 'internal set theory'.

Better, read Enderton's beautifully written 'A Mathematical Introduction To Logic' in which he has a wonderfully clear and concise section on non-standard analysis. (It's been a while since I studied it, so I might not be able to immediately answer all questions about the details. Anyway, to get up to the section on non-standard analysis, one needs to first comprehend the material leading up to, starting from the first page, which is the best favor anyone could do for oneself if one were sincerely interested in topics such as this one.)

Bigger than any countable number. — MoK

Greater than any countable number or greater than any finite number?

Better, read Enderton's beautifully written 'A Mathematical Introduction To Logic' — TonesInDeepFreeze

PDF here:
https://sistemas.fciencias.unam.mx/~lokylog/images/Notas/la_aldea_de_la_logica/Libros_notas_varios/L_03_ENDERTON_A%20Mathematical%20Introduction%20to%20Logic,%20Second%202Ed.pdf

Looks like a mighty good book. Alas, no cheap used copies available.

I have the first edition in print. I don't have an errata sheet for it. But I do have pencil marks for the errata I caught.

There is an errata sheet for the second edition; I think it might be online too.

Anyway, the second edition online is gold, free for the taking.

The course I recommend is, in order:

(1) Logic: Techniques of Formal Reasoning 2nd ed. - Kalish, Montague and Mar
or either of these:
Introduction To Logic - Suppes
Elementary Logic - Mates

(Lately I've been thinking that Suppes is the best choice, especially for its treatment of the subject of definitions.)

(2) Elements of Set Theory - Enderton (the errata sheet might be online)
or
Axiomatic Set Theory - Suppes

(3) A Mathematical Introduction to Logic - Enderton

(4) Introduction to Mathematical Logic (just for the Introduction chapter) - Church (the Introduction chapter is the best overview of the primary considerations I've found)

For a book with both classical and intuitionistic logic:

Logic and Structure - van Dalen

For an overview of many alternative logics:

An Introduction to Non-Classical Logic - Priest

/

And there are so many other great books, especially Smullyan's books 'First-Order Logic' and 'Godel's Incompleteness Theorems' (Smullyan writes so beautifully and his formulations are so clever and elegant).

Two great tomes:

Mathematical Logic - Monk

Fundamentals of Mathematical Logic - Hinman

I mean, they really are tomes. And they are remarkably rigorous with notation and extensive with details as you're likely to find, especially Hinman which goes the whole nine yards in the way it makes explicit which symbols are in the object language and which are in the meta-language.

And more advanced books:

Model Theory - Chang & Keisler (a tome and the OG of model theory textbooks)

Set Theory - Jech (a tome)

Set Theory - Kunen (I like Kunen's "philosophical/heuristic" framework that tends toward "formalism")

These are all beauties.

So we have division, which is a binary operation. But we also prove that for every positive real number r, there exists a function f whose domain is the set of positive natural numbers and such that, for every positive natural number n:

f(1) = r
f(n+1) = f(n)/2

f is a function from the set of positive natural numbers into the set of real numbers. — TonesInDeepFreeze

Is f(infinity) a member of the above sequence? If yes, what is its value? If not, how could the sequence be an infinite one?

Notice that it is trivial to prove that for no n is it the case that f(n) = 0. — TonesInDeepFreeze

Let's wait for your answer to the previous questions.