Infinity and 0

[studiot]

OK I'll have another go.

 

 

Which of these is closer to your claim:

 

d.

 

 

 

Now I have answered a question of yours please try one of mine.

 

Take the collection of set axioms of your choice (pun intended for your pun collection)

Remove the axiom of infinity.

 

Now my question is

 

Is the reduced collection of axioms

 

a) Compatible with the existence of infinity

 

b) Incompatible with the existence of infinity

 

 

 

I see this as equivalent to the question

 

Given that Mr Jines has never lived but is only a character in an excercise book.

 

Is the perimeter of Mr Jines' garden

 

a) 160 feet

 

b) Indeterminate

 

c) Some other value.

Edited April 25, 2016 by studiot

[wtf]

d.

Ok, then what is your position exactly? You said I don't understand what you're saying -- and I agree. But when I ask you directly to state your thesis, you don't. That's why I don't understand your thesis. You have not clearly stated it.

 

 

 

Take the collection of set axioms of your choice (pun intended for your pun collection)

Remove the axiom of infinity.

For purposes of this question, I'll start with standard ZF and remove AxInf. So I'm working in ZF minus AxInf.

 

Now my question is

 

Is the reduced collection of axioms

 

a) Compatible with the existence of infinity

No, in ZF minus AxInf, the nonexistence of an infinite set is provable.

 

But I do want to make one refinement. You say "infinity" but that's imprecise. I am using the phrase, "an infinite set," and that is precise. Under the negation of AxInf, an infinite set does not exist, provably.

 

So please define what you mean by "infinity." Do you mean an infinite set? Or do you mean something else, and if so, what?

 

b) Incompatible with the existence of infinity

The negation of AxInf is incompatible with the existence of an infinite set.

 

Since you haven't defined what you mean by "infinity," I can't answer.

 

 

Regarding your question about Mr. Jines, I take it you are referring to factual assertions about fictional entities. For example, Ahab is captain of the Pequod. True. Ahab is the cabin boy. False. So we can make true/false claims about fictional entities. But please, let's focus on what you mean by "infinity," because I'm starting to suspect that when you say "infinity" you mean something OTHER than an infinite set; and that would indeed cause confusion until you define what you mean.

Edited April 25, 2016 by wtf

[studiot]

 

No, in ZF minus AxInf, the nonexistence of an infinite set is provable.

 

Can you outline or link to the proof please?

 

Since you want me to define infinity, I take it you are prepared to define an infinite set?

[wtf]

Can you outline or link to the proof please?

https://en.wikipedia.org/wiki/Hereditarily_finite_set

 

Also: http://plato.stanford.edu/entries/set-theory/, from which I quote:

 

The theory of the hereditarily-finite sets, namely those finite sets whose elements are also finite sets, the elements of which are also finite, and so on, is formally equivalent to arithmetic.

 

Also http://www.isa-afp.org/entries/HereditarilyFinite.shtml, from which I quote:

 

An HF set is a finite collection of other HF sets; they enjoy an induction principle and satisfy all the axioms of ZF set theory apart from the axiom of infinity, which is negated. All constructions that are possible in ZF set theory (Cartesian products, disjoint sums, natural numbers, functions) without using infinite sets are possible here.

 

 

 

 

 

 

 

 

 

Since you want me to define infinity, I take it you are prepared to define an infinite set?

There are two common definitions of an infinite set.

 

* (Infinite) A set is called finite if it's bijectable to some natural number. Otherwise it's called infinite.

 

* (Dedekind infinite) A set is Dedekind-infinite if it's bijectable to a proper subset of itself.

 

https://en.wikipedia.org/wiki/Dedekind-infinite_set

 

There are some subtleties and considerations I'd like to mention.

 

* What do I mean "bijectable to a natural number?" In what way is the set {Socrates, Plato} bijectable to the number 2? Good question! The answer is that in set theory, the number 0 is defined as the empty set; the number 1 is defined as the set containing 0; the number 2 is defined as the set containing 0 and 1, and so forth. So in fact 2 = {0,1}, a set that conveniently has exactly 2 elements. This definition by the way is due to John von Neumann.

