[FOM] Re: The Omega Express (Set Theory and Ordinary Language)
jeremyraclark at hotmail.com
Tue Dec 31 10:21:40 EST 2002
> 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
> 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,
can generate a model of ZF with an element c which is a 'natural number' but
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
> contradiction, but it seems odd to me.
I would argue that we are *not* actually capable of grasping natural numbers
all-inclusive sense that you are talking about. We seem to be pretty good at
individual natural numbers but I have never seen a general natural language
of natural number which isn't severely question-begging.
I certainly don't think that ZF is a good approximation to 'natural
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,
> 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
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.
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