Maths is constructed. One can do with it as one pleases with the symbols involved. We make the rules up as we go, and we can and do go back and change them as we like. They are not tied to instruments or forms or anything other than themselves. — Banno

We make rules for a purpose. The rules are instruments, used for obtaining our goals. Rules do not exist independently from their use, as if they are things in "themselves". And "use" implies purpose.

"Infinity" in mathematics is just a name, a symbol that could be replaced with any other symbol — SophistiCat

More of a concept, actually. An expression "becomes infinite" if it grows without bound. No need for a symbol. But it's there to be used if one wishes.

Do you understand Turing's answer to the Halting problem? Just as Cantor's diagonal argument shows that not every infinite set of numbers can be put into 1-to-1 correspondence with the Natural numbers, so do the various undecidability results, starting from Church-Turing thesis, show that indeed there are mathematical objects that cannot computed. Not everything can be calculated/computed by a Turing Machine. — ssu

Recall that the proofs of Godel and Cantor correspond to Turing-computable algorithms. Are you meaning to suggest that a deterministic machine can follow a set of rules to prove that there exists a theorem that cannot be deterministically derived by those rules? Or that proves in a finite number of steps the existence of a literally uncountable universe? That's what the common (mis)interpretation of Godel's and Cantor's results amounts to.

It is certainly possible that nature is really random in the sense of falsifying any proposed theory-of-everything. But this is to speculate about nature, rather than to deduce a logical conclusion.

So changing the axioms isn't changing the way think about math? — ssu

That's the first thing you've gotten right. And the fact that you will disagree is why you won't ever understand what I am saying.

Axioms don't define "the way we think about math." They define areas ("fields") within the framework of "how we think about math." There are three different fields that use contradictory Parallel Axioms (hyperbolic, elliptic, and Euclidean geometry), yet the way we think about them in math is the same. Each field is valid, and each Parallel Axiom is true within its field, and false in the others.. Each becomes a false statement (not an Axiom) in the others.

So are against something the idea that if something is inconsistent (in math/logic),it is false,...

I can repeat this as often as you ignore it, but I'm running out of ways to make it sound different from what you've ignored before. You keep using the indexical word "something" without indicating what it refers to, the statement or the set. Some part of what you say is clearly wrong each time you use the word, but how it is wrong depends on what you mean. And if you understand the difference.

Even if you prove that a ****SET**** of Axioms is inconsistent, that does not mean that any one Axiom in it is "false." And a ****SET**** can't be false, just inconsistent.

Qualities a *****SET***** of Axioms can have include "consistent" and "inconsistent," but not "true" or "false."

Qualities a ****STATEMENT**** can have include "true in the context of a ****SET**** of Axioms," and "false in the context of a ****SET**** of Axioms," but not "true outside the context of a ****SET**** of Axioms," of "false outside the context of a ****SET**** of Axioms."

An Axiom is a special kind of statement that assumed to be true to define the ****SET***. The only quality an Axiom can have is ""true because we assume it in this ****SET**** of Axioms." Outside the context of any ****SET**** - that is, in the absolute or universal sense you deny you are using - it is neither true nor false. It is merely a set of words.

That's the first thing you've gotten right. And the fact that you will disagree is why you won't ever understand what I am saying. — JeffJo

At least I'm trying to understand your point. (Which you think is impossible, I guess)

There are three different fields that use contradictory Parallel Axioms (hyperbolic, elliptic, and Euclidean geometry), yet the way we think about them in math is the same. — JeffJo

That geometry is different in two dimensions and more dimensions is evident yes. Yet we do speak of Geometry, even when there is Euclidean and non-Euclidean geometry.

Qualities a *****SET***** of Axioms can have include "consistent" and "inconsistent," but not "true" or "false." — JeffJo

Fine. So in this case we will you just the definitions of "consistent" and "inconsistent".

So how much do you do with "inconsistent" axiomatic systems, or as you wrote, "a *****SET***** of Axioms" that is inconsistent?

At least I'm trying to understand your point. — ssu

There is no evidence of it.

