### Independence of the axiom of infinity

In my last post, I introduced the 'Ockham' version of the axiom of infinity

But it is not so simple as that, and I will complete my argument by returning to the point I made in my first post on this subject, about the difference between Ockham sets and mathematical ones. The key difference is that, given the existence of the elements corresponding to an Ockham set, the set itself has to exist, for the set is identical with its members. This is Malezieu's principle: if twenty men exist, it is

Thus, given a universe with an infinite number of elements but no oset corresponding to them, can we wave a magic wand and bring that corresponding oset into existence, as we can easily do in ordinary set theory? I suggest we can't. For an oset simply is its members. If we bring an oset into existence, we must bring at least one of its elements into existence. But we can't do that here. By hypothesis, every member of the infinitely large universe exists already. What is lacking is the oset of all of them. If we now create the oset, what exactly are we creating? Nothing. There are no things to create, for they are already there. Nor can we suppose that there is any other possible universe corresponding to this possible one, containing identical elements, but also containing the oset itself. So, if a universe consisting of these elements is possible, it is necessarily the case that there is no oset containing all its elements. And in general, there can be no universe containing an infinite number of elements, together with the oset of all of them. If every possible world is such that p, then necessarily p.

This is a surprising result. The alternative is to reject (Ockinf) above, and thus reject the possibility of an infinite universe without a corresponding oset. But that is equally surprising, for it would amount to rejecting the independence of the axiom of infinity. We could merely define 'finite oset' as we did before, assert that to every finite oset there is an (equally finite) oset containing one extra element, and then deduce the existence of an infinite oset, based on the logical impossibility of (Ockinf) above.

Comments welcome.

(Ockinf) For any X's, there is some y such that y is not one of the X's.which, for a suitable definition of infinity, gives us an infinitely large universe, but no infinitely large Ockham set to correspond to it. Every Ockham set is finite, yet the universe is infinite. As 'Belette' has spotted, this may be no surprise, for it is well known that the axiom of infinity cannot be derived from the rest of the axioms of Zermelo Fraenkel set theory, and we can even construct a model of the axioms where the axiom of infinity is replaced by its negation (called ZF-INF).

But it is not so simple as that, and I will complete my argument by returning to the point I made in my first post on this subject, about the difference between Ockham sets and mathematical ones. The key difference is that, given the existence of the elements corresponding to an Ockham set, the set itself has to exist, for the set is identical with its members. This is Malezieu's principle: if twenty men exist, it is

*because*the first, the second, the third man etc., exist. If a exists and b exists, then the Ockham set*a and b*exists also. By contrast, in set theory we have to posit this using an axiom such as the axiom of pairs.Thus, given a universe with an infinite number of elements but no oset corresponding to them, can we wave a magic wand and bring that corresponding oset into existence, as we can easily do in ordinary set theory? I suggest we can't. For an oset simply is its members. If we bring an oset into existence, we must bring at least one of its elements into existence. But we can't do that here. By hypothesis, every member of the infinitely large universe exists already. What is lacking is the oset of all of them. If we now create the oset, what exactly are we creating? Nothing. There are no things to create, for they are already there. Nor can we suppose that there is any other possible universe corresponding to this possible one, containing identical elements, but also containing the oset itself. So, if a universe consisting of these elements is possible, it is necessarily the case that there is no oset containing all its elements. And in general, there can be no universe containing an infinite number of elements, together with the oset of all of them. If every possible world is such that p, then necessarily p.

This is a surprising result. The alternative is to reject (Ockinf) above, and thus reject the possibility of an infinite universe without a corresponding oset. But that is equally surprising, for it would amount to rejecting the independence of the axiom of infinity. We could merely define 'finite oset' as we did before, assert that to every finite oset there is an (equally finite) oset containing one extra element, and then deduce the existence of an infinite oset, based on the logical impossibility of (Ockinf) above.

Comments welcome.

Labels: infinity, set theory

## 14 Comments:

From the earlier 'Are there infinite sets?':

>> And if the Ockham set does not exist, at least one of its members fails to exist.

On the face of it this is a strange thing to say. Compare with 'And if the golf club does not exist, at least one of its members fails to exist. ' Or 'And if the forest does not exist, at least one of its trees fails to exist.'

I take it that the project here is to extend RL with plural terms? So the sense in which some oset fails to exist is that some plural term, say 'Zeus and Bacchus', refers to nothing because its member singular terms individually fail to refer? In which case there is a confusing equivocation in the use of 'oset' between the plural terms themselves and the things to which they refer.

To try to clarify this, suppose we have a language L that can be used to talk about some domain of things D. Then I suggest that the osets of L over D are those subsets of D (in the ordinary maths sense of 'subset') that the terms of L can refer to. The osets clearly depend on the plural referencing powers inherent in L. And there may be no term in L that refers to all of D.

Am I barking up the right tree here? If so, it would appear to be an 'empirical' question, requiring some sort of mathematical investigation into the referencing capabilities of L, whether D itself is among the osets of L?

But if this is right I don't understand what you mean by 'bringing an oset into existence'.

>>Compare with 'And if the golf club does not exist, at least one of its members fails to exist

But a golf club is probably something over and above its members, like a mathematical set. But if one of the dozen eggs is broken, then there is no longer a dozen eggs, no?

>>Then I suggest that the osets of L over D are those subsets of D (in the ordinary maths sense of 'subset') that the terms of L can refer to. The osets clearly depend on the plural referencing powers inherent in L. And there may be no term in L that refers to all of D.

Yes that's the idea.

>>But if this is right I don't understand what you mean by 'bringing an oset into existence'.

Well, causing it to exist, then. We first suppose that the universe contains infinitely many things, but no oset containing everything. Now could it be that the universe could change simply by the addition of an oset containing it.

