the set of numbers between 1 and 2, is a smaller set of infinite numbers between 1 and 3 — Vaskane

I previously corrected that:

The set of real numbers between 1 and 2 has the same cardinality as the set of real numbers between 1 and 3.

The set of rational numbers between 1 and 2 has the same cardinality as the set of rational numbers between 1 and 3.

Even still infinity as a noun is relevant to mathematics in numerous ways like [...] — Vaskane

Those other ways haven't been the context in which 'infinity' is used as a noun here. The gravamen of the original poster has been that there are not different "sizes of infinity". The notion that there are infinite sets with different cardinalities is a set theoretic notion in which context it is crucial not to speak as if there is an object named 'infinity'.

It makes no difference. — Philosopher19

It makes a real difference. By saying 'infinity' as a noun and then that there are different sizes of infinity is to picture an object that has different sizes. There is no such object in mathematics.

You keep saying I ignore your points, but rightly or wrongly, I also think you have not read or considered what I've written with sufficient attention to detail. — Philosopher19

I've answered that. I have read and re-read and thought about what you've posted. What you have posted is in ignorance of the mathematics you criticize, mixed up, and dogmatic. You commit a non sequitur by inferring from the fact that I have corrected you on certain crucial points that I haven't read and considered what you posted.

And even if I had not read, re-read and thought about what you've posted, it would not change that you have continued to ignore the information given you. Please stop saying 'infinity' as you do in context of the mathematics you're criticize. The unthinking and habitual use of 'infinity' in that context both reflects a misunderstanding of that which you criticize and contributes to even more misunderstanding of it.

I do think that I am being sincere and honest in this discussion (as well as not closed-minded). — Philosopher19

Good faith in posting a critique of mathematics would entail at least knowing something about it.

If you understand that our brains are churning out stand alone theories that work fine in a certain context but don't all work together in every context [...] — Mark Nyquist

Set theory is not standalone in the sense that it has no application outside itself.

I am very well aware that that set theory does not account for all contexts of human knowledge. I have posted about that previously. And it is a point that is obvious most especially even to those who work with set theory.

I don't mean to use Existence loosely/abstractly. By "Existence" I mean that which encompasses all things physical or otherwise (if otherwise is possible). — Philosopher19

OK.

If the universe is expanding, it is expanding in something. — Philosopher19

I guess this "something" is "space", right? Like a balloon ...
But this seems impossible since space is part of the universe itself; it cannot be larger than it. E.g. like the space around a balloon that is inflated ...

So to me, Infinity/Existence is the reason that something can expand forever or go on forever. As for the thing that expands (like the universe), it is a part of the Infinite. It is not itself infinite. — Philosopher19

OK.

As I said, astronomers and cosmologists are more suitable for answering these questions ...

On the basis of mathematics it's quite easy to detail larger and smaller sets of infinite numbers, like the set of numbers between 1 and 2, is a smaller set of infinite numbers between 1 and 3. [emphasis added] — Vaskane

In ordinary mathematics the idea of size is formalized as cardinality.

I've said that one may propose whatever other concept or alternative mathematics one wishes to propose. But when you say "on the basis of mathematics" that would ordinarily be understood not to be some unspecified personal alternative you have but rather ordinary mathematics.

By Being. Existence just Is. It just is the case that triangles are triangular or that Existence is Infinite or that 1 plus 1 = 2 — Philosopher19

So for you, the nature of existence accounts for the awareness that Euclidian triangles sum up 180º degrees. That is what you said in your two comments.
You have not specific what the nature of existence is besides a brute fact. If so, how do brute facts account for our knowledge of something that is not verified empirically — Euclidian triangles?
You said here:

The angles in a true triangle add up to 180 degrees because that is the nature of Existence

But you did not say how.

Cardinality and Size aren't the same. (..) The interval between 1 and 3 is 2 and thus larger than the interval between 1 and 2 which is only 1.

Can you provide a formal criterion for what constitutes the size of an infinite set, beyond its cardinality?

When taking about intervals in ℝ, the cardinalities of the [1,2] and the [1,3] intervals are exactly the same, namely, the cardinality of the continuum. The reason is that there are bijective functions linking the two. Take, for instance, a function f : [1,3] → [1,2] with the rule f(x) = (x+1) / 2.

Hey, to you I'm just words on a screen, but I'm an actual person. Sorry if I misunderstood you, there are a lot of comments in this thread and I'm not up to speed with the whole context.

