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# Infinity and 0

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If numbers have a beginning witch is one they necessarily have an ending

If they don't .....

Is the problem in the way we address or write numbers ?

Let say numbers are a circle

1 is a rayon in a circle

2 is a diameter

3 is 3 rayon dividing the circle in equal surfaces

4 is 4 rayon ..... etc

In the end we are going to draw a disk

But when?

When are the surfaces between rayons are going to be equal to 0?

One day for sure

Conclusion :

Infinity doesn't exist and numbers begin with 0 which is a centre in the circle

And the infinity? well its calculable but how much exacly? Is it Allah (1665) that the Muslims..etc worship? And what is the obvious way to get it ?

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If numbers have a beginning witch is one they necessarily have an ending

If they don't .....

Numbers do not have a beginning or an end. The real numbers are like an infinitely long piece of string. However, the structure of the real numbers suggest a natural 'starting place' to count forwards or backwards: the number 0.

Is the problem in the way we address or write numbers ?

I do not think there is a problem here.

Let say numbers are a circle

Look up modular arthritic. This is in a sense close to your imagination.

Infinity doesn't exist and numbers begin with 0 which is a centre in the circle

Like a disk? How do you associate any given number with a point on the disk?

Is it Allah (1665) that the Muslims..etc worship? And what is the obvious way to get it ?

Forget Allah and similar, we are discussing mathematics and not religion

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I am going to or will forget him

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If numbers have a beginning witch is one they necessarily have an ending

You mean the natural numbers then. Just because they have a beginning (0 or 1) doesn't mean they must have an end (they don't).

And the infinity? well its calculable but how much exacly?

You should look at Cantor's diagonal argument. Not only does infinity exist, but there are different types of infinity; for example, the set of real numbers is infinitely larger than the set of integers.

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Not only does infinity exist, but there are different types of infinity;

Indeed, but being careful here (just to make sure sooroor does not fall into this trap), all these notions of infinity do not give a real number. We cannot think of infinity as a number in the standard sense, it does not obey the rules of numbers.

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AHHHHh😲 I ddidn't ssay dthat😠!

?

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AHHHHh I ddidn't ssay dthat!

?

Good, but it is something that people always get confused about.

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AHHHHh I ddidn't ssay dthat!

?

!

Moderator Note

By using the quote function, you can let everyone know what it is exactly that you didn't say. It's a button at the bottom of each post marked "Quote".

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!

Moderator Note

By using the quote function, you can let everyone know what it is exactly that you didn't say. It's a button at the bottom of each post

marked "Quote".

Hhhhhhhh oh yeah

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You should look at Cantor's diagonal argument. Not only does infinity exist, but there are different types of infinity; for example, the set of real numbers is infinitely larger than the set of integers.

How does Cantor's argument show that infinity exists?

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How does Cantor's argument show that infinity exists?

Good point. It doesn't.

(But I didn't say it does. )

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Good point. It doesn't.

(But I didn't say it does. )

I may have misunderstood.

More generally, what Cantor showed was that if we believe in an infinite set at all, we have to believe in different types of infinite sets. Not just the cardinals, but also the ordinals, which are less widely known but equally important.

But why do we believe in the mathematical existence of infinite sets at all? They're an arbitrary assumption taken more for convenience. The leap is going from 1, 2, 3, ... which continues without end; to the idea that we can form the completed set of all of them taken at once. It would be logically consistent to deny the mathematical existence of an infinite set. That's an ontological point not always appreciated. There is nothing necessary about infinite sets. They have only fictional existence in our minds, as far as we know. If tomorrow morning the physicists discover an infinite set, that would be different. Till then, infinite sets have the same ontological status as Star Wars characters. Interesting fictions.

Now the interesting question (to me) is: Why do our rational and presumably finite minds so readily conceive of the infinite? Why are we able to reason so precisely about something that does not exist in our world?

In short, what is Cantor's work telling us about our own minds and about the world?

Edited by wtf
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Why are we able to reason so precisely about something that does not exist in our world?

No idea, but that seems to be the beauty of mathematics!

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Why are we able to reason so precisely about something that does not exist in our world?

That is true of many mathematical structures, I suppose. It relates to the old question of whether mathematics is discovered or invented. But do we know that infinity does not exist in our world? The universe may be infinite, after all.