>>I take it that the project here is to extend RL with plural terms?

Yes. Or rather, to extend the nominalist project by introducing expressions that would deal with sets, without sets themselves. So that the only things in our ontology are singulars.

>> Now could it be that the universe could change simply by the addition of an oset containing it.

Not sure I understand this! The answer is obviously and trivially No because an oset is not a thing in the universe?

>> But if one of the dozen eggs is broken, then there is no longer a dozen eggs, no?

OK, perhaps I begin to see whence you come. Our ability to see a unity in a multiplicity. *A* dozen has become *an* eleven. This sounds like a special case of the more general ship of Theseus problem.

By 'creating the oset' do you mean something like rearranging the individuals of the world so that it becomes true to say that there is a dozen eggs in this basket? But I don't see the relevance of this. Either there is a term in the language L that encompasses the whole domain or there isn't and no amount of rearranging the deckchairs changes this. Of course, we can always *invent* such a term and add it to the language. But then we have a different language. And hasn't this happened to natural languages? After all English now has terms like 'the world', 'the universe', which I would have thought refute your argument.

Suppose we have a universe with three things in it. Can I then make the thought experiment of supposing I add a further thing to the universe, or change the universe in some way, in order to see whether a contradiction results?

This method of making imaginary changes to the universe is fundamental to much philosophical enquiry. Ockham begins it by asking whether God 'in his power' could make certain changes.

If you are saying there is something fundamentally wrong or bankrupt with this method, that is pretty serious.

That's something of a Doomsday reply! Of course I think that's a reasonable procedure. But as you have said that an oset is not a thing I'm having trouble grasping what you mean by 'creating an oset' or 'adding a further oset'. To me it can only mean beefing up the language to give it greater referential coverage. Apologies if I'm being more than usually dense!

>>But as you have said that an oset is not a thing I'm having trouble grasping what you mean by 'creating an oset' or 'adding a further oset'.

<<

Ockham set (oset) was defined here.

Actually the limiting case is when the oset contains one element, in which case it is a single thing, i.e. that element, because it is by definition identical with its elements and thus identical with that member. If it has two elements, it is identical with those two things - one pair of things, two things.

By creating on oset, I mean creating the things which are its elements - or creating enough things to ensure that its elements exist. Does that make more sense?

>> *A* dozen has become *an* eleven.

No that can't happen. A dozen is always a dozen. Since the oset of a dozen things simply is those dozen things, that oset ceases to exist if any of the dozen ceases to exist.

>> No that can't happen.

Fair enough. We would certainly say 'I no longer have a dozen eggs in my basket'.

>> By creating on oset, I mean creating the things which are its elements - or creating enough things to ensure that its elements exist. Does that make more sense?

This seems to point toward an intensional aspect of osets. We have the idea of some oset and rearrange the world to bring it into existence. My understanding is that an oset is just the extension of some plural referring term. If I have an empty basket then 'the eggs in my basket' is non-referring---we might say the corresponding oset does not exist, though I'm uncomfortable with this use of 'exist'. If I put at least one egg in my basket then the phrase becomes referring and denotes an oset. Is this what you mean?

Suppose we allow not only names that denote no objects but also objects that have no names. This reflects our epistemic condition. If every oset is the denotation of a phrase of the form 'a and b and ...' for some finite number of names, then no oset can cover the domain and by the definition in 'Ockham sets:preliminaries' it's not the case that 'there are finitely many things'. But the domain could easily be finite in the ordinary sense. If OckInf correctly captures our sense of infinity this is a counter-intuitive result and perhaps 'explains' why natural languages have more general plural referring terms?

>>Fair enough. We would certainly say 'I no longer have a dozen eggs in my basket'.

Careful. If I break one of the eggs, I no longer have

thatdozen. But if I replace the broken one with a new one, I still have a dozen. However, it’sthisdozen, notthat.>>My understanding is that an oset is just the extension of some plural referring term. If I have an empty basket then 'the eggs in my basket' is non-referring---we might say the corresponding oset does not exist, though I'm uncomfortable with this use of 'exist'. If I put at least one egg in my basket then the phrase becomes referring and denotes an oset. Is this what you mean?

As before. A plural referring phrase should be rigid in that it refers to the same objects in all possible worlds where the objects actually are the same. For example ‘Obama and Cameron’ is a plural referring phrase. ‘Obama and Cameron might not have been politicians’ refers to the very same two people in some possible state of affairs. As does ‘those two people’, or ‘that pair’.

>>If every oset is the denotation of a phrase of the form 'a and b and ...' for some finite number of names, then no oset can cover the domain and by the definition in 'Ockham sets:preliminaries' it's not the case that 'there are finitely many things'.

This wasn’t the intention. After all, if we can have infinitely many natural numbers, why not infinitely many proper names?

>>If OckInf correctly captures our sense of infinity this is a counter-intuitive result and perhaps 'explains' why natural languages have more general plural referring terms?

I didn’t follow this point.

> This is Malezieu's principle: if twenty men exist, it is because the first, the second, the third man etc., exist.

As you state it, M's principle applies to finite "sets" only. Was that your intent?

>>As you state it, M's principle applies to finite "sets" only. Was that your intent?

No, it clearly applies to any oset whatsoever. "Twenty men" is the example used by Malezieu (or rather by Hume).

Then you have a problem, because your demonstration / enunciation of it is "it is because the first, the second, the third man etc., exist." You can't do that with infinite "sets", since yuo can't count through them.

>>You can't do that with infinite "sets", since yuo can't count through them.

You don't have to count them, any more than you have to count the twenty men, in order that they exist. Even if no one has counted them, it is still true that the 20 exist because one exists (any one you like), another exists (different from the first), another exists (different from the first two) etc.

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