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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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'.

The above post is correct and mentions a crucial point. Therefore, it will be of no interest to too many people.

for mathematicians to be using the label "infinity" — Philosopher19

So your method of conversation is to ignore when someone informs you nearly a dozen times on a point:

In this context, mathematics doesn't use 'infinity' as a noun, as if there is an object named 'infinity', but rather 'is infinite' as an adjective to name a property. That distinction is crucial to understanding the subject matter.

from what I've seen of mathematicians — Philosopher19

What you've seen is what you've allowed yourself to see, which is virtually nothing about the actual mathematics you've not even bothered looked up.

just like it is — Mark Nyquist

Nothing at all like it is.

infinity is a concept considering continuity, not size. — Vaskane

Again, in mathematics, the concept is 'is infinite' as an adjective, not 'infinity' as a noun. And continuity is a different idea, while the idea of "size" is approached by the formulation of the idea of cardinality.

In mathematics, with or without the axiom of extensionality, '=' means identity.

Of course notions of infinity pertain in different areas of study. But just to bear in mind, the original post is a challenge to the idea that there are "different sizes" of infinity, which is primarily a set theoretic idea, and the post mentioned mathematics regarding that. So among philosophers, physicists, astronomers, cosmologists and theologists, we will still have mathematicians to consider.

So your qualification is noted. But what you wrote originally naturally would be taken as a pejorative. "just mental showboating" would not ordinarily be understood as approbation.

Some of the advanced math theories are maybe just some mental showboating of things the math people can do with their brains. — Mark Nyquist

A person hasn't studied the pertinent mathematics, doesn't know anything about it, doesn't understand it. So their response to it is to say that it might be just a bunch of "mental showboating" anyway.

To add to the above:

If X := X->Y then X <-> (X->Y).

But we don' t have the converse that if X <-> (X->Y) then X := X->Y.

So X := X->Y is not equivalent with X <-> (X->Y).

So we can't dispense the paradox by incorrectly saying that it reduces to X <-> (X -> Y).

Notation I'll use:

'iff' for 'if and only if'

df. x is equinumerous with y iff there is a one-to-one correspondence between x and y

df. x is countable iff (x is finite or x is equinumerous with the set of natural numbers)

df. x is denumerable iff x is equinumerous with the set of natural numbers

'*' for cardinal multiplication

Your scenario can be boiled down to this:

There are denumerably many rooms. And if we multiply the number of rooms by 2, then there are still denumerably many rooms.

That reflects the set theoretic fact that if H is a denumerable cardinal and K is a countable cardinal, then H*K = H.

In this case, H is the number of rooms in the original hotel and K = 2.

To put your musings in perspective, here are the mathematical facts:

The set of rational numbers between any two natural numbers is not sequenced by the ordinary less-than relation on rational numbers.

Between any two natural numbers there is a denumerable sequence of the set of rational numbers between the natural numbers, but it is not isomorphic with the ordinary less-than relation on rational numbers.

The set of irrational numbers between any two natural numbers is not sequenced by any countable ordinal.

The set of irrational numbers between any two natural numbers is sequenced by some uncountable ordinal if we have the axiom of choice.

The set of rational numbers between any two natural numbers is equinumerous with the set of all natural numbers.

The set of irrational numbers between any two natural numbers is not equinumerous with the set of natural numbers.

It is not the case that there are more rational numbers between 0 and 2 than between 0 and 1.

It is not the case that there are more irrational numbers between 0 and 2 than between 0 and 1.

The set of rational numbers is equinumerous with the set of natural numbers.

Any infinite subset of the set of rational numbers is equinumerous with the set of natural numbers.

Any infinite subset of the set of rational numbers is equinumerous with the set of rational numbers.

The set of irrational numbers is not equinumerous with the set of natural numbers.

The set of irrational numbers is not equinumerous with the set of rational numbers.

There are infinite subsets of the set of irrational numbers that are equinumerous with the set of natural numbers.

There are infinite subsets of the set of irrational numbers that are equinumerous with the set of irrational numbers.

Any infinite subset of the set of natural numbers is equinumerous with the set of natural numbers.

If x is an infinite subset of the set of natural numbers and y is a finite subset of the set of natural numbers, then {n | n is in x and n is not in y} is equinumerous with the set of natural numbers.

If x is an infinite subset of the set of rational numbers and y is a finite subset of the set of rational numbers, then {r | r is in x and r is not in y} is equinumerous with the set of rational numbers.

If x is an infinite subset of the set of natural numbers and y is a finite subset of the set of natural numbers, then the union of x and y is equinumerous with the set of natural numbers.

If x is an infinite subset of the set of rational numbers and y is a finite subset of the set of rational numbers, then the union of x and y is equinumerous with the set of natural numbers.

'distance' is defined by the absolute value of the difference between points, not by cardinality. The distance between 0 and 1 is 1, no matter what about the cardinalities of the set of irrationals and the set of irrationals between 0 and 1.