What I think, is that you only try to see what contradicts your predetermined idea of universal truth. And then you say the first thing that comes to your mind, that seems to imply what I said as wrong. Example:

That geometry is different in two dimensions and more dimensions is evident yes. Yet we do speak of Geometry, even when there is Euclidean and non-Euclidean geometry.

This is a classic example of a strawman argument.

I described three different examples of two-dimensional geometry. They can be expanded into more dimensions. Yet I mentioned no such higher dimensions, and their existence is completely irrelevant. There are ***THREE*** ***DIFFERENT***, mutually contradictory versions of ***PLANE*** geometry. Each is a consistent field of mathematics. And the point is that there is no such thing as this universal, capital-G Geometry as you imply. Just consistent fields of geometry based on different sets of axioms. One axiom in particular is different in each field, making the other two false in that field. Yet none of them is true, or false, outside of a set of axioms that describes it.

If you truly are trying to understand my point, understand that one. And then try to see that your claim is preposterous, because it says the opposite. The claim that the Axiom of Infinity can be considered to be true, or false, outside of a field of mathematics that has either accepts it as true, or as false, without justification or proof.

So how much do you do with "inconsistent" axiomatic systems, or as you wrote, "a *****SET***** of Axioms" that is inconsistent?

?????
I don't "do" any quantity of whatever it is you are implying with them. I also don't suppose that they could be inconsistent because they contradict a "universal truth" that I want to others to accept as blindly as you do, and dismiss an individual Axiom in the set solely on the basis of that unsupported supposition. Which is exactly what you are doing.

I don't "do" any quantity of whatever it is you are implying with them. — JeffJo

Right. So you don't do anything with them. Well, neither do I.

And that was my point. But from the following it's obvious you don't get it.

I also don't suppose that they could be inconsistent because they contradict a "universal truth" that I want to others to accept as blindly as you do, and dismiss an individual Axiom in the set solely on the basis of that unsupported supposition. Which is exactly what you are doing. — JeffJo

No, the set of axioms are inconsistent when they aren't consistent with each other. You don't compare two different axiomatic systems to each other.

But perhaps for you even to mention that there is the law of non-contradiction is too much like an "universal truth", which you oppose.

A sample portion of an infinite list as proposed by Cantor.


s1 000001...

s2 101010...

s3 001100...

s4 100000...

s5 100001...

s6 100111...

s7 101100...

s8 000110...

s9 101010...


Diagonal sequence formed from list starting with s1, then s2, then s3, etc.


d1 110110...

d2 011101...

d3 111011...

...

If d1 is not in the list, neither is d2, and all that follow.
For every s there is a corresponding d missing.
Therefore half of the list is missing!

No, the set of axioms are inconsistent when they aren't consistent with each other. You don't compare two different axiomatic systems to each other. — ssu

Which is what I have been saying. When the set of axioms lead to an inconsistency, it is the set is that is inconsistent. No one axiom is inconsistent, or false. Nor is any one axiom inconsistent with another. The set itself is inconsistent.

And I never compared two systems to each other. I listed three different, consistent sets that include three contradictory versions of the parallel postulate. The point wasn't to compare the sets, it was to show that none of these parallel postulates could be called "true" or "false." But I'm beginning to suspect you know this, and are deliberately arguing around the point.

And you still have not demonstrated an inconsistency with Zermelo–Fraenkel set theory, You have supposed it could be inconsistent, and blamed it on the Axiom of Infinity possibly being false. Which is preposterous.

Which is what I have been saying. When the set of axioms lead to an inconsistency, it is the set is that is inconsistent. No one axiom is inconsistent, or false. Nor is any one axiom inconsistent with another. The set itself is inconsistent.

And I never compared two systems to each other. — JeffJo

Great, we both agree on something.

And you still have not demonstrated an inconsistency with Zermelo–Fraenkel set theory, You have supposed it could be inconsistent, and blamed it on the Axiom of Infinity possibly being false. Which is preposterous. — JeffJo

And notice the word "could". Could doesn't have the same meaning as is. I've only said it could be a possibility that in the future it is shown to be inconsistent. You see, there was a purpose for ZF - set theory to be made: It was to avoid the Paradoxes. It was made to avoid the pitfall that Frege's naive set theory had fallen to. I don't blame the axiom, in my view Infinity (and hence an axiom for it) is an integral part of mathematics. All I've said that we haven't understood infinity well. Even if ZF doesn't directly answer Cantor's hierarchial system of ever larger infinities, it's still there. Yet how much has there been use for Aleph-2, for Aleph-3, or Aleph-4? Cantor, a very religious man, thought that there could be an Absolute Infinity, but that was only for God to know. All I'm saying is that there could be surprises and new insights in this issue.

