[FOM] ZC vs. ZFC

[Jeremy Bem]

On Sun, Jan 31, 2010 at 4:43 AM, Thomas Forster
<T.Forster at dpmms.cam.ac.uk> wrote:

> Harvey Friedman has a catalogue of nice facts about sets of low rank that
> can be proved only by reasoning about sets of high rank.   This *ought*
> to shut down the debate, but for some reason it doesn't.

Why ought it to?  Wouldn't this also be an argument for including all
large cardinal axioms?  Presumably part of the debate is how much
sense these high-rank sets make.

> This brings me to what - for me at least - is one of the major reasons for
> accepting replacement.  There are at least three halfway-sensible ways of
> thinking of natural numbers as sets that have been proposed: there are the
> Zermelo naturals, the von Neumann naturals, and there are Scott's trick
> naturals.  It is a fact so important and so basic that it tends not to get
> spelled out that *it doesn't matter which implementation we use*. Certainly
> the von Neumann implementation is cuter than the others, and more generally
> used, but we must not lose sight of the fundamental insight that *it
> shouldn't matter which one we use*.  Actually i don't think we are in danger
> of denying this point, so much as *overlooking* it.  And here is the key
> point:
>
>  *** If we want to maintain a stance of lofty indifference about      ***
>  *** our choice of implementation then we have to assume replacement. ***

I don't understand your key point.  If this is really a general fact,
can you help me understand how it would apply to the reals?  One might
wish to be loftily indifferent to Cauchy sequences versus Dedekind
cuts.  Does replacement help with this somehow?

In both systems, of course, one is limited to "implementations" that
one is able to construct.  For example, in ZFC, I couldn't use "the
least n-huge cardinal, for each n" as my implementation of the natural
numbers.

-Jeremy