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• 4 weeks later...

well numbers does not start from 0...

they have no beginning and we call it NEGATIVE INFINITY

and they does not have any end and we call it POSITIVE INFINITY

if u give me a huge negative number then i will subtract 1 from it and will give u an even smaller number

same goes for larger

the disk thing is something we made to visualize the positive number line but it have its limitations as clearly stated in your question

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well numbers does not start from 0...

that depends on what you're considering or working with, the smallest number in the set of non negative integers is indeed 0 so that'd be a nice beginning. but yes, the reals don't have a "beginning."

and as ajb mentions 0 is a convenient spot in the reals to count forward or negative.

Edited by andrewcellini
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• 1 month later...

I may have misunderstood.

More generally, what Cantor showed was that if we believe in an infinite set at all, we have to believe in different types of infinite sets. Not just the cardinals, but also the ordinals, which are less widely known but equally important.

But why do we believe in the mathematical existence of infinite sets at all? They're an arbitrary assumption taken more for convenience. The leap is going from 1, 2, 3, ... which continues without end; to the idea that we can form the completed set of all of them taken at once. It would be logically consistent to deny the mathematical existence of an infinite set. That's an ontological point not always appreciated. There is nothing necessary about infinite sets. They have only fictional existence in our minds, as far as we know. If tomorrow morning the physicists discover an infinite set, that would be different. Till then, infinite sets have the same ontological status as Star Wars characters. Interesting fictions.

Now the interesting question (to me) is: Why do our rational and presumably finite minds so readily conceive of the infinite? Why are we able to reason so precisely about something that does not exist in our world?

In short, what is Cantor's work telling us about our own minds and about the world?

Wait I do not get why one can't wrap his head around infinity. Isn't it simply the idea or concept of always, continuously increasing our numbers endlessly. I know that is a terrible definition giving the fact that when you take theinfinte set of all real numbers...you know what happens...

It is possible with our number system, especially with the study of prime numbers, we can see that our current number system is capable of this concept, it just cannot be used to assign "olafs". As in, we cannot figure out what level of infinity our certain set is. What is their not to believe about infinity...Believe me I am not trying to make a point in an argument...i just want to make sure i am not seeing something that other people see that is essential in the concept of infinity and makes them doubt its existence.

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wtf

How does Cantor's argument show that infinity exists?

Of course it does.

But equally of course

ajb

Forget Allah and similar, we are discussing mathematics and not religion

The mathematical statement "there exists" does not mean you can go down to the supermarket and buy a pound of "infinity".

It means that the properties of a mathematically defined object called "infinity" is consistent with the axioms and theorems already available.

This is more in later with your later statement, which I agree with.

wtf

More generally, what Cantor showed was that if we believe in an infinite set at all, we have to believe in different types of infinite sets. Not just the cardinals, but also the ordinals, which are less widely known but equally important.

However we should be aware that the words cardinal and ordinal have different usage in English and mathematics.

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Of course it does.

I envy people who have such certainty. Certainty is the fashion these days. You can't be thoughtful on cable tv. Opinions must be loud, certain, and extreme. But when we're discussing a philosophically loaded aspect of math such as infinity, may we dispense with absolutism and perhaps acknowledge the difficulty of the question? Or are we on cable tv here too?

If we accept the axiom of infinity, then an infinite set exists.

If we reject the axiom of infinity, then no infinite set exists. Note that we still have all the natural numbers 1, 2, 3, 4, 5, ... What we don't have is a set that contains all of them. This is a perfectly valid system, known as the hereditarily finite sets.

Ever since Russell's paradox, we learned that we can only apply the word "set" to very specific collections that satisfy some formal axioms. We can't just put any old collection of objects into a set just because we want to. We can't just say, "Well I can imagine 1, 2, 3, ..." therefore there's an infinite set. In fact that's not a valid logical deduction. All we have is an infinite collection that may or may not be a set.

If it's logically consistent to accept the axiom of infinity and logically consistent to reject it, why do we generally accept it? Simply for pragmatic reasons of convenience. It's a lot easier to do math if we allow infinite sets, so we allow them. This has of course no bearing whatever on the truth of the matter, if it's even meaningful to speak of truth in this regard.

Accepting infinity into mathematics is simply a convenience, nothing more.

