Wednesday, September 14, 2011

Intuitions about infinity

I must confess my intuition, which rarely lets me down, fails me in the present case.  The question is whether the 'Ockham set' axiom of infinity is possible or not:
(Ockinf) For any x's, there is some y such that y is not one of the x's.
Against: if we can speak of 'any x' in an infinite domain, then surely we can speak of 'all the x's'.  But if the axiom above is true, it follows that we cannot speak of all x's.

For: there is no logical impossibility here. Indeed, if we take x's as being an ordinary mathematical set, the statement corresponds to a version of standard set theory (being part of ZF-inf).  Why should a different in the interpretation of the terms lead to a difference in truth value?

So I shall leave this one for now (but any comments or ideas gratefully accepted).  It's connected with a wider question of whether there could be a nominalist version of set theory, and whether it would differ in any way from standard set theory.

14 comments:

David Brightly said...

Isn't there a parallel here with Euclid's fifth postulate?

Belette said...

I think the lesson is that attempting to reason about infinity using ordinary language words with imprecise or implied definitions doesn't work. You end up snarled up. It only works in maths because of precise definitions and no ambiguity.

As we've said before, and as you comment in your "for" case, this basically corresponds to a standard axiom. So in std set theory, it is possible to have this axiom, or not, as you please.

So all we need to do is dispose of your "against" case. I think the problem is that, to make your against case valid, you're attempting to say: "I can construct a oset containing everything". But you've given no rules on how osets are to be constructed. If for, example, your rule on oset construction was "given any oset, and an object not in it, then I'm allowed to make a new oset by adding that object" then your "against" argument would fail.

Edward Ockham said...

So all we need to do is dispose of your "against" case. I think the problem is that, to make your against case valid, you're attempting to say: "I can construct a oset containing everything". But you've given no rules on how osets are to be constructed. If for, example, your rule on oset construction was "given any oset, and an object not in it, then I'm allowed to make a new oset by adding that object" then your "against" argument would fail.

>> If for, example, your rule on oset construction was "given any oset, and an object not in it, then I'm allowed to make a new oset by adding that object" then your "against" argument would fail.

This is wrong. Given the rule, the argument succeeds. But you have not precisely specified the rule. The rule should be that, given any oset of things (whether finite or infinite) – the x’s – and given the existence of any individual object y that is not one of those things, then you can construct the oset of the x’s and y. No problem. But this does not dispose of the "against" case. For there may not be such an object y – and this would be precisely the case where we are able to refer to all the things that exist, and so there cannot exist anything which is not one of those things, and so the “against” case does not fail at all.

Edward Ockham said...

>>Isn't there a parallel here with Euclid's fifth postulate?

There are two parallels. The first that Euclid's postulate is independent of the other four, in exactly the way that Axinf is independent in standard set theory. It is clear that in oset theory, there is no such independence.

But there is a second parallel (which I think is the one you mean) in that we cannot decide whether Ockinf is true or not. This is a different issue.

Belette said...

> This is wrong

I disagree (so much so that I think we must be misunderstanding each other). You are trying to assert "then surely we can speak of 'all the x's'." Ie, that you can construct an oset with "all the x's". You can't do that merely by starting with a finite set and (finitely) adding more x's (trivially). If you want to extend the adding method to an infinite number of adds, then I think you're just begging the question. And if your universe (reverting to std.maths for a moment since we know where we are there) is non-denumerably infinite, then trivially you can't do it even if you allow (countably) infinite number of adds.

Edward Ockham said...

Perhaps I misunderstood your argument. Your argument was not that the conclusion of the ‘against’ argument was false. Rather, that the argument is inconclusive.

But then the ‘against’ is not an argument, and certainly not an argument of the kind you are just describing. It is appealing to a basic and fundamental intuition about plural reference, and thus axiomatic in character. It is the intuition that whenever there are things, even though we don’t know how many, we can always take those things.

For example, suppose there are infinitely many atoms in the universe. Then surely I can talk about ‘all the atoms in the universe’. No proof required. Surely it is self-evident.

But then I am wondering whether it is self-evident.