It's fine, I can see you're a nice guy.

it's quite easy to detail larger and smaller sets of infinite numbers, like the set of numbers between 1 and 2, is a smaller set of infinite numbers between 1 and 3. — Vaskane

Unfortunately for you the interval between 1 and 3 is 2 and thus larger than the interval between 1 and 2 which is only 1, and thus quite simple to show that having two intervals of infinity is twice the size of one interval of infinity. — Vaskane

Unfortunately for you, you have conflated distance of an interval with the size of the infinite set of numbers in that interval.

The distance between 1 and 2 is smaller than the distance between 1 and 3.

But the cardinality of the infinite set of numbers in those intervals is the same.

First, you took exception to me taking 'size' as cardinality, as you said that cardinality is not the only sense of 'size'. So I pointed out to you that you yourself said the context was mathematics. Then you didn't recognize that but instead ...

Second, you conflated distance of an interval with the size of the infinite set of numbers in that interval.

It stands that "the set of numbers between 1 and 2, is a smaller set of infinite numbers between 1 and 3" deserved being corrected. Or, sure, shoot the messenger if you prefer.

Consider it a contextual error of considering area as size? — Vaskane

First, incorrect objection to cardinality.

Second, conflating distance with cardinality.

Third, citing area when there is no area involved.

It's a line. Area is not involved.

Your claim "the set of numbers between 1 and 2, is a smaller set of infinite numbers between 1 and 3" is plainly wrong and deserves being corrected by whatever "dude" extends the favor of correcting it.

Funny thread.

https://jlmartin.ku.edu/courses/math410-S09/cantor.pdf

Doesn't infinity mean endless? i.e. unreachable eternal continuation in concept? — Corvus

I think Infinity is why something can go on forever. But if something goes on forever (or keeps going without end) it will not become infinite (just as if I keep counting without end, I will not reach Infinity)

To me, the only thing that is Infinite, is Existence. And Existence has always Existed and will always Exist (so It has no beginning and no end whilst all ends and beginning are within It. And if something goes on forever within It like a number sequence or a forever expanding universe, then that thing will never reach Infinity/Infiniteness

If it was reachable, then it wouldn't be infinity. Any set or size would be unknowable, if it were infinity. Therefore talking about different size, set or number of infinity, is it not a nonsense? — Corvus

I think it's nonsense to say Infinity comes in various sizes. But the semantic of Infinity itself is not nonsense because it is clearly meaningful. As for sets, the only thing that can be the set of all cardinalities or houses or any other meaningful/imaginable/understandable thing, is Existence/Infinity. Since Existence is Infinity, it allows for there to be no end to the number of numbers possible (because you can always add one and this can go on forever without Infinity being reached or exhausted).

That's a matter of perspective really, I was a Navy Cryptologist, that you only view numbers as a row and now both rows and columns, again revolves back to your objective perspective. — Vaskane

I said nothing about rows and columns.

First, you incorrectly objected to taking size as cardinality when you said yourself that the context is mathematics.

Second, you conflated distance with cardinality.

Third, you claimed it's about area, though area is not involved.

Fourth, you were a Navy cryptologist and something about me regarding rows and columns, though I said nothing about rows and columns.

Will there be a fifth attempt to evade the plain fact below?:

"the set of numbers between 1 and 2, is a smaller set of infinite numbers between 1 and 3" is incorrect.

I would think it would be much less wear and tear on your credibility to instead just think about the correction given you and then recognize it.

I think it's nonsense to say Infinity comes in various sizes. — Philosopher19

He did it again! He still persists in mischaracterizing mathematics as claiming that there is an "Infinity" [capitalized, no less] that has different sizes. That is after the mistake of that has been pointed out at least a dozen times, and as he protests that he is posting in good faith.

Then we have people saying that there are contexts other than set theory so one should be tolerant not to demand that set theory is the only context we may consider. Quite so. But the context of the original poster is not just a proposal for another concept but a claim that the mathematical set theoretic concept "is nonsense". And that is intolerance. To know nothing about the mathematics behind the concept of cardinality but instead just call it nonsense. That is egregious intolerance.