 

Using this definition, we can implement a model of the Peano axioms within ZF, with or without AxInf. Without AxInf we have 1, 2, 3, ... but not a set containing them all. AxInf allows us to write {1,2,3,...}, a set that contains all the natural numbers. As the references I linked indicate, we don't need an infinite set to do arithmetic and mathematical induction.

 

* If we assume the axiom of choice (AC) then a set is infinite if and only if it's Dedekind-infinite. But strangely, if we assume the negation of AC, these definitions are not equivalent. That is, in the absence of AC, there is a set that is not bijectable to any natural number; yet still has no bijection to a proper subset of itself.

 

[in passing I note that people sometimes dislike AC because it leads to counterintuitive results. But the negation of AC also leads to counterintuitive results. Pick your poison. We generally assume AC because it's convenient to do so.]

 

* Finally, a point I made earlier. A bijection is defined in set theory as a type of relation that maps a set to a set. Without an infinite set of natural numbers, you don't have a bijection from the natural numbers to a subset of themselves. In the absence of AxInf there is no infinite set and no Dedekind-infinite set.

 

Now, here I think is the nub of the matter.

 

Consider the collection (not set) 1, 2, 3, 4, 5, ... I think we agree intuitively that is an infinite collection, even in the absence of AxInf. It may not be an infinite set, but it's an infinite something.

 

In set theory we have a name for a collection that's "too big" to be a set: a proper class. So in ZF-minus-AxInf, the natural numbers form a proper class. In standard ZF, which includes AxInf, the natural numbers form a set. A more familiar proper class is the class of all sets. It's too big to be a set by Russell's paradox.

 

Aristotle made the distinction between potential and actual infinity. I believe that this is the exact distinction between an infinite collection and an infinite set. The sequence 1, 2, 3, ... is a potential infinity. The set {1, 2, 3, ...} is an actual infinity. This terminology is NOT used in math, only in philosophy; but perhaps this is the point you are making.

 

Hope this is helpful. If you are saying that even in the absence of AxInf, if we believe in the natural numbers then there is an intuitively infinite proper class, then I agree with you. But do note that the ultrafinitists don't even grant that much! They don't believe in the full collection of natural numbers.

Edited April 25, 2016 by wtf

[studiot]

Written whilst you were still editing.

 

Thank you for post 29, it is most interesting and illuminating, so I hope you won't take my comments the wrong way.

 

Not in any special order.

 

 

I have been discussing this topic with a couple of pure math professors elsewhere, who very definitely use real and potential infinities, however see my SCIAM link.

 

Your definition (1) of an infinite set is the one I usually adopt, except that I have seen it said that since 'infinite' can possibly be misinterpreted the use of either not-finite or transfinite is preferred.

 

I have heard (and the SCIAM link attributes) your definition (2) attributed to Cantor, not Dedekind, though I understand Cantor was good at acknowledging the work of others, unlike some.

 

As far as I can see, you have not proven that ZFC minus the axiom of infinity disbars the existence of an infinite set, merely that it does not provide one.

I did wonder if your statment referred to the 2010 Cohen proof?

 

http://www.scientificamerican.com/article/infinity-logic-law/

[wtf]

Written whilst you were still editing.

Just getting the quoting right takes all I've got!

 

Thank you for post 29, it is most interesting and illuminating, so I hope you won't take my comments the wrong way.

Not at all, I got some of my own thinking straight too. In particular, I realized that without AxInf, the natural numbers form a proper class in ZF-minus-AxInf. However that still would not satisfy the ultrafinitists.

 

I have been discussing this topic with a couple of pure math professors elsewhere, who very definitely use real and potential infinities, however see my SCIAM link.

That link is about extremely advanced set theory and has no relevance to our conversation. The state-of-the-art thinking about the Continuum hypothesis is far beyond anything we're talking about.

 

 

 

I have heard (and the SCIAM link attributes) your definition (2) attributed to Cantor, not Dedekind, though I understand Cantor was good at acknowledging the work of others, unlike some.