One can go farther. There are ultrafinitists who reject not only infinite sets; but even sufficiently large finite sets. The late Ed Nelson was an ultrafinitist who was a serious, well-respected mathematician.

In summary, there is no absolute answer to the question of whether mathematical infinity exists. It's purely a matter of personal choice and convenience.

Or did you mean perhaps that Cantor's work shows that infinite sets exist? Sadly not. There's a lot Cantor didn't know about set theory, just as there's a lot the Wright brothers didn't know about airplanes. Set theory in its modern form took a good fifty years to develop after Cantor's work in the 1870's. What Cantor thought of as sets wouldn't pass muster in undergrad set theory class today. That doesn't diminish his genius, but it does indicate the logic traps we may fall into if we take Cantor as the last word about set theory rather than the first.

But equally of course

Sorry, couldn't parse that. What does that refer to?

The mathematical statement "there exists" does not mean you can go down to the supermarket and buy a pound of "infinity".

It means that the properties of a mathematically defined object called "infinity" is consistent with the axioms and theorems already available.

Which axioms? You said "of course" to the question of whether mathematical infinity exists. But now you admit that it's purely a matter of which axioms one adopts and that there is no actual truth of the matter (or that the truth of the matter is not presently known). Which is your position? You are not being clear. You appear to be contradicting yourself in two different paragraphs.

However we should be aware that the words cardinal and ordinal have different usage in English and mathematics.

Well yes, just as the word vector has a technical meaning in math that's subtly different than the meaning in physics which is a lot different than the meaning in biology. Just as calculus is the stuff the dentist scrapes off your teeth. We're having a math discussion. You know the old joke. Q: What do you get when you cross a mosquito with a rock climber? A: Nothing. You can't cross a vector with a scalar.

Edited by wtf
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wtf

You know the old joke. Q: What do you get when you cross a mosquito with a rock climber? A: Nothing. You can't cross a vector with a scalar.

No, I've never come across that one.

Thank you.

wtf

Well yes, just as the word vector has a technical meaning in math that's subtly different than the meaning in physics which is a lot different than the meaning in biology. Just as calculus is the stuff the dentist scrapes off your teeth. We're having a math discussion.

Yes I couldn't agree more, I've even started a thread here asking for examples of words which cause confusion because of multiple meanings in different disciplines.

Thank you for the calculus example, I will add it to my list.

wtf

This has of course no bearing whatever on the truth of the matter, if it's even meaningful to speak of truth in this regard.

Accepting infinity into mathematics is simply a convenience, nothing more.

I agree that truth is one such word, and therfore perhaps best avoided.

Come now I think "just a convieniece is a bit weak", don't you?

wtf

If it's logically consistent to accept the axiom of infinity and logically consistent to reject it, why do we generally accept it? Simply for pragmatic reasons of convenience. It's a lot easier to do math if we allow infinite sets, so we allow them. This has of course no bearing whatever on the truth of the matter, if it's even meaningful to speak of truth in this regard.

Well what of Euclid's 5th axiom then?

It's logical to accept it and logical to reject it, but we don't have all this ho-hah about Euclidian v non Euclidian geometry.

Both are equally accepted into mathematics as consistent.

For the rest I think you are manufacturing an argument, where none exists.

I made it perfectly plain that the mathematical statement

"There exists an n such that n+2 = 3" doesn't give physical embodiment to the phrase "there exists", as does English,

it actually means that n = 1 is consistent with the rules of arithmetic.

By the same token the mathematical statement

There exists an infinite object means that we can demonstrate a mathematical object which can be placed into one-to-one correspondence with a part of itself, that is not inconsistent with stated mathematical rules, though it may contravene others that we are not employing. It does not mean we can, as I said, buy a pound of it in Tescos.

wtf

Sorry, couldn't parse that. What does that refer to?

Sorry if you couldn't make that out.

It referred to my previous sentence.

Roughly translated I was saying

Yes I assert that mathematical infinity exists, but on the other hand remember that mathematical infinity has a different meaning than the one you are perhaps used to in English.

Edited by studiot
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No, I've never come across that one.

Thank you.

You're welcome! I'm addicted to bad puns and lame jokes, I got a million of 'em.

Yes I couldn't agree more, I've even started a thread here asking for examples of words which cause confusion because of multiple meanings in different disciplines.

Thank you for the calculus example, I will add it to my list.

