[FOM] Re: FOM Digest, Vol 14, Issue 29

[Robert Owen]

> From: Stephen Lavelle <icecube at maths.tcd.ie>
> 
> i know it's mathematically a rather pointless arguement (pun
> unintended), becuase mathematically what i have called sets above are
> called countable sets and what i have called a continuum is viewed as
> being an uncountable set, but in terms of understanding the intuitive
> "nature" of such objects i think the traditional definitions are very
> misleading.

What you're saying is, I think, entirely formalizable within ZFC by
considering the set HC of hereditarily countable sets.  HC has the
naturals, integers and rationals but lacks the reals and, in fact, any
fundamentally uncountable object.  Better yet, HC models ZFC - Power, so
you really can do (much) of mathematics in there if you're so inclined.

[Formally, Con(ZFC) -> Con( ZFC - Power + Ax (x is countable) ) by
considering HC.]

I'm not convinced of the philosophical utility in thinking of sets like
that -- or, set-theoretically, of working in HC -- but there could be
pedagogical reasons that I'm not seeing.

Cheers,

Robert Owen
owen at math.wisc.edu