I guess this "something" is "space", right? Like a balloon ...
But this seems impossible since space is part of the universe itself; it cannot be larger than it. E.g. like the space around a balloon that is inflated ... — Alkis Piskas

I would just say if the universe is expanding, then it is expanding in Existence (as opposed to 'space-like-the-space-in-our-universe')

If scientists have in fact measured the size of our universe, then our universe is finite (meaning that our universe is not all there is to Existence. I hear there's been more talk of parallel universes lately).

We know by way of pure reason that the universe cannot be expanding in non-Existence. Such a thing is not conceivable at all, therefore, it is not observable at all. And this is not an unknown like a 10th sense which some superior being may have that we can't comprehend. This is a clear case of something contradictory (like a round square) that no being would make sense of because it is a known contradiction as opposed to an unknown.

Do you agree that lines have length? — Vaskane

Whether you intend it or not, that is a trollish question.

Of course, lines have length. That is implied by my talking about distance.

I already explained to you that there is a difference between the distance of an interval and the cardinality of the infinite set of numbers in the interval. But instead of recognizing that information, you hit me with a mindlessly posed quiz.

So I suppose I need to lay it out for you in even greater detail:

The real line is the set of numbers ordered by the usual less-than relation on the set of real numbers.

The intervals [1 2] and [1 3] are segments of that real line.

For any pair of numbers there is the distance between them, which is the absolute value of their difference. The distance between 1 and 2 is 1, and the distance between 1 and 3 is 2.

But the intervals are a set of numbers, not just their max and min. The interval [1 2] is the set of real numbers that are greater than or equal to 1 and less than or equal to 2. The interval [1 3] is the set of real numbers that are greater than or equal to 1 and less than or equal to 3. Both of those sets have the same cardinality, i.e. both those intervals have the same cardinality.

Your non sequitur is to infer that intervals having different distances implies that they have different sizes. (And I say that because you still have not said that you didn't mean what you said when you said the context here is mathematics.)

Since this keeps getting lost with you, for the third time:

Intervals may have different distances (lengths, if you prefer) but the same cardinality.

He still persists in mischaracterizing mathematics as claiming that there is an "Infinity" [capitalized, no less] that has different sizes — TonesInDeepFreeze

To my understanding, mainstream maths claims:
There are infinites of various sizes (or at least infinite sets of various sizes, but that amounts to the same thing)
The set of all sets is contradictory

Is my understanding wrong?

That is egregious intolerance — TonesInDeepFreeze

If someone came to me and said I've seen a triangular square, I would say to them that that's nonsense to me and that it is impossible for them to have seen such a thing. It's in the semantic of square that it can't be triangular. I would not call this intolerance, but perhaps I could be more tolerant by trying to understand the person better. Perhaps what they really mean is that they saw some shape, that the best way that they could label it was "triangular square". Maybe they saw some kind of trapezium and did not know the label/word for the semantic of trapezium.

Similarly, if someone came to me and said they have demonstrated how infinity comes in various sizes, I would say to them that that's nonsense to me. It's in the semantic of Infinity that It does not come in various sizes.

It makes a real difference. By saying 'infinity' as a noun and then that there are different sizes of infinity is to picture an object that has different sizes. There is no such object in mathematics. — TonesInDeepFreeze

I don't think I'm picturing an object. I think I'm just focused on the semantic of Infinity.

Good faith in posting a critique of mathematics would entail at least knowing something about it. — TonesInDeepFreeze

I think it is from all that I have seen and heard that I said the following:

But from what I've seen of mathematicians, they either have no part for infinity, or they're using infinity wrong. I believe they're doing the latter which leads to the former (which I think is why I have heard it said before that "maths is incomplete") — Philosopher19

Whether all that I have seen or heard is enough, is another matter. You don't think I have. I think I have.

If I've understood him right, Cantor treats a number sequence that goes on forever as being infinite. But something going on forever does not make it infinite (if my counting to infinity goes on forever, that neither makes my counting infinite, nor does it mean I will eventually reach infinity). It also makes no sense to say something like "assume that your counting to infinity is completed such that you have counted the set of all natural numbers and have successfully proven that there are an infinite number of natural numbers" and then label this as {1,2,3,4,...}

If Cantor did not do this, he would not then be forced to conclude as he did with his diagonal argument. Now I feel the following is relevant:

But from what I've seen of mathematicians, they either have no part for infinity, or they're using infinity wrongly. I believe they're doing the latter which leads to the former (which I think is why I have heard it said before that "maths is incomplete") — Philosopher19

To my understanding, mainstream maths claims:
There are infinites of various sizes (or at least infinite sets of various sizes, but that amounts to the same thing)
The set of all sets is contradictory — Philosopher19

(1)

There is no object called 'Infinity' in the sense you have been using it.

