Cantor avoids the paradox simply by having larger and larger infinities and not referring to the set of sets as you do above.

The paradox is rigorously avoided also by the axioms of ZF-logic. Some of the axioms are there basically only to deal with the paradox.

Of course when you think of it, the issue of the nature of infinities is open as there is no true answer to the Continuum Hypothesis, which basically shows we don't have any idea of just how infities work and we simply take it as an axiom as it's very useful in math.

Axiom of restricted comprehension sorts that out. You can't form sets out of arbitrary predicates, the only time you can form a set using a predicate is when you apply a predicate to a set which has been established to exist by other means.

Cantor avoids the paradox simply by having larger and larger infinities and not referring to the set of sets. — ssu

A reasonable, working definition of infinity:

‘A number bigger than any other number’

It is clear then that there can only be one such number - if there was a second infinity then both would have to be larger than the other. Once it is excepted that there can be only one infinity, transfinite maths falls apart.

The paradox is rigorously avoided also by the axioms of ZF-logic. Some of the axioms are there basically only to deal with the paradox. — ssu

Exactly. Its a hack. Infinite sets are unmeasurable so they do not have a size.

A reasonable, working definition of infinity:

‘A number bigger than any other number’ — Devans99

Well… what is your definition of a number? Numbers you see are used to measure something and when you have something that isn't measurable / countable, you have bit of a problem.

It is clear then that there can only be one such number - if there was a second infinity then both would have to be larger than the other. — Devans99

Well, I'm a proponent of Absolute Infinity, but before going into that, a question:

So what do you then think about Cantor's finding that there are more real numbers than natural numbers? Or said another way, that you cannot put into 1-to-1 correspondence the real numbers with the natural numbers, as you can put the rational numbers with the natural numbers?

This is the cornerstone observation that lead Cantor to argue that there are bigger and bigger infinities.

Well… what is your definition of a number? Numbers you see are used to measure something and when you have something that isn't measurable / countable, you have bit of a problem — ssu

I'd associate size with something measurable. If its unmeasurable, I would not try to measure it. Unmeasurable things can have no size. Infinite sets have no cardinality. Making up fictitious numbers is not the way to go. For example, the rules of transfinite arithmetic assert that:

∞+1=∞

This assertion says in english:

’There exists something that when changed, does not change’

That is deeply illogical.

Or:

∞/2=∞

'The whole is greater than the parts' is flaunted.

So these fictitious numbers, the transfinites, have no basis in logic IMO.

So what do you then think about Cantor's finding that there are more real numbers than natural numbers? Or said another way, that you cannot put into 1-to-1 correspondence the real numbers with the natural numbers, as you can put the rational numbers with the natural numbers? — ssu

I do not agree with the bijection procedure as a valid way to compare two things. It produces results like the size of the set of naturals is the same as the size of the set of rationals or the number of squares is the same as the number of non-squares (Galileo’s paradox):

- For each natural number there is clearly an infinite number of rationals so the two sets cannot be the same size
- By induction we know that for any reasonably sized sample, the number of non-squares is greater than the number of squares.

How can a procedure that is meant to demonstrate equality produce such obviously wrong results? It is because infinity has no size (it is unmeasurable) so it is impossible to compare the size of infinite sets.

Sets are just ideas. They're something we made up. Can we make up ideas that run into consistency problems? Sure. And then we can make up modifications or restrictions to avoid the consistency problems.

Sets are just ideas. They're something we made up. Can we make up ideas that run into consistency problems? Sure. And then we can make up modifications or restrictions to avoid the consistency problems. — Terrapin Station

But if you start with clean ideas (non contradictory axioms) you get clean theories.

Set theory is all the fudges and hacks because the axiom of infinity is wrong - infinite things exist in our minds only, they cannot have real existence.

Only a madman would claim infinity is measurable. It has no size. Infinite sets do not have a cardinality.

But if you start with clean ideas (non contradictory axioms) you get clean theories. — Devans99

You can try to be careful about avoiding contradictions, but they can creep up unexpectedly. The more complex any "system" is, the more likely it is to have contradictions.

We are in the modern day, awash with complexity. Maths and science are layer upon layer of complexity. Some of the foundations are wrong; that is where the some of the complexity comes from. If something is wrong in the foundation and it is not spotted, people over elaborate in higher layers to compensate. I've seen this effect 1st hand in computer systems, but it applies equally to maths/physics/cosmology as well.

Infinite sets have no cardinality. — Devans99

Wrong. They do. The cardinality of aleph-null and aleph-1 is not the same.

I do not agree with the bijection procedure as a valid way to compare two things. — Devans99

Really?

So the bijection 1+2=3 you don't agree with? That isn't valid?