So please understand my point of view: we have gotten new insights on mathematics in history and our understanding of math has greatly changed from what it was during Ancient times and what it is now. Hence what is preposterous is then to think that a) no new insights will be made in mathematics in the future and b) these new insights won't change our understanding from the one we currently have. In science we admit this and talk just about theorems.

Yes, you might argue in your formalism that then these new insights would be just are new axiomatic systems separate of others. But if something is shown to be inconsistent, some people would dare to say there's something wrong then, it's false. Some would even dare to say that it would change our understanding of math as ZF is commonly seen (at least by some) as part of the foundations of mathematics. I do understand your point if you disagree with this, but still argue that this is a philosophical disagreement we have here.

So the basic argument we have had has been about inconsistency and falsehood.

we have gotten new insights on mathematics in history and our understanding of math has greatly changed from what it was during Ancient times and what it is now. Hence what is preposterous is then to think that a) no new insights will be made in mathematics in the future and b) these new insights won't change our understanding from the one we currently have. — ssu

Not only do the foundations shift, but mathematics rolls along like a giant intellectual snowball, gathering layer after layer of new concepts and theory, a plethora of results that can be bewildering even to an expert in a specific area. I was in a classical area, complex analysis, for years, and still dabble in elementary research, but these days I can hardly understand the titles of papers in that subject.

And notice the word "could". Could doesn't have the same meaning as is. I've only said it could be a possibility that in the future it is shown to be inconsistent. — ssu

Did you notice the word "could" also? Anything "could" happen.The sun could explode tomorrow, ending life as we know it. Do you discuss that possibility? I personally am not even considering emptying my bank account, to use it before I lose it. mathematics is not about what "could" be true, it is about what is shown to be true within a given framework of axioms.

And did you understand that finding that a ****SET**** of axioms is inconsistent does not mean that any one axiom in the set is "false"? Or that claiming that an axiom could be false is an oxymoron?

I don't blame the axiom, in my view Infinity (and hence an axiom for it) is an integral part of mathematics. All I've said that we haven't understood infinity well.

And all I've said is that this is a nonsensical statement. You, on the other hand, said:

Axiom of Infinity is anything but established and self-evidently true. The discussion here ought to show it. — ssu

That would be a pointless discussion. An axiom is not, and cannot be, inherently "self-evidently true." We cannot "prove" it, and no amount of discussion will shed any light on it. It is because it cannot be shown to be self-evidently true, or false, that we assume it is self-evidently true. So we can lay the groundwork for a specific field of mathematics.

we have gotten new insights on mathematics in history and our understanding of math has greatly changed from what it was during Ancient times and what it is now.

The insights you refer to all apply within a particular set of axiom, The only insights on mathematics in general that we have gained, are how to demonstrate that a field is internally consistent, and that there is no such thing as truth *in* a field from supposed truths *outside of* that field.

Not only do the foundations shift, but mathematics rolls along like a giant intellectual snowball, gathering layer after layer of new concepts and theory, a plethora of results that can be bewildering even to an expert in a specific area. I was in a classical area, complex analysis, for years, and still dabble in elementary research, but these days I can hardly understand the titles of papers in that subject. — jgill

I agree.

Even if math follows it's own logic (no pun intended), it's still something that people do and it does evolve. Early 19th Century Mathematics and present day mathematics are taught differently and are different, even if a large part is totally same. It would be naive to think that the intellectual snowball, as you put it, would now stop. Obscure fields of mathematics can become important once people can use the field to build models and compute things. As one physicist once remarked, he just hopes math will give us new tools to use. I'm optimistic that those new tools will be invented/found.

Even if math follows it's own logic (no pun intended), it's still something that people do and it does evolve. — ssu

Nobody has said otherwise. (Well, other than "what people do" is completely ambiguous.)

