[FOM] Why would one prefer ZFC to ZC?

[Andrej Bauer]

On Sat, Jan 30, 2010 at 8:24 AM,  <T.Forster at dpmms.cam.ac.uk> wrote:
> You can take the axiom of infinity in at least three forms
>
> (i)   The Von Neumann omega exists
> (ii)  Thee is a Dedekind-infinite set
> (iii) V_\omega exists
>
>
> In ZF they are all equivalent. Without replacement they are all
> inequivalent. See e.g. Adrian Mathias' *slim models* paper in the JSL.

If I understand the discussion correctly, this is supposed to be a
reason why replacement is desirable, i.e., we want different notions
of infinity to be equivalent. But why do we want such a thing?

For example, in computable mathematics it is beneficial to distinguish
between different notions of finiteness because they naturally arise
and are not computably equivalent. For a specific example, consider
the set of complex zeroes of a general polynomial of degree at most n
in two cases:

(a) when the polynomial has integer coefficients

(b) when the polynomial has (computable) real coefficients

In case (a) the set of zeroes can be computably listed without
repetitions, because algebraic numbers form a decidable field. In case
(b) we cannot list them computably, even with repetitions, but we can
compute for every epsilon > 0 a list of at most n approximate zeroes
such that the Hausdorff distance from the actual zeroes is less than
epsilon.

The way I see it, the desire for just one notion of infinity is just a
reflection of the deeper desire for just one eternal and absolute
universe of sets in which we can all live happily ever after. But
that's a form of mysticism.

It is entirely possible to adopt a standpoint in which one appreciates
a wealth of different notions, even when they cannot be all put
together in a single consistent picture. This way we get to study a
gallery of pictures rather than just a single one.

With kind regards,

Andrej