How can a procedure that is meant to demonstrate equality produce such obviously wrong results? It is because infinity has no size (it is unmeasurable) so it is impossible to compare the size of infinite sets. — Devans99

Because the proof is a reductio ad absurdum proof.

It has no size. Infinite sets do not have a cardinality. — Devans99

So now you are dismissing totally set theory. Good luck with that.

Ok. Is infinity bigger than 54? Does 54 have size? No?
If 54 has size, then where does infinity loose it?

You see it's two different thing to a) have 'a size' and b) to be measurable. You see, unmeasurability doesn't make other things dissappear.

I'll take another example with the Sorites Paradox. If from a heap of sand you start taking a single grain at a time, at what exact point does it cease to be considered a heap?

Ok, so the answer is that the term 'heap' is vague. But what you do with your argument that there is no size, would be similar to say that there is no weight with the heap. That if you have a grain of sand it has a weight and add one or more to it, the weight increses, however with 'heap' the weight would dissappear as...it's definition vague and hence unmeasurable.

So the bijection 1+2=3 you don't agree with? — ssu

I don't believe you can use bijection with infinite sets - infinite sets are by definition not fully defined; IE they are UNDEFINED and you cannot operate with them logically.

So now you are dismissing totally set theory. Good luck with that. — ssu

Finite set theory is OK. The rest is shot through with holes. The definition of a set is polymorphic:

- A finite set may be specified as a list of items
- A infinite set maybe specified by selection criteria such as ‘all real numbers’

However, this is not a valid polymorphism. An infinite set is not a-kind-of finite set and vice-versa. The two object types have very different properties:

- An infinite set clearly does not have a cardinality property. Cardinality or size implies the ability to measure something. Infinity is by definition unmeasurable so infinite sets have no size property.

- A finite set has a completely defined list of members. An infinite set does not have this property.

These are very different types of objects; to try to treat them the same is like trying to force a square peg into a round hole. An infinite set is just a partial description of a set - it is the selection criteria for the set: ‘all natural numbers’ does not completely define a set, it just describes what type of objects go in the set. Contrast that to a finite set, which is fully described and defined.

It is never possible to fully define an infinite set - there is not enough paper in the world - so when working with infinite sets we are always working with a partly defined IE UNDEFINED objects. This is one of the reasons why so many paradoxes occur with infinite sets - they are not fully defined logical entities.

What has been done in set theory is an abomination to the principles of sound design; instead of treating finite and infinite sets as different objects each having different operations and properties, Cantor simply made up fictitious numbers (the transfinites) to represent the nonexistent cardinality property of infinite sets.

Ok. Is infinity bigger than 54? Does 54 have size? No?
If 54 has size, then where does infinity loose it? — ssu

Infinity only exists in our minds, not in reality. Conceptually (and it is a broken concept) infinity is bigger than everything but has no size itself.

What has been done in set theory is an abomination to the principles of sound design — Devans99

Well, you simply have to prove it in mathematics. If you show that either all or some the axioms of ZF are incorrect, then that is that's a positive breakthrough.

Infinity only exists in our minds, not in reality. — Devans99

Does the number 54 exist in reality? Show me where the real 54 is.

Besides, I think infinity is used a lot in math and is a very useful, very logical mathematical object, which is inherent to mathematics in order for it to be logical.

It is never possible to fully define an infinite set - there is not enough paper in the world — Devans99

REALLY? You think that defining something in math is something like 'writing it down'?

This is a very illogical idea: there isn't enough paper to write the finite numbers from 1 to googolplex, hence your reasoning also states that big finite numbers aren't possible. That's illogical.

Well, you simply have to prove it[/i] in mathematics. If you show that either all or some the axioms of ZF are incorrect, then that is that's a positive breakthrough. — ssu

Axioms are commonly evaluated by two criteria:

First, the axiom should not lead to logical inconsistencies in the system under question. I think the discussion above on transfinite arithmetic is enough to show that the axiom of infinity does lead to logical inconsistencies.

Second, axioms are chosen because they are inductively very likely to be true. We have strong reasons for believing in our axioms. A problem with axiomatically defining infinity to exist is that it is not clear that infinity exists:

1. We have no examples from nature of infinity
2. Constructing anything infinity large is impossible; not enough time
3. Constructing anything infinity small is impossible; one would never finish chopping it up
4. Basic arithmetic says infinity is not a number. If it were a number, it would be a number X greater than all other numbers. But X+1>X

Bearing in mind the above doubts, is the assumption of the existence of infinity a good axiom? A house rests upon its foundations. Set theory rests upon the decidedly shaky foundation of the axiom of infinity.