What was said, is that Math accepts no absolute truths.The entire point is that there should be no need to discuss what is, or is not, self-evidently true.

Math says "If the statements in the set of axioms A are accepted as true, then the statements in the set of theorems T follow logically from them. The point of the evolution you misinterpret is to determine if the set T is internally consistent. If it is not, then the set A is invalid as a set of Axioms. But no one axiom in A is invalid, and any one of them can be in another set A' that is consistent.

Look, we've got your point, Jeffjo. Hope you would get the point of others too and not just make strawman arguments.

The point of the evolution you misinterpret is to determine if the set T is internally consistent. — JeffJo

No. It's the inconsistency between two or more axioms in the axiomatic system, which make the system inconsistent. Your assumptions what others think are incorrect here.

What was said, is that Math accepts no absolute truths. — JeffJo

Hmm. And in just what category would you put your idea presented here btw? :wink:

I think you fail to get the point so this discussion isn't productive. There is a thing called the philosophy of mathematics and there are various schools of thought in philosophy of math, you know.

Look, we've got your point, Jeffjo. — ssu

No, I really don't think you do. Or at least, you have shown no evidence of it.

The point of the evolution you misinterpret is to determine if the set T is internally consistent. — JeffJo

No. It's the inconsistency between two or more axioms in the axiomatic system, which make the system inconsistent.

And how is that not what I said?

But my point, that you have not shown you understand, is that finding such an inconsistency does not mean any of these two-or-more axioms is false.

There is a thing called the philosophy of mathematics and there are various schools of thought in philosophy of math, you know.

And it is that there are no pre-determined truths, only truths that follow from one's axioms which are assumed to be true without proof.

And in just what category would you put your idea presented here btw

It is a statement about philosophy, not a statement in math. "True" statments in math are either axioms, or theorems that follow from axioms. Unlike what you want here:

Axiom of Infinity is anything but established and self-evidently true. The discussion here ought to show it. — ssu

Look, we've got your point, Jeffjo. — ssu

No, I really don't think you do. Or at least, you have shown no evidence of it.

The point of the evolution you misinterpret is to determine if the set T is internally consistent. — JeffJo

No. It's the inconsistency between two or more axioms in the axiomatic system, which make the system inconsistent. — ssu

And how is that not what I said? — JeffJo — JeffJo

Yeah, Jeffjo, how isn't it what you said? (Hint: see first line in the quote above)

It is a statement about philosophy, not a statement in math. — JeffJo

Great! So you admit that what you said was a philosophical statement.

FYI, it's the Logic & Philosophy of Mathematics part of the Philosophy Forum, so you shouldn't be surprised that we debate here questions of philosophy.

Now even if the string is infinite in length it will still terminate on a multiple of a 1/2. — Umonsarmon

Now I measure the distance from A to E. This distance will be some multiple of a 1/2 x some a/b

We know this has to be the case because we are always dividing our distances by 1/2 so the final distance will be some multiple of a 1/2 x a/b. This will be true even if the number has an infinite number of digits — Umonsarmon

Here is where your mistake lies. The numbers a and b can become ever bigger as the string gets longer. When the string gets endlessly long, so can the numbers a and b. But for a/b to lie in the countable set of rationals, a and b must both be finite. Therefore, what you say is untrue for some strings of endless length. That’s the case for irrational numbers, such as the square root of 2. The latter’s irrationalness can be very easily proven, which was done more than two thousand years ago. In particular, the distance corresponding to sqrt(2) is not a rational multiple of ½.

Cantor’s diagonal proof, on the other hand, is perfectly sound. So, Cantor most certainly is right after all.

I am not so sure Tristan. You are assuming a priori that countable infinities are not equal to uncountable. That's one of your premises, but it's left unproven. IF it's true, the diagonal proof works, but not otherwise. Aristotle said all infinities are equal, so i've read

Four infinites, from our pespective anyway for each corner of a square a symmetrical infinity.

There can potentially be more or less. I would say four is a powerful number.

Infinity is also a wider concept of a beginning and middle and end.

It's end being that which is progression from a point that can continue infinitely, or it's a illogical theory of infinity. Such as a breathing, it reaches a point and that's it's end, but breathing continues because the shape is infinite.

