Saturday, September 10, 2011

Ockham sets and infinity

In my previous post about infinity, I distinguished 'Ockham sets' or osets from ordinary mathematical sets.  An Ockham set is like a pair or a dozen.  It is not a thing, but rather a set of things, just as a dozen things is not a thirteenth thing separate from the twelve things it is a dozen of.

Now I shall ask whether it is possible that there could be infinitely many things, without there being an oset of those things.  My aim is to show that while it is possible that there are infinitely many things, it is impossible that there should be an Ockham set of those infinitely many things.  And my first step will be to ask whether it even could be possible that there were infinitely many things, without there being an Ockham set of them.

Assume that at least one thing exists.  Then ask whether the following could be true:
(Ockinf) For any X's, there is some y such that y is not one of the X's.
If it is true, then clearly there are infinitely many things.  For if there were finitely many, we could count through all the things there were, and come to a halt at some point.  Then all the things we had counted would fall within the range of the 'for any X's' above, and there would be at least one y that was not one of the things we had counted, which contradicts the assumption that we had counted all the things.  So there are not finitely many things, if Ockinf is true.  Therefore, if it is true, there are infinitely many things.

But is that even possible?  It is possible in standard set theory. Indeed, there is even a version of set theory where the axiom of infinity is denied.  But is possible in Ockham world?  I believe it is, but it runs counter to our natural assumption that we can talk about or refer to 'all the things there are'. For if the assumption above is possible we can't refer to 'all things'.  Suppose we can.  But the statement above says that for any X's, i.e. for any things whatsoever, there is at least one thing that is not one of them.  Hence there would be at least one thing that was not one of all the things, which is contradictory, for 'all things' includes absolutely everything, leaving no thing out.

You might argue that Ockinf above is impossible, because it is self-evident that, how many things there are, we can always refer to all of them.  But can we?  The statement above denies precisely that.  Is it false in virtue of its meaning?  I don't think so (although I'm not sure either).  Clearly 'any x' must be satisfiable by any x in the domain.  The lasso of a singular variable must be far reaching enough to get its singular loop round any object.  There can't be some x that isn't any x.  But any X's?  The plural variable lasso loops around numbers of objects in the domain.  There can't be some X's that aren't any x's.  But the statement above doesn't claim this.  It says that there are not some plural X's such that they include every singular x.

The lasso of plural quantification is flexible enough to rope around any number of finite things.  But it is not big enough to capture all of infinitely many things.  It will always fall at least one short.  And if we pull it a bit to get that one thing, we find to our frustration that there is yet another one that we missed!

In conclusion, it seems possible that there are infinitely many things, without there being an Ockham set that includes all those things.  Or at least the obvious arguments that would suggest it wasn't possible, are invalid.  In the next post, I shall argue that if this is possible in some domain, then it is so in every domain, i.e. there cannot possibly be any Ockham set that includes infinitely many things.

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5 Comments:

Blogger Belette said...

You appear to be discussing infinity without having defined it. That way lies madness, or at least endless pointless words.

9:13 am  
Blogger Belette said...

Oh, and also it isn't clear how to parse "For any X's, there is some y such that y is not one of the X's.". I might have expected capital X to be an oset, and y to be an element. But then you say "y not one of the X's" which puts y at the same level as X, so the membership implication fails.

Anyway, I don't know what you mean.

9:16 am  
Blogger Edward Ockham said...

N.B. I deleted some remarks so ignore anything apart from this comment, that you may have seen. Let's start with your point about definition:

>>You appear to be discussing infinity without having defined it.

OK, let's start with 'finity' first, and define recursively.

1. Any one thing is finite in umber

2. If any Xs are finite in number, then those Xs and some y (where y is not one of the Xs) is also finite in number.

Thus Peter is finite in number, hence Peter and Paul are finite in number, hence Peter and Paul and John are finite in number, and so on.

Then 'Infinite' is whatever is not finite.

I haven't defined the terms 'one of ' and 'and'.

The connective 'and' is primitive, and corresponds to the ordinary English 'and' which connects singular terms. Thus 'Peter and Paul preached in Athens and Jerusalem'. The relation 'is one of' can be defined in terms of 'and'. Thus

a is one of Xs iff for some Ys, Xs = a and Ys

Thus let the Xs be Peter and Paul and John. Then there are Ys, namely Paul and John, such that

Xs = Peter and Ys.

I.e. Peter is one of those three people, Peter and Paul and John.

Note that the '=' here is a plural =, and should be read 'equal', third person plural. The mathematical equivalent, by contrast, is only singular.

10:54 am  
Blogger Anthony said...

I can't figure out your definition of "things".

2:24 pm  
Blogger Edward Ockham said...

The logical definition of 'thing'. I.e. for any x = for any thing

4:48 pm  

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