Thursday, September 08, 2011

Are there infinite sets?

I shall argue that there are there no infinite sets. Which I shall immediately qualify by distinguishing between mathematical sets, and Ockham sets. A mathematical set is a thing that can contain things. The set {a, b} contains two things – the elements a and b – and is itself a third thing separate from them both. Since I hold that there are no such things as mathematical sets (which I will not argue here, let’s assume that for now), it immediately follows that there are no such things as infinite mathematical sets, but do not worry about that.

An Ockham set, by contrast, is a set of things that is not itself a thing. Let me explain. We speak of ‘a dozen’ or ‘one dozen’ things. If they are dozen, then they are 12. The ‘one’ dozen is not one thing in addition to those 12 things, but rather ‘a dozen’ is a collective noun for twelve things (and nothing else). I claim that there are no infinite Ockham sets.

This may take a few posts, so I will begin with a feature of Ockham sets that does not apply to ordinary (mathematical sets), and which will be essential to my argument. Since an Ockham set is its elements (i.e. the dozen things is or are those twelve things), it follows that if each element exists. I.e. if a exists and b exists, then a and b exist, i.e. the plural expression ‘a and b’ refers to some things. Therefore the Ockham set – the reference of ‘a and b’ or ‘those two things’ or ‘that pair of things’ also exists. If twelve eggs exist, then a dozen eggs exist, and so on. I shall call this Malezieu’s principle, after the French geometer about whom we know almost nothing except what Hume says about him:

I may subjoin another argument proposed by a noted author [Mons. MALEZIEU], which seems to me very strong and beautiful. It is evident, that existence in itself belongs only to unity, and is never applicable to number, but on account of the unites, of which the number is composed. Twenty men may be said to exist; but it is only because one, two, three, four, &c. are existent, and if you deny the existence of the latter, that of the former falls of course. [link]
I.e. if the elements exist, the Ockham set exists. And if the Ockham set does not exist, at least one of its members fails to exist. This principle is not true of ordinary sets. If a is in the domain, and b is also, we need an assumption – the axiom of pairs – that allows us to assume the exist of the ordinary set {a, b} containing as elements a and b, and nothing else.

More tomorrow.

William M. Connolley said...

Maths only has infinite sets if you ask it to: http://en.wikipedia.org/wiki/Axiom_of_infinity

But do go on.

Michael Sullivan said...

I've never understood how a mathematical set could be anything other than a relation of reason.

Edward Ockham said...

>>Maths only has infinite sets if you ask it to: http://en.wikipedia.org/wiki/Axiom_of_infinity

That would be a mathematical set. The question is whether we can ask the same thing of an Ockham set. Can there be infinite Ockham sets?

More later.

William M. Connolley said...

Yes, indeed it is. I was only pointing out that in maths, whether you have infinite sets or not depends on the axioms you choose. Likely the same is true for you.

You don't give your full axioms, but "is a set of things that is not itself a thing" appears to exclude osets containing other osets. In that case, the formulation of the AoI that I linked to wouldn't generate any infinite osets.

William M. Connolley said...

Incidentally, don't forget to define infinity very carefully in your world.

Edward Ockham said...

I'm liking this word 'oset'. It sounds like a variety of furry mammal that lives in the uplands somewhere.

>>"is a set of things that is not itself a thing" appears to exclude osets containing other osets.

That is correct, but depends on definition of 'contain'. In the sense of being a subset, then there can be containment. E.g. 'this pair is one of this dozen things' means that these 2 things are 2 of these 12 things.

Alternatively 'this pair is one on this dozen pairs' means that these 2 things are 2 of these 24 things.

>>Incidentally, don't forget to define infinity very carefully in your world.

I was coming to that.

William M. Connolley said...

A osubset is an oset, of course. So if A = {a, b, c} (with a, b and c being real things) then A is an oset, and so is B = {b, c}. Then B is-a-subset-of A, but B is-not-a-member-of A. So that is OK. But B' = {a, {B}} isn't an oset.

However, the definition of infinity that I know and love involves functions, which are of course subsets of set of pairs, so watching you try to construct an ofunction will be fun. Unless you're going to use normal set theory to define infinity; but if you have to do that, you'd be confessing the limited utility of your osets.

Edward Ockham said...

>>A osubset is an oset, of course. So if A = {a, b, c} (with a, b and c being real things) then A is an oset, and so is B = {b, c}.
<<

Yes

>>Then B is-a-subset-of A, but B is-not-a-member-of A.
<<

Correct.

>>But B' = {a, {B}} isn't an oset.

Doesn't mean anything in oset world. {a} and a are identical. Or rather, {a} is ill-formed. {a, b} means "a and b", which take a verb in the plural. But {a}, if it meant anything, would mean a singular subject that took a plural verb.

>>However, the definition of infinity that I know and love involves functions, which are of course subsets of set of pairs, so watching you try to construct an ofunction will be fun. Unless you're going to use normal set theory to define infinity; but if you have to do that, you'd be confessing the limited utility of your osets.
<<

Patience.

William M. Connolley said...

> Doesn't mean anything in oset world.

Agreed.

> {a} is ill-formed

Why? Do you have a special rule prohibiting one-element osets? If so, why?

Edward Ockham said...

>>Why? Do you have a special rule prohibiting one-element osets?

Well an oset is strictly identical with its elements, rather than a thing containing them. So nothing is prohibited. I suppose we could make sense of {a} in the same way we make sense of 'a pair of', and so have 'a single of' or 'a one of' shoes. But unlike set theory, where {a} is never identical with a, in oset theory a single of a is simply a itself. But that would be pointless. And you would prohibit the empty set, because 'a nothing of Fs' cannot be identical with anything.