A Klien bottle is a type of abstract infinity of air.

The proof that sqrt(2) is irrational and thus not a rational multiple of 1/2 doesn't depend on the existence of uncountable infinities or their difference from countable ones. Therefore, the OP Umonsarmon’s argument is invalid and flawed.

Regarding uncountable infinities:
First of all, “uncountable” means “not countable” by definition, so what you want to say likely is that I am assuming the existence of uncountable infinities. That’s not the case at all. Cantor proved that the set {0, 1}^IN of all functions from the set IN of natural numbers to the finite set {0, 1} is uncountable in the sense that there cannot be a bijection between {0, 1}^IN and IN. So, the only thing that I assume is that IN and {0, 1}^IN exist, and with this assumption, the diagonal proof works. More generally, the diagonal proof shows that if every set has a power set (which is a very reasonable assumption that does not postulate the existence of infinities, let alone differences between them), then there is no bijection between that power set and the original set. Thus, since there is obviously an injection from the original set to its power set, the power set is bigger than the original one. That’s how size for sets is defined. Therefore, if there is any infinite set at all (and we can be pretty sure that IN exists and is infinite), then the power set of that infinite set must be infinite and bigger than the original set.

I trust this hard mathematical, logical and sound reasoning more than what some philosopher who advocated severe ethnocentrism and a geocentric world view and who rejected atomism said more than two thousand years ago (that’s an attack aimed not at you, but at Aristotle. I find your search for unproven premises very useful and important, and I would like you to correct me if my reasoning regarding different infinities contains other premises which I’ve not explicitly stated or if my reasoning went wrong somewhere). In fact, Aristotle even ruled out the existence of an actual infinite on the grounds that such a thing would be “bigger than the heavens” – a very logical argument (scoffing), which also shows that Aristotle was thinking in much too concrete a way (mathematical objects are abstract and cannot be compared in size to the heavens any more than oddness can be compared to an orange). However, as long as every natural number is real and has a successor bigger than itself and all smaller natural numbers, the set of all natural numbers is also real and actually infinite. In fact, being abstract, it is at least as real as every concrete object.

Summing up, the diagonal proof that there are different infinities only rests on the premise that there is an infinite set which has a power set. Do you agree?

No, by definition, x is uncountable if and only if x is not countable. That's simply a definition of 'uncountable'.

Then we prove that there do indeed exist sets that are uncountable. And we prove that, in particular, the set of real numbers is uncountable.

The diagonal proof does not assume what it proves. Rather, it assumes some quite basic axioms of set theory, then it proves from those axioms that the set of real numbers is not countable (i.e. that it is uncountable). One may wish not to accept the axioms used, but that is a different matter.

No, I don't agree with your argument. The odds numbers don't line up with the whole numbers (you say), but you say they are equal infinities. You can prove "uncountable" infinities don't line up with the whole numbers either, but maybe they are equal as well. Until you prove that "uncountable" cannot be lined up with the wholes you haven't proven Cantor right. The diagonal shows that there are numbers not in the wholes, but there are evens not in the odds. I don't see the argument for why you can't just start at zero and line any infinity up with any other

The odds numbers don't line up with the whole numbers (you say), but you say they are equal infinities. — Gregory

We DEFINE them to have equal cardinality because there is at least one way of matching them up bijectively.

You can prove "uncountable" infinities don't line up with the whole numbers either, but maybe they are equal as well. Until you prove that "uncountable" cannot be lined up with the wholes you haven't proven Cantor right. — Gregory

Again, this follows by definition. If a countable set is defined as one that can be bijected to the natural numbers, then BY DEFINITION an uncountable set is one that can't. So if a set is uncountable, by definition it can not be bijected with the naturals.

That doesn't in and of itself prove that there ARE any uncountable sets; only that if there were one, it could not be bijected to the naturals. And of course Cantor proved that the reals are one such uncountable set.

How much math must one know to understand this Catorian proof? It seems to me infinity is everywhere and nowhere, speaking of abstract infinity that is. You might not know how to start a bijection of the reals to the wholes, but I say start with any member, and then another and so we have bijection to 1 and 2. Send them off infinity like you do comparing whole to odd, and walla we have Aristotle's result

How much math must one know to understand this Catorian proof? — Gregory

Virtually none. There are proofs all over the Internet.