Set! That's causing the confusion right here!! Set has a very technical meaning, it couldn't be more different than the everyday idea of a collection of things. Likewise group, ring, field, module, manifold (the thing on your car's engine) etc. I can't think of any word in math that does have its usual meaning. I didn't see your other thread, if you link to it I'll be happy to add to it.

I agree that truth is one such word, and therfore perhaps best avoided.

I hope it's not me you're agreeing with, since I never said that the word truth should be avoided. Truth has a very specific meaning in mathematical logic and it's been intensively studied. And it's the subject of our conversation: Whether it's true that infinite sets exist in math, or just a convenient assumption. I'm arguging the latter. I believe you're claiming the former but not putting forth an argument.

Come now I think "just a convieniece is a bit weak", don't you?

I would say it's a strong statement. I am making a very specific point. Infinite sets are widely accepted in math because they're useful and interesting. Any claim of truth beyond that requires careful analysis and exposition. I'm sure one could make a case that infinite sets "truly" exist in math; just as well as one could make the opposite claim. We accept infinite sets because they're useful. This is a fact. It's counterintuitive because the axiom of infinity is always assumed in ZF[C], so it becomes like the air. We don't notice it unless we are explicitly discussing axiomatic set theory. But isn't that what we're discussing here? If not, maybe that's a source of confusion.

To put this in a contemporary context, intuitionism and constructivism are making a comeback, due to the influence of computer science and also category theory. (That's a big statement which would take us too far afield to go into right now in any detail). There are a lot of people these days who feel that a mathematical object exists only insofar as it can be explicitly constructed by an algorithm. So pi and the square root of 3 exist; but not the uncomputable numbers. There are a lot of people doing serious work in this area and it's far from clear what our ideas of the real numbers will look like in the future.

If one believes in full powersets, one goes immediately from the naturals to the reals. But most subsets of the reals can not be generated by an algorithm and have a far weaker claim to existence than, say, the number 6 or the set of even numbers.

By the way I myself am not a constructivist. I believe in full ZFC. That doesn't mean I think it's true. It means I think it's convenient! But I'm aware that the constructivists are making a lot of progress lately and it's far from clear which point of view will prevail. Math is a historically contingent human activity, whether or not there's an ultimate truth "out there."

Well what of Euclid's 5th axiom then?

It's logical to accept it and logical to reject it, but we don't have all this ho-hah about Euclidian v non Euclidian geometry.

Both are equally accepted into mathematics as consistent.

What a perfect example supporting my point. When the young Gauss discovered the consistency of non-Euclidean geometry in the 1820's he did not publish, fearing the negative impact on his career. Imagine that, Gauss himself afraid to publish. That gives us an idea of just how controversial non-Euclidean geometry was. When Riemann and others established non-Euclidean geometry in the 1840's it was VERY controversial and shocking to people. It was regarded as a mathematical curiosity having nothing to do with reality. It wasn't till Einstein came along with his crazy theory of relativity, and Minkowski said, "Hey, math has just the right gadget in our back pocket." That's when people started taking non-Euclidean geometry seriously.

There was a MASSIVE hoo-hah about non-Euclidean geometry before that. Simply massive. It was really upsetting to people. Think about it. One day, math and physics are the same thing. The next day, they're not. This was one of the most profound intellectual developments in history. People are STILL arguing about the extent to which math must be based on reality.

For the rest I think you are manufacturing an argument, where none exists.

I'm making an important point of mathematical ontology that you're failing to understand or engage with. If you claim that infinite sets exist in mathematics independent of the axiom of infinity, make a case. Or if you think that they do depend on the axiom of infinity but that the axiom of infinity is somehow more correct, more natural, more true, then make that case. I claim it's simply more convenient. If you disagree, don't just repeat your claim. Make an argument. Maybe I'll learn something.

I made it perfectly plain that the mathematical statement

"There exists an n such that n+2 = 3" doesn't give physical embodiment to the phrase "there exists", as does English,

it actually means that n = 1 is consistent with the rules of arithmetic.

I'm not entirely sure what this has to do with the discussion. I also don't recall you making this statement. This is a two month old thread that someone resurrected and I haven't read through from the beginning, would that help?