That's all ordinary mathematics, proven from the ordinary axioms.

One is free to propose different axioms that prove differently.

What evidence do you have that fishfry left because of this thread?

The reason technical content should not be shunted elsewhere is that if the philosophical discussion is ABOUT the mathematics, especially a critique of the mathematics, then the mathematics should not be misrepresented, otherwise (1) Misinformation about mathematics is permitted to thrive and (3) The philosophical claims themselves are errant for not even being correctly about what they are supposed to be about.

Anyway, the recent deletions were not about the earlier exchanges with fishfry.

Q is enumerable and so is the set of finite sequences of members of Q. The set of infinite sequences of members of Q and R are not enumerable. Okay, some things are enumerable and others aren't.
— TonesInDeepFreeze

I know all that, already said it, spent years writing proofs for professors. Not asking random internet guy about the basics of analysis. Tried to ask you about 'subjective' (maybe philosophical) responses to all the symbols that swim like fish in those textbooks you mentioned. You gave a disappointing response, like you are deaf and mute to anything that isn't mere chatbot correctness. I have loved math as a meaningful 'science' of form(s) with some intuitive validity. I care about various formalisms only because they strive to mean something, capture something beyond them. The continuum is a endlessly fascinating beast that great thinkers have wrestled with for centuries. I don't know if you know or care much about mathematical history, but I love the drama. But I'll save that for others who aren't satisfied with the relatively trivial (however difficult at times ) syntactical part. — plaque flag

I said I don't have anything immediate to say about subjective impressions of mathematics. I am not thereby like a "chatbot" that is not interested in anything other than the syntactical aspects of mathematics. Specifically, the fact that I don't have anything immediate to say about "R is mostly a black and seamless sea in the darkness" does not entail that I am uninterested in intuitions involved in the construction of the real number system. And the poster quoted above had said that he had been reading my posts over time; but in some of my posts I had written that indeed I am interested in intuitions about mathematics.

It's not complicated.

Definitions:

f is an injection iff f is a one-to-one function. We may also say 'f is an injective function'.

f is an injection from x into y iff (f is an injection and the domain of f is x and the range of f is a subset of y).

f is surjection from x onto y iff (f is a function and the domain of f is x and the range of f is y).

(So every function is a surjection from its domain onto its range.)

f is a bijection from x onto y iff (f is an injection and f is a surjection from x onto y).

(So every injection is a bijection from its domain onto its range.)

/

It seems to me that your feet-head is just an example of multiplication. Two feet for each of n number of people is 2*n feet.

Or it could be a bijection between the number of pairs of feet and number of people. There are the same number of pairs of feet as there is the number of people.

A clearer example would be just two lines of people such that we could see that there is a one-to-one correspondence between the people in one line and the people in the other line.

Posts are missing from this thread, including some of my own. What happened?

And a post of mine was deleted in another thread. So I listed the reasons why it and the posts in this thread should not be deleted. Then that post was deleted and an admin made a post saying where discussions of deletions should be, but when I came back to read the admin's post and take note of where the admin had in mind for discussion of deletions, that post by the admin was also deleted. Thus, I didn't get to save my own post about the deletions to instead put it in the location suggested, and I didn't even get to read the admin's post about the suggested location. That doesn't even include all the posts by me and others in this thread that were wiped out, without notification or stated reason.

Note that 'countable' in mathematics does not mean that any human being can count every member of the set. Rather, 'countable' in mathematics only means that the set has a one-to-one correspondence with a natural number or the set of natural numbers.

Also, AGAIN, there is no object in set theory called 'the infinite'.

And it is not the case that set theory says that every infinite set is countable.

And a bijection is a certain kind of function. And set theory doesn't have a term 'counting' so set theory does not say anything about whether bijections are a form of counting.

For about the sixth time, and this is one of the points you keep refusing to address:

I don't begrudge anyone from having whatever concept and definition of infinitude they wish to have.

But having a different concept and definition of infinitude doesn't thereby entail that there is a contradiction in set theory or mathematics.

Again, yes, there may be a contradiction between set theory and certain other formulations. But that does not entail that there is a contradiction within set theory.

Again, for emphasis yet again, since you keep skipping this point, no one should deny you from having whatever concepts and definitions you would like to have, and if thereby set theory does not suit you or does not make sense to you, then so be it, but that doesn't entail that set theory leads to any contradiction in itself.

Yes, set theory does not have the same concept of infinitude that you have. As well as, which you also keep skipping, set theory does not refer to an object named 'infinity' but rather to the property of being infinite, which is a crucial distinction.

In this thread, there was discussion about set theory and that discussion had important errors. So I provided a systematic synopsis of the area in discussion as that synopsis corrected the errors and explained why they are in error. Then replies came to my posts, but certain of those replies still had misconceptions about set theory.

Again, espouse whatever concept of infinitude you wish. But that does not justify an incorrect and misinformed critique of set theory.

And whether the thread is or is not a negative experience for anyone, it still stands that your posting has been an absurd loop.