>>You are trying to assert "then surely we can speak of 'all the x's'." Ie, that you can construct an oset with "all the x's".

No, I am showing that we can construct things such that there are infinitely many things, by building up from finite sets. And then I am asserting, or rather 'against' is asserting, as self-evident and not requiring proof, that in such a case we can talk of 'all the things'. And since an oset (unlike an ordinary set) is the reference of a plural noun phrase, I can therefore speak of the oset of those things, just as, given the existence of these twelve eggs, I can infer the existence of 'one' dozen.

Belette said...

But this is part of the lessons of std set theory: arguments from intuition fail ("obviously" there are more integers than their are primes; oops, contradiction, as we can number the primes. Oh dear, well, we'd better understand what we mean by "as many as" then. Oh good, all contradictions fall away).

> surely I can talk about ‘all the atoms in the universe’

Of course you can. You just did. But all you did was connect some English words together in a way that both you and I and many other people would understand.

But your question, beginning "(Ockinf) For any x's..." is a question depending on the nature of x's, where x's are osets. If you haven't defined the rules for constructing them, how can you hope to answer it?

Edward Ockham said...

>>If you haven't defined the rules for constructing them, how can you hope to answer it?

I gave precise rules for constructing them in this post. These rules are enough to construct any finite set, and also enough to construct an infinite universe, although not to construct an oset containing every element in the universe. This would involve modifying Ockinf above to "any finite number of x's".

I thought you had accepted that idea.

The question of whether we can then go on to assume the existence of an oset corresponding to "all the elements" is what I am wondering about. Is it a logical truth (rather than an assumption to be determined by an axiom) that such a set exists?

>>Of course you can. You just did. But all you did was connect some English words together in a way that both you and I and many other people would understand.

In that case, what was it that you were understanding?

Belette said...

> I gave precise rules for constructing them

Not terribly explicitly. Even having re-read it I'm not sure which bit you mean. Is it "{x: apostle(x)}" type stuff? If so, then "{x: atom(x)}" gets you all the atoms in the universe, whether than is infinite or not.

But if that is your rule, then the question has been trivially solved. So I can only assume you mean something else.

Edward Ockham said...

>>Is it "{x: apostle(x)}" type stuff? If so, then "{x: atom(x)}" gets you all the atoms in the universe, whether that is infinite or not.
<<

The question is whether, if we assign the right meaning to {x: Fx}, we can deduce its existence automatically. You can't in standard set theory, indeed you get into deep trouble if F is something like 'not in x'.

But as I have been saying, the question is whether we can do this if we assign some other meaning to the symbols. Our intuition is that if there are any F's at at all, then we can automatically refer to 'all the Fs'.

Belette said...

> if we assign the right meaning to {x: Fx}

Well yes. That is what I meant when I said "you've given no rules on how osets are to be constructed". Perhaps I should have said "your rules on how osets are to be constructed are vague and imprecise". But then you replied "I gave precise rules for constructing them". I don't understand how you can say that your rules are precise, and yet not have a meaning for {x: Fx}.

Edward Ockham said...

>>That is what I meant when I said "you've given no rules on how osets are to be constructed

But I have given some rules, which clearly contradicts your statement that I have given no rules. It is the comprehension rule where I am having trouble.

Belette said...

I've already said I was wrong to assert you'd given no rules. I continued:

Perhaps I should have said "your rules on how osets are to be constructed are vague and imprecise". But then you replied "I gave precise rules for constructing them".

So we're agreed that you didn't give no rules, and we're agreed you didn't give precise rules.

That allows us to step back to your comment labelled "5:08 PM", which we're now agreed is wrong.

You've given some vague an intuitive rules; these are not adequate to allow us to decide questions of infinity, as you have written them. We're back to my comment of "9:46 PM". Your rules are too imprecise to allow determination of your question.

Edward Ockham said...

>>You've given some vague an intuitive rules

In my latest post titled 'on set construction' I have included assumptions 1-4 which should be sufficient to 'generate' an infinite sequence of finite osets. Can you say whether these are precise enough?

The first part of that post clarifies why statements about osets do not involve 'construction', except for expressions.