 

Here's the Wiki page for Dedekind-infinite_sets.

Whether this is a case of the wrong person getting credit I cannot say. The Wiki page says Dedekind came up with the idea and/or name in 1888, so I imagine he and Cantor must have exchanged ideas.

 

 

As far as I can see, you have not proven that ZFC minus the axiom of infinity disbars the existence of an infinite set, merely that it does not provide one.

You're correct. I owe you that proof that if there's no inductive infinite set (which is what AxInf gives) then there's no infinite set at all. But the natural numbers (as I defined them earlier) are an inductive set; and the negation of AxInf says there is no inductive set. So ~AxInf definitely says that there is no set that models the natural numbers. Let me think about the full proof that there can be no infinite set at all, not even a non-inductive one.

 

 

I did wonder if your statment referred to the 2010 Cohen proof?

Paul Cohen died in 2007 though his spirit lives on. Not clear what you're referring to. In any event none of this is related to Cohen's famous proof of the independence of AC and CH, which uses advanced set theory and is not related to our much more elementary concerns.

 

 

 

 

 

 

 

 

 

 

 

 

 

http://www.scientificamerican.com/article/infinity-logic-law/

That article refers to the absolute bleeding edge of modern set theory, and is not relevant to our concerns. We are in ZF and at most ZFC. The theories they're talking about involve the addition of many so-called large cardinals, which go far (far far far ...) beyond any sets we are contemplating. Large cardinals are sets that are too big to live in ZFC and require new axioms for their existence. What that article is about is the question of how many and what type of large cardinals can you add to ZF without making set theory inconsistent, yet resolving the Continuum hypothesis. Woodin has an idea but his proof's not even finished as far as I know.

 

An analogy would be that we are discussing the fact that molecules are made of atoms and atoms are made of protons, neutrons, and electrons. And the SciAm article is about the most advanced as-yet-unproven work in string theory. There's a gap of a century of professional research between atoms and strings, and between elementary ZF and the type of set theory that article is discussing.

 

 

 

(Added later ...)

 

Ok, about that proof I owe you ... it's trickier than it looks.

 

The question is whether the negation of AxInf, which says there is no infinite inductive set, implies that there is no infinite set.

 

First, proof by Wiki. From https://en.wikipedia.org/wiki/Infinite_set:

 

The set of natural numbers (whose existence is postulated by the axiom of infinity) is infinite. It is the only set that is directly required by the axioms to be infinite. The existence of any other infinite set can be proved in ZermeloFraenkel set theory (ZFC) only by showing that it follows from the existence of the natural numbers.

 

Of course that's not actually a proof, it's a claim without a proof. And it's interesting that they explicitly use ZFC (ZF plus the axiom of choice) and not just ZF. Could it be that this proof requires choice?

 

 

I found this lovely little proof here. It's Henning Makholm's comment to the original question.

 

Theorem: There is no infinite set in ZFC-AxInf.

 

Pf: If X is any set in ZFC-AxInf, by the axiom of choice we can well-order X. That means that X is order-isomorphic to some ordinal; and in particular, X is bijective to some ordinal.

 

Since ~AxInf says that there is no infinite inductive set, it follows that there are no infinite ordinals. (Inductive sets are ordinals). Therefore X must be bijectable to some finite ordinal, ie a natural number.

 

QED

 

In order to drill that proof down to include a detailed explanation of ordinals and inductive sets, the post length would quickly get out of hand. I started doing that then realized it would just be another way-too-long post. Perhaps it's better if I leave it as-is and respond to specific questions.

 

To sum up: There are no infinite sets in ZFC-AxInf.

 

Now what about ZF-AxInf? That's ZF minus AxInf and also without the axiom of choice. Well, I can't find a proof and I can't think of one!

 

I wonder if perhaps it's not even true. When you take infinite sets without choice, you get weird consequences, such as an infinite set that's not Dedekind-infinite. Perhaps there's some pathalogical counterexample in ZF-AxInf of an infinite set. I will continue to look around.

Edited April 26, 2016 by wtf