By the same token the mathematical statement

There exists an infinite object means that we can demonstrate a mathematical object which can be placed into one-to-one correspondence with a part of itself, that is not inconsistent with stated mathematical rules, though it may contravene others that we are not employing. It does not mean we can, as I said, buy a pound of it in Tescos.

An infinite object? Oh yes. An infinite SET? Requires the axiom of infinity. Did you notice that you swapped in the word "object" when we are talking about sets? The word set has a lot of technical baggage around it these days. One can agree that there is an infinite object without agreeing that there is an infinite set.

Let me sketch a proof that in the absence of the axiom of infinity, there is no such bijection. I'll grant you the existence of a collection of natural numbers 1, 2, 3, 4, 5, ... Now you say there's a bijection between ... well, between what, exactly? A bijection is a type of function. What is a function? It's a mapping that goes from a set to a set. That's the formal definition of a function. So now that we haven't got a set of natural numbers, we can't biject anything to it. You see how tricky this can be. Saying something is a collection is fine. Saying it's a set means that we can define functions on it, take powersets, etc. Without the axiom of infinity we can't do any of those things to the collection of natural numbers.

I hope this is clear, it's the heart of the matter. The axioms tell us exactly what operations may be performed on sets. If we have a collection that's not a set, we can't do those operations on it. You haven't got a bijection between the natural numbers and anything, because without the axiom of infinity, you haven't got a set on which to define a function. You may well have an intuition about these things, so do I. But if we are being careful to follow the rules of set theory, we have no bijection.

Yes I assert that mathematical infinity exists

You can not possibly justify that statement without adding context. Are you saying an infinite SET exists? If so, that requires the axiom of infinity; and it's perfectly consistent to deny the existence of infinite sets; and the only reason we accept infinite sets in everyday math is that they are convenient. There is no other reason. If you have an argument to the contrary, you are free to make it but so far you have not done so.

, but on the other hand remember that mathematical infinity has a different meaning than the one you are perhaps used to in English.

I wonder what I could ever have written that leads you to think I'm confused on that point?

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wtfI didn't see your other thread,

if you link to it I'll be happy to add to it.

Thank you for that offer. I look forward to your contributions.

http://www.scienceforums.net/topic/88336-differences-between-mathematics-and-physicsengineering/

wtf

You can not possibly justify that statement without adding context. Are you saying an infinite SET exists?

I think we are still at cross purposes.

The only meaning of "there exists an X such that" in mathematics is, as I have already indicated, that the stated properties of X are consistent with the definitions, axioms and deduced theorems we are currently employing.

[aside] Don't forget it has been shown that there is no comprehensive list of definitions, axioms and theorems for the whole of mathematics at once. The progroms of Hilbert and Frege are unrealisble dreams.[/aside]

It does not matter whether we have 'discovered' X or just invented it or whether X is an abstract construct in our minds or has physical manifestation, mathematically X 'exists'.

I could invent pink elephants with green spots, for the purpose of counting spots on an animal. Mathematically, such an animal would exist.

This is really no different from saying Mr Jines' garden is 50' by 30' in a problem in a maths book.

There is no necessity for Mr Jines to physically exist or have a garden if he does.

Therefore I assert that at least one infinite set exists, and therefore at least one infinite 'object', since object is a more general noun that includes set.

I do not know of any inconsistency with the current definitions, axioms and theorems.

If you can offer one or more, please post them.

[aside] Sometimes we find that our rules conflict with experience. As such we need to re-examine our rules. Such was the case with the introductuion of i, which I hope you will grant mathematical existence to. [/aside]

Edited by studiot
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I think we are still at cross purposes.

To rectify that situation I will state my purpose.

I am explaining. I assume you are interested in learning.

Why would I say that? Because in other posts you have said that your background in math is practical and applied. Mine is pure and abstract.

If you are talking to me about set theory, I must assume you have curiousity about the subject. I'm the right person to explain it.

Of course if we were discussing applied math, the situation would be reversed. I would not presume to argue with you about Fourier series, PDEs, or even integration by parts. When I taught calculus most of my students could do integration problems better than I could. That's only a slight exaggeration.

But when it comes to basic set theory, I'm authoritative.

That's where I'm coming from, for better or worse. I assume if you're talking to me at all, it's with the intention of trying to learn how pure math sees the subject of infinity. What other reason could there be?

So ... where are you coming from?