I said exactly what points you are not addressing, and now you just come back to insist that you are addressing them though you are not. And you said, "I believe you are not reading all of it. See my last post to you" before I even had a chance to reply to it, as then I did reply to it. How absurd to charge someone with not reading a post before he has even had a chance to reply to it in the order it was posted. Your posting is an absurd loop.

I can say it is an informal notation for the set of all and only the natural numbers because that is exactly what it is an informal notation for. The question of how a notation is used is settled empirically, by just looking at the way it is used. And in mathematics, when '{0 1 2 3 ...}' is used, it is used to stand for the set of all and only the set of natural numbers.

A separate question is whether in mathematics there exists such a set. And I addressed that, but you skipped what I wrote.

I'm no pot, since I haven't ignored what the poster said.

But you are such a pot, as recently you ignored what I said.

Mark Nyquist:

Now for the third time:

Anyone can have whatever concept of mathematics they want to have. But having a different concept from set theory doesn't entail that set theory itself proves any contradictions.

As to subtraction with infinite cardinals, again I say, just start by defining it.

And you skipped what I said about removing.

Philospher19, again ignored for the second time what I wrote about "contradiction".

Another misconception:

"saying 1,2,3,4 ad infinitum or {1,2,3,4,...} does not mean one has shown an infinite number of natural numbers."

Yes, saying that for every natural number there is a greater natural number does not in and of itself imply that there is an infinite set of natural numbers. But {0 1 2 3 ...} is not notation that for every natural number there is a greater natural number, but rather it is an informal notation to stand for the set of all and only the natural numbers.

The way set theory proves there exists a set with all and only the natural numbers is by an axiom from which we prove that there exists a set with all and only the natural numbers.

So the objection that we can't extrapolate from "for every natural number there is a greater natural number" to "there is a set with all the natural numbers" is true but a huge strawman since set theory does NOT claim that we can extrapolate that way.

Again, such objections are a product of sheer unfamiliarity with the subject matter.

It is by tolerance that there are forums such as this that allow such spreading of confusion.

I am paying close attention to what you are saying. While I just now showed specifically and fundamentally how you skipped what I said.

And if you want to discontinue your posting in this thread, then you can discontinue it. No one is stopping you.

Then you believe an untruth.

And for about the fifth time, there is no object named by 'infinity'. So there is no object named by 'the quantity of infinity'.

There is least infinite cardinal, which is the cardinality of the set of the infinite set of natural numbers. And there are cardinals greater than the least infinite cardinal. Moreover, for each cardinal, whether it is a finite cardinal or infinite cardinal, there is a greater cardinal.

And when someone says "contradictory to the semantic of infinity" here it indicates that one did not read, or chose to ignore, what was written about that.

I'll repeat: Study of set theory is not a commitment to adherence to all the many meanings of 'infinite' and 'infinity' in everyday discourse, in philosophy and even other science. Rather, the adjective 'is infinite' has a special and very particular definition in set theory. It some ways it is compatible to other non set theoretic meanings of notions, but is not compatible with certain other non set theoretic meanings or notions.

No one disputes that the set theoretical definition might not accord with anyone's other notions. But that does make set theory inconsistent. A theory is inconsistent if and only if it proves some sentence P and not-P.

"There is a cardinality greater than the cardinality of the set of natural numbers" and your notion "There is no cardinality greater than the cardinality of the set of natural numbers" is not a contradiction of set theory, because set theory does not prove "There is no cardinality greater than the cardinality of the set of natural numbers" despite that that sentence is one you believe in your non set theoretic notions.

Time well spent would be to learn some mathematics rather than claiming untrue things about it.

Anyway, did someone say "beyond infinity"? 'Beyond infinity' has no apparent meaning. First, again, 'infinity' should not be used as a noun in this context. Set theory does not define 'infinity'. Second, the set theoretic fact is that for any infinite cardinality there is a greater infinite cardinality. So, yes, in that sense, there is an infinite cardinality "beyond" the cardinality of the set of natural numbers, but there is no "beyond infinity".

If one claimed that the definition of 'equinumerous' must lead to a definition of cardinal subtraction, moreover a requirement that a definition of cardinal subtraction adheres in certain ways to integer subtraction, then one would be mistaken.

But to be positive about this, I would suggest: What definition of cardinal subtraction does anyone here have in mind? That is to provide the formula P in the following (where P has no free variables other than x, y and z; and for each <x y> there is a unique z such that P):

If x and y are cardinals, then x-y = z <-> P.

Someone said 'This sentence is false' doesn't indicate what sentence is being referred to.

In 'This sentence is false', 'this sentence' is referring to 'This sentence is false'.

Not recognizing that is just waving away the problem.

What invalid statement was implied by mathematics formalized in first order logic? By definition, all non-logical axioms are invalid (i.e. not validities, not true in every interpretation). And unrestricted comprehension implies Russell's paradox, which is not 'this sentence is false'.