Does the number 54 exist in reality? Show me where the real 54 is. — ssu

54 stones can exist. An infinite number of stones cannot. Spacetime is a creation so it must be finite and discrete.

Besides, I think infinity is used a lot in math and is a very useful, very logical mathematical object, which is inherent to mathematics in order for it to be logical. — ssu

Potential Infinity (limits in calculus) is useful. Actual Infinity (transfinites) is not useful and misleads people. We have a whole bunch of cosmologists out there thinking that infinity is a grounded, logical mathematical concept when it is no such thing. They are wasting time on infinite universe models that could be better employed on the correct models (finite models).

REALLY? You think that defining something in math is something like 'writing it down'? — ssu

If you can't even say what size something is or iterate a complete list of members, I think calling it defined is a massive stretch and that stretch leads to paradoxes. The OP is just one of many paradoxes that go away once it is realised that infinity has no size. Set theory is rife with paradoxes because of infinity.

54 stones can exist. An infinite number of stones cannot. — Devans99

Stones can exist. Yet Again you have the same illogical idea here: two googolplex of stones cannot exist. And where in reality exists this '54'?

Potential Infinity (limits in calculus) is useful. — Devans99

Congratulations! You've made it to Aristotle with accepting potential infinity.

Set theory is rife with paradoxes because of infinity. — Devans99

The whole error is then to deny the existence of the paradoxes and think that everything in mathematics is fine and dandy if we a) don't approach this question or b) ban it.

The existence of the paradoxes show simply that our understanding of infinity is still lacking. Mathematics as a system isn't finished. We have still these issues with it that we cannot understand, just like the Greeks had problems with irrationals and at least for some time had this idea that all numbers were rational.

That's my view: paradoxes aren't a problem to be solved, but something that shows that the base of mathematics isn't yet finished. Paradoxes are more likely to be answers to be understood. And it's linked to our understanding of the infinite.

Stones can exist. Yet Again you have the same illogical idea here: two googolplex of stones cannot exist. And where in reality exists this '54'? — ssu

I'm not sure what you mean? Two googolplex of stones can exist IMO.

The existence of the paradoxes show simply that our understanding of infinity is still lacking. — ssu

My view is that paradoxes indicates that there is an error in the explicit/implicit assumption underlying the problem. In this case the problem is the assumption that an infinite set has a size that leads to the paradox. A paradox is just a contradiction so it is a form of proof via contradiction that infinite sets do not have sizes / infinity does not exist.

I'm not sure what you mean? Two googolplex of stones can exist IMO. — Devans99

At this level, it is estimated that the there are far less than a googol of atoms in the observable universe. As stones consist of more than one atom, obviously two googolplex of stones cannot exist.

My view is that paradoxes indicates that there is an error in the explicit/implicit assumption underlying the problem. — Devans99

I agree with this. We makes assumptions that are contradictory to each other. So what are we lacking? That's the interesting question.

A paradox is just a contradiction so it is a form of proof via contradiction that infinite sets do not have sizes / infinity does not exist. — Devans99

I have to disagree with you in this one. The set of natural numbers N does exist in the Mathematical realm. It is an infinite set as it surely isn't a finite set of numbers.

Does this have something to do with Godel's incompleteness theorems? Reminds me of Heisenberg's uncertainty principle - basically that we can have only one of two things, both which we desire.

Clearly the set of all sets does exist — Devans99

Beware the words 'clearly' and 'obviously'. When used, they are nearly always wrong. That is the case here. If you think otherwise, try to prove that a set of all sets exists!

Beware the words 'clearly' and 'obviously'. When used, they are nearly always wrong. That is the case here. If you think otherwise, try to prove that a set of all sets exists! — andrewk

I still maintain that infinity is unmeasurable so has no size - that is the real cause of most of the paradoxes of infinity.

At this level, it is estimated that the there are far less than a googol of atoms in the observable universe. As stones consist of more than one atom, obviously two googolplex of stones cannot exist. — ssu

OK fair point, but my meaning was if sufficient stones existed, the a googolplex of stones would be possible.

I have to disagree with you in this one. The set of natural numbers N does exist in the Mathematical realm. It is an infinite set as it surely isn't a finite set of numbers. — ssu

The set of natural numbers exists in our heads only; in does not and can cannot exist in reality. Infinity has conceptual existence in our minds but so do talking trees; so conceptual existence is not enough to prove real existence.

Godel's incompleteness theorems are discussed here:

https://thephilosophyforum.com/discussion/5602/the-liar-paradox-is-it-even-a-valid-statement

I still maintain that infinity is unmeasurable so has no size - that is the real cause of most of the paradoxes of infinity. — Devans99

It's fine for you to do that. But realise that most people do not share your opinion, so their beliefs will differ from yours. From what I have seen of your posts on infinity, the paradoxes you think you see stem from that belief, so they are not paradoxes for other people.