The only meaning of "there exists an X such that" in mathematics is, as I have already indicated, that the stated properties of X are consistent with the definitions, axioms and deduced theorems we are currently employing.

That's right. If you assume the axiom of infinity (AxInf from now on) then there is an infinite set. If you assume the negation of AxInf, there is not an infinite set. Note that there may still be an unending sequence of natural numbers 1, 2, 3, ... But there is no (completed) set of them.

I use the word completed because that was Aristotle's clever idea, to distinguish between the potential and the actual infinity. However note that the terms potential/actual do not occur in modern math, only in the philosophy of math.

[aside] Don't forget it has been shown that there is no comprehensive list of definitions, axioms and theorems for the whole of mathematics at once. The progroms of Hilbert and Frege are unrealisble dreams.[/aside]

Non sequitur. Totally irrelevant to our discussion. "Argument from random Wiki pages without understanding."

It does not matter whether we have 'discovered' X or just invented it or whether X is an abstract construct in our minds or has physical manifestation, mathematically X 'exists'.

I could invent pink elephants with green spots, for the purpose of counting spots on an animal. Mathematically, such an animal would exist.

This is really no different from saying Mr Jines' garden is 50' by 30' in a problem in a maths book.

There is no necessity for Mr Jines to physically exist or have a garden if he does.

Right. Assume AxInf and an infinite set exists. Deny AxInf and no infinite set exists. I am puzzled as to why you keep ignoring this point.

To be more accurate: Assume AxInf and the existence of an infinite set is provable. Deny AxInf and the existence of an infinite set is not provable. The "truth" of the matter is unknown, nor is it clear that there is a truth of the matter at all.

Therefore I assert that at least one infinite set exists, and therefore at least one infinite 'object', since object is a more general noun that includes set.

I do not know of any inconsistency with the current definitions, axioms and theorems.

Yes you keep ASSERTING. What you have not yet done, and what I have repeatedly invited you do to, is to put forth an actual ARGUMENT in support of your position.

What "current definitions, axioms and theorems" do you refer to?

In standard ZF, an infinite set exists. In ZF minus AxInf, no infinite set exists. Why do you persist in ignoring this point?

If you can offer one or more, please post them.

I have repeatedly stated the argument. The system ZF minus AxInf is a consistent set theory known as the hereditarily finite sets. It has W Wikipedia entry that I've already posted.

Tell me: What do the hereditarily finite sets mean to you? Do you understand their significance? Please read this page. https://en.wikipedia.org/wiki/Hereditarily_finite_set

[aside] Sometimes we find that our rules conflict with experience. As such we need to re-examine our rules. Such was the case with the introductuion of i, which I hope you will grant mathematical existence to. [/aside]

More irrelevance. I don't see an argument here. In fact in math when the rules are logically consistent yet conflict with our experience, it's the rules that matter. Such was the case with non-Euclidean geometry (an example you brought up earlier). Such is the case with the mathematics of the infinite.

Please tell me what your point is. We actually have made no progress at all since our previous posts. You still have not put forth an argument that an infinite set exists even in the absence of AxInf. Is that in fact your claim? I can't figure out what you are claiming.

ps -- Perhaps to clarify in my mind exactly what it is you are saying, which of these are you saying:

a) An infinite set exists if and only if we assume AxInf; or

b) An infinite set exists whether or not we assume AxInf. If we don't assume AxInf then we can't PROVE an infinite set exists, but in the Platonic realm an infinite set does in fact exist; or

c) Something else.

Edited by wtf
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I can't figure out what you are claiming

That is the problem.

You appear to have reverted to

quoting tracts of what I say,

Declaring it false.

But not actually addressing the content.

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That is the problem.

You appear to have reverted to

quoting tracts of what I say,

Declaring it false.

But not actually addressing the content.

You might have posted that before my last edit. You are right, I don't understand what you're saying. Which of these is closer to your claim:

a) An infinite set exists if and only if we assume AxInf [that's my position]; or

b) An infinite set exists whether or not we assume AxInf. If we don't assume AxInf then we can't PROVE an infinite set exists, but in the Platonic realm an infinite set does in fact exist [that's not my position, because I'm not a Platonist]; or

c) Whether or not we assume AxInf, an infinite set exists, not just in the Platonic world but ... well, where exactly?

d) Something else.

Edited by wtf

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