Here is a way to say what you want to say:

In mathematics, there are sets that are infinite but that have different cardinality from one another.

Better yet:

If x is infinite then there is a y that is infinite and y has greater cardinality than x.

(2)

The statement that there exists a set z such that every set is a member of z is inconsistent with the axioms of set theory; and set theory proves the negation of the statement that there is a set z such that very set is a member of z.

Better yet:

'There exists a z such that for all y, y is a member of z' contradicts this instance of the axiom schema of separation: For all z, there is a x such that for all y, y is a member of x iff (y is a member of z and yis not a member of y).

And we prove that easily:

First, we prove:

There is no x such that for all y, y is a member of x iff y is not a member of y. Proof:

Suppose, toward a contradiction, that there is a such and x. Then x is a member of x and x is not a member of x. Contradiction. So there is no such that x for all y, y is a member of x iff y is not a member of y.

Next, Suppose, toward a contradiction, that there is a z such that for all y, y is a member of z.

Then from separation we have an x whose members are all and only those y that are both a member of z and not a member of y.

But since every y is a member of z, we have x whose members are all only those y that are not a member of y. That contradicts that there is no x whose members are all only those y that are not a member of y. So there is no z such that for all y, y is a member of z.

if someone came to me and said they have demonstrated how infinity comes in various sizes, I would say to them that that's nonsense to me. — Philosopher19

And it is fine that the concept is nonsense to you. That is not at issue. But that something strikes you as nonsense doesn't thereby render it nonsense.

You don't know anything about the mathematics, so you are in no position to try to convince other people that it is nonsense. The only thing you could fairly say is, "I don't know anything about the mathematics, so I can't fairly opine on it, but my own concept of infinity does not allow that there are sets of different sizes of infinity".

But then you keep harping on this "semantic" issue you have. If I hadn't said in this thread, I'll say what I've said in other threads:

(1) Words have different meanings and senses in different areas of study. In biology the word 'cell' means one thing and in criminology it means another. Even when the senses are closely related and in closely related or even overlapping fields, the senses can be different. When mathematicians talk about 'infinite' they don't thereby declare that that mathematical sense applies to all other areas including philosophy, cosmology or theology. However, yes, the mathematical sense may be applied in other areas, but it's up to the author to make clear what sense they mean. At the very least, mathematicians ought to be allowed to talk about 'infinite' in the mathematical sense without someone who knows virtually nothing about the mathematics arrogating to convince people that it is "nonsense" and even to claim to "prove" that it is nonsense with unstated standards of "proof" quite different from mathematical proof.

(2) Even if you don't have the least bit of reasonability to grant (1), then we could say, "Fine, from now on consider every book and article and post in mathematics as if the word 'infinite' were replaced with 'zinfinite'. The formal mathematics would be just the same except the word you arrogate to your own meaning would not be used and you could rest easy that mathematics does not impinge on you own concepts.

(3) Again, formal contradiction is any statement P asserted along with the assertion of the negation of P. There is no known contradiction in set theory. A DIFFERENT matter is that in certain senses set theory is not compatible with your own philosophy, and again that a certain word is used in mathematics differently than you use it.

Cantor’s Theorem and the Unending Hierarchy of Infinities: still hasn't been disproven;
"For instance, by iteratively taking the power set of an infinite set and applying Cantor's theorem,we obtain an endless hierarchy of infinite cardinals, each strictly larger than the one before it. Consequently, the theorem implies that there is no largest cardinal number (colloquially, "there's no largest infinity")." — Vaskane

Yes, it's the colloquial part that is so often abused by people who know virtually nothing about the subject. Especially among beginners in the subject, if we refrain from that unfortunate usage, then (1) We avoid having set theory look ridiculous as if it claims that there is an object that has different cardinalities. (2) We adhere to the way the actual mathematics is couched, which is that is the property 'is infinite' but not an object that is 'infinity'.

