[FOM] Re: FOM: Natural numbers and inductive sets

[Lasse]

Jeff Hirst wrote:

> ...
> >Nearly all proceed as follows. Define an inductive set S as being such
that
> >0 is in S and for all x, if x is in S the successor of x is in S.
> >Show there exists a "smallest" inductive set by constructing the set Z
of
> >all inductive sets (invoking power set).  Show the "smallest" set is
the
> >intersection of Z.  One proof argues that Axiom of Choice is necessary.
>
> The collection of all inductive sets includes all the limit ordinals,
> and is therefore a proper class whose existence is not provable in ZFC.

I expect Dean meant to say: the smallest inductive subset of S. I guess
that he is observing the rather obvious fact that, if S is a set and P
is a property (of subsets of S), the intersection of all subsets of S
with property P (i.e., the smallest such subset in the case where P is
stable under intersection) can be defined without resorting to power
set.

Whether it makes the construction of the natural numbers clearer is
another question; I personally find taking the intersection easier to
understand, and if we have power set at our disposal, we might as well
use it if it helps intuition. But then again, I am an analyst and
usually take the axiom of choice for granted, so I'm possibly a bit
ignorant on this front anyway ;)

Lasse


-- 
Lasse Rempe
Currently: Mathematics Institute, University of Warwick, Coventry CV4
7AL
Email: lasse at maths.warwick.ac.uk, lasse at math.uni-kiel.de
http://analysis.math.uni-kiel.de/lasse