Re: Definition of finite.

On 3 May 2007 14:00:23 -0700, MoeBlee <jazzmobe@xxxxxxxxxxx> wrote:


[the] definition is

x is finite <-> Ey(y is a natural number & x equinumerous to y).


More formally:

finite x :<-> En(n e w & x ~ n)

Now, as to the particular definition you just mentioned [...],
it's not circular as long as 'is a natural number' and 'equi-
numerous' have been previously defined without use of 'finite'.

Right.


Now Let y be a natural number, How do I use [this] definition
to show that y is finite, without being involved in some circu-
larity?

I guess, he wanted to stress the word "show" in this question.

That's indeed an interesting "problem", imho. Of course, you should
not interpret "show" here as "prove" (as one might be tempted to
do), but rather with: "show that this is _really_ the case".

I guess, his point is the following: Just assume _for the sake of
the argument_ that there were infinite sets in w (here I'll use an
intuitive notion of infinite!) - since we don't have a definition
of finite/infinite so far, this might very well be the case, at the
present stage of the development of our formal theory (i.e. we
don't have a proof that this is not the case). If we _define_ now
/finite/ (in our theory) the way we did, we could _prove_ (i.e. we
would get), that all sets in w are _finite_ (which _actually_ would
not be the case).

So proving that all n in w are finite by "assuming" that they are
finite is "circular" (from this point of view).

Actually, *I* personally would consider this as asome sort of
argument for NOT using the mentioned definition as our "basic"
definition for "finite" in set theory (say ZFC) - despite its
simplicity (and hence its attractiveness).

So one might claim (if you are to follow the argumentation from
above) that our "basic" definition of /finite sets/ should be via
_Dedekind's_ definition of /infinite sets/.

Of course, then we can show (at least in ZF_C_) that these two
definitions of "finite" are equivalent. So we might claim that the
natural numbers are _really_ finite. (And this might be considered
as a justification for using of the definition mentioned above.)


F.

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