Re: an important set theory post

On Jul 25, 9:04 am, David C. Ullrich <dullr...@xxxxxxxxxxx> wrote:
On Thu, 24 Jul 2008 08:46:11 -0700 (PDT), calvin
<cri...@xxxxxxxxxxxxxx> wrote:
On Jul 24, 11:27 am, Dave Seaman <dsea...@xxxxxxxxxxxx> wrote:
You are confusing ordinals and cardinals here.  What you described is the
successor function for ordinals, which in this case should be written:

        succ(omega_0) = omega_0 + 1

or simply

        succ(w) = w+1.

I'm trying not to use unreasonable-sounding words
now, but I'll try to tell you the problem I have
with that.  Adding 1 to something infinite doesn't
produce anything, does it?  It's still just as
infinite, no more, no less, than before you
added the one, I thought.  Where am I going wrong
here?

Where you're going wrong is misunderstanding the
notion of a _definition_. You have the (not unreasonable)
impression that you know what addition is, and then
object that what's above doesn't seem right.

But in fact what's above is simply the _definition_
of omega + 1.

So, omega+1 is merely notation for something that
has a clear meaning? If so, then it would seem to me
to be in the same category as the notation, 2^aleph_0,
which stands for 'the cardinality of the set of all
subsets of a set that has cardinality aleph_0.'
.

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