[FOM] Re: The Omega Express (Set Theory and Ordinary Language)

[Jeremy Clark]

Dear Dean,

You wrote:
>
> But the analogue of positing the set of stations in ZF, is postulating the
> existence of a set of natural numbers, i.e. the Axiom of Infinity.  We
can't
> define the concept "- is a natural number" in ZF without postulating the
> existence of an infinite set of numbers.

But even *with* the axiom of infinity, we get only a local definition
*within* ZF of a natural
number. Compactness arguments can still be applied: If ZF is consistent,
then we
can generate a model of ZF with an element c which is a 'natural number' but
which
satisfies 'c != 0', 'c!=1', ... in your model.

>
> Yet we can define the same concept (or one that looks very similar ) in
> ordinary language.  With our finite minds we are able to grasp something
> whose definition requires an infinitude of things.  That's not necessarily
a
> contradiction, but it seems odd to me.

I would argue that we are *not* actually capable of grasping natural numbers
in the
all-inclusive sense that you are talking about. We seem to be pretty good at
grasping
individual natural numbers but I have never seen a general natural language
definition
of natural number which isn't severely question-begging.

I certainly don't think that ZF is a good approximation to 'natural
language' definitions,
or even a good way of approaching 'real' objects like natural numbers. The
Axiom of Infinity
is there to ensure the existence of infinite *sets*.

>
> That's my problem.
>
> I think I see a solution (involving the fact that, in ordinary language,
we
> do not quantify over "numbers" at all), but before that, does this set the
> problem up in the right way?  Is it in fact true that we can't define the
> concept "is a number" in ZF without AxInf?

Well, we could take the following formula in the language of ZF:

F(x) = 'x transitive' ^ 'x linearly ordered by \in' ^ 'x not equinumerous
with a proper subset of itself'.

This formula, along with the axiom of regularity, seems to me to do a pretty
good job,
as good (or as bad, see above) as the usual way of defining natural numbers.
But F(x) defines
a class within your model of ZF, not a set (or at least, not without AxInf).

I hasten to add at this point that my set theory / model theory is very
rusty, and that my views on
the philosophy of mathematics are unorthodox.

Regards,

Jeremy Clark