So those who post uninformed, intellectually bigoted and confused lashing out against set theory would not have the slippery wedge of presenting mathematics as if it is itself absurd. People who are unfamiliar with the mathematics ordinarily don't think of infinitude the way mathematicians do. In everyday life and conversation, and even academically in certain contexts, it would strike as extremely odd to hear that there are "greater and greater infinities". But if it is said instead, "In mathematics, 'finite' and 'infinite' are properties of sets, and there are infinite sets, and there are greater and greater sizes, such as there are more real numbers than natural numbers" then it may strike one as much more sensible and mathematics is not made to look ridiculous as if it has an object that is infinity but that it comes in different sizes.

By the way, Cantor did not work axiomatically. The results of set theory are on much firmer ground now as we work axiomatically now.

Cantor treats a number sequence that goes on forever as being infinite. But something going on forever does not make it infinite (if my counting to infinity goes on forever, that neither makes my counting infinite, nor does it mean I will eventually reach infinity). It also makes no sense to say something like "assume that your counting to infinity is completed such that you have counted the set of all natural numbers and have successfully proven that there are an infinite number of natural numbers" and then label this as {1,2,3,4,...} — Philosopher19

I answered that exactly already.

You truly are not in good faith.

You make claims about a subject of which you are ignorant. Then when it is explained to you exactly what your confusion is, you ignore that explanation and instead just go on to make the confused claim again.

So I'll give you the explanation yet again so you can ignore it again:

We do NOT claim that from "after each natural number there is a next number" and "there is no greatest natural number" that we can infer that there is a set of all the natural numbers. Indeed such an inference IS a non sequitur. And every mathematician and logician knows it is a non sequitur. So, we recognize that to have a set with all the natural numbers we need an AXIOM for that, which is NOT an inference.

Also, it is good to study Cantor for historical context, to appreciate his intellectual power, and to gain insights into the concepts. But Cantor has been supplanted for 125 year or whatever by axiomatic set theory. If you are sincere in wanting to fairly critique the mathematics then you would get a book on set theory and read it.

And that's why I make a great analyst because I have an ability to understand concepts without even knowing of them — Vaskane

Damn, I knew there was something special about you! :starstruck:

But from what I've seen of mathematicians, they either have no part for infinity, or they're using infinity wrongly. I believe they're doing the latter which leads to the former (which I think is why I have heard it said before that "maths is incomplete") — Philosopher19

A great many of us never go beyond using "unbounded". But we use the symbol for infinity. As for transfinite math, it rarely if ever comes up in classical analysis. But Foundations and Set Theory mathematicians follow the basic axioms and explore what lies beyond. You are in way over your head.

I like your comment on comic relief.
I think I ended yesterday laughing at all this.
It's not always as good. Maybe a winter pastime for some of us.

Yo, like I said, I came here making a comic relief joke, to which it tumbled into the argument on the size of infinities, and lo and behold I have an actual theorem that shows sets of cardinals being larger than the last, that backs me up and it's a theorem that has had impact upon reality my friend: Theory of Computation: Cantor's diagonal argument, used to prove the existence of different sizes of infinity, inspired Alan Turing's work on the undecidability of the halting problem, a foundational result in the theory of computation and computer science. — Vaskane

What?

(1) Whatever jokes you made, you also made the incorrect claim about the size of the set of numbers in the intervals. You still haven't recognized the the thorough explanations why your claim is incorrect.

(2) Yes, theorems:

If C is a cardinal then there is a cardinal greater than C.

If C is a cardinal then there is an infinite set of cardinals greater than C.

If C is an infinite set of infinite cardinals, then there is an infinite set of cardinals greater than C and such that it has infinitely many members that each is greater than every member of C.

There is no set k such that every cardinal is a member of k.

We already know that. It's not something that somehow vindicates your incorrect claim about intervals or anything else you might have said.

(3) Yes, we already know that Turing used a diagonal argument and that the diagonal technique was made prominent by Cantor. Moreover, we know that infinitistic set theory applies to the theory of computability. I don't how in the world you think think any of this some how "backs you up" in terms of any controversy there's been with you.

Still Cantor proves Cardinals do indeed have varying sizes. — Vaskane

Of course we prove that there are cardinals of different size. We know the proof well.

That doesn't even the least bit suggest that there is a mathematical object called 'Infinity' that has different sizes.

Maybe a winter pastime for some of us. — Mark Nyquist

It is all-seasonal and perennial I assure you. People spouting hyper-opinionated uninformed and confused misinformation about this subject goes on constantly and forever on the Internet.

Still.... especially good (or bad) recently.