The diagonal argument is the one people usually see. But I think the proof of Cantor's theorem is simpler and more beautiful. It shows by way of a short and simple argument that there can never be a bijection between a set and its powerset.

So the powerset of the natural numbers must be uncountable. It's not difficult to show that there is a bijection between the powerset of the naturals and the set of real numbers (think binary strings) so this shows that the reals are uncountable without the confusion usually generated by the diagonal argument.

Fishfry, if the set is infinite, it's like saying there are the same infinity of points in the pineal gland as in the whole body. That's how it appears to me. I just know infinity from Hegel. He says the finite is the infinite thrown from itself. For him the infinite must be one, not four or whatever

Fishfry, if the set is infinite, it's like saying there are the same infinity of points in the pineal gland as in the whole body. That's how it appears to me. I just know infinity from Hegel. He says the finite is the infinite thrown from itself. For him the infinite must be one, not four or whatever — Gregory

I'm afraid I don't know Hegel. I've heard he's difficult to read. I'm not much for the classical philosophers, my limitation. That said, I think it's better, when studying mathematical infinity, to put aside prior notions of philosophical conceptions of infinity.

Mathematical infinity starts from the counting numbers 0, 1, 2, 3, ... that we all have an intuition of as being unending. Given that, there are interesting things we can say. But none of this is intended to resolve any philosophical issues regarding being or the world or heavy things like that. It's only math.

If you can take the math on its own terms, the study of mathematical infinity is interesting and beautiful. Those are the criteria for what mathematicians care about.

It's true that Cantor himself believed that after Aleph-0, Aleph-1, and so forth, was an "absolute infinity" that he called God. Today, Cantor's religious beliefs are not much remembered except historically.

All in all, when approaching this material it's better to put aside all philosophical preconceptions. Mathematical infinity is not attempting to resolve any philosophical issues about the world or God or metaphysics.

In particular, there's no reason to believe the world is made up of dimensionless mathematical points in the same way the real number line or Euclidean space are. The question of how many points are in your pineal gland is meaningless. The pineal gland is made up of organic molecules, or atoms, or electrons, or quarks, or quantum probability waves; depending on the level of discourse. But nothing in the body, or in the world, is made up of mathematical points. Mathematical points are purely conceptual entities, like justice; or fictional entities like chess pieces.

On the subject of religion, will two people in heaven have less eternity than three? I feel like bijection is invalid. With the natural numbers, you have to step every odd numbers back in order to biject them and who knows what that does to the infinity on the other side. I'm probably not smart enough to understand the coming response,.but I like this subject.

On the subject of religion, will two people in heaven have less eternity than three? — Gregory

Is this for me? Can you please Quote a bit of my text by selecting it and hitting the Quote button that appears? That way I get notified rather than having to keep coming back to the thread. Thanks.

I just said that mathematical infinity has nothing to do with heaven or eternity. Or people for that matter. Or religion. So if this post was for me, I surely don't have any idea.

I feel like bijection is invalid. With the natural numbers, you have to step every odd numbers back in order to biject them and who knows what that does to the infinity on the other side. — Gregory

In 1638 Galileo noted that you can put the whole numbers 0, 1, 2, 3, 4, ... into 1-1 correspondence with the perfect squares 0, 1, 3, 9, 16, ... You can see that this is true, right? Without overthinking it. You can line up the whole numbers and line up the square numbers and connect them with lines such that every whole corresponds to a unique square and vice versa. Without trying to figure it out or overthink it, you can see this is true, right?

I'm probably not smart enough to understand the coming response,.but I like this subject. — Gregory

We're all on that path. I'm not smart enough to understand all the math I wish I knew but I like to read about it anyway. Remember Wiles spent seven years of evenings after his regular day job as a math professor, working on Fermat's last theorem. Seven years of confusion and hard work and struggling to learn all the areas of math he needed in order to figure the thing out. Math is beyond everyone that way. You have to work at it. As Euclid said when the King asked him for an easy way to understand Euclid's great book The Elements: "Sire, there is no royal road to geometry."

Liking it's a good place to be.