I still maintain that infinity is unmeasurable so has no size — Devans99

Sorry to butt in like this but I think the correct term is interminable and not unmeasurable. The difference is that the former captures infinity as a quantity while the latter seems to treat infinity as a quality.

I'm not a mathematician yet I feel Cantor didn't do anything so radical as to be unacceptable in his treatment of infinity. The principle of bijection (1-1 correspondence) is quite natural in mathematics of the finite. Cantor applied this basic principle to infinity and discovered some counterintuitive ''truths''.

If you wish to critique Cantor then you'll have to first prove that bijection (1-1 correspondence) is inapplicable to infinity. I'd like to hear your thoughts on that. Thanks.

OK fair point, but my meaning was if sufficient stones existed, the a googolplex of stones would be possible. — Devans99

So now it would be possible. But it isn't possible.

The set of natural numbers exists in our heads only — Devans99

Numbers exist in our heads only. And likely some animals use a mathematical system of "nothing, 1,2,3, many.) which is a totally functional system if you don't have an issue with or the need to count to something more than three. So likely this whole system of counting isn't only limited to humans. Yet in the physical realm there is no number 54. 54 doesn't exist physically. So it doesn't exist.

I've tried to make my point with the example of googolplex of anything that you cannot make the basis of mathematics in what physically exists in the universe or otherwise you end up with a totally illogical idea of finite numbers exist until they don't.

Mathematical proofs are done in the realm of mathematics and logic. Mathematical entities are not proven by starting to look at what we have observed by science and physics to exist. This is a very typical confusion I have noticed people have.

Hence the set of natural numbers exists in Mathematics. And it is an infinite set. And there is the axiom of infinity. And just disagreeing with this won't get you anywhere. You have to prove what is illogical in the mathematical system.

I still maintain that infinity is unmeasurable so has no size - that is the real cause of most of the paradoxes of infinity. — Devans99

Infinity has the size of infinity. An infinity. One infinity.

Infinite infinities? Let's just divide infinity in to infinite pieces.
But we're still dividing an infinity. One infinity.

Combination or unification, whichever you prefer, amounts to one.
Division amounts to many.

Sorry to butt in like this but I think the correct term is interminable and not unmeasurable. The difference is that the former captures infinity as a quantity while the latter seems to treat infinity as a quality. — TheMadFool

Infinity is not a number/quantity:

1. Basic arithmetic says infinity is not a quantity. If it were a quantity, it would be a quantity X greater than all other quantities. But X+1>X
2. Quantities are magnitudes or sizes. Infinity is not measurable so has no size; it is not a quantity
3. Numbers have a fixed value; that is their defining characteristic. Numbers are not variable. Infinity has no fixed value so is not a number.

Infinity is not a number/quantity: — Devans99

How about the set of odd numbers? We can count it as so: 1 one, 3 two, 5 three, 7 four, and so on. I don't see how quantification such as I described suddenly ceases to be one. Another thing I want to draw your attention to is time. We don't have to worry about its beginning but it seems to me that it'll extend into the infinity of the future. We do quantify time don't we? I don't see how it ceases to be a quantity in the infinity of the future.

One thing I do want to say is that when we talk of infinity we usually do so using mathematical infinities like the set of naturals/reals/integers, etc. Here's where I think the liar's paradox is relevant because it's a case of self-reference - numbers counting numbers. Do you think this self-reference is important? Does it result in the paradoxes we see in the math of infinities?

Personally, I don't think self-reference is problematic here because we're using bijection (1-1 correspondence) which means that we aren't really counting, just matching one element of a set with another element of another set.

We don't have to worry about its beginning but it seems to me that it'll extend into the infinity of the future — TheMadFool

I am mainly eternalist and finitist so I think time may have an end.

Do you think this self-reference is important? Does it result in the paradoxes we see in the math of infinities? — TheMadFool

The infinity paradoxes tend to be related to trying to measure the unmeasurable. Galileo's paradox (the number of squares is less than the number of non-squares yet each number has a square) is typical - there is no valid way to compare the size of two infinite sets and trying to do so yields contradictory results as Galileo noted.

Self-Reference is a large class of paradoxes. If it is recognised that the liar statement (and similar constructs) is not actually a statement, I believe this class of paradoxes will cease to be paradoxes.

eternalist and finitist — Devans99

This seems paradoxical: eternal and finite.

You can resolve that by considering eternity isn't bigger than itself; so it's finite.

You can resolve that by considering eternity isn't bigger than itself; so it's finite. — Shamshir

Difficult to understand but ok.