Re: An uncountable countable set

MoeBlee wrote:
Tony Orlow wrote:
I'm reading Non-Standard Analysis instead.

What book?

You really would be better served by having your basics in set theory
and mathematical logic in order first and then taking on non-standard
analysis.

Robinson agrees there's no
smallest infinity,

Then that is not the same as the ordering of the ordinals we're talking
about. I very much doubt that Robinson claims that there is not a least
infinite ordinal.

Please tell me exactly what passages or theorems you are referring to
in Robinson's work so that I can see exactly what it is you are talking
about.

And this reminds me that you never did come to understand the
difference between cardinality and ordering. I and others have pointed
out to you that you conflate these. One doesn't even need non-standard
analysis to provide an ordering in which there is a set S with no least
member yet with every member of S greater than some set with no
greatest member. But that ordering is not an ordering by cardinality.
Yes, PA has models in which there are different orderings so that
objects are called 'infinite' per these orderings, but this is NOT the
same sense of 'infinite' as that of the cardinality sense. And we can
define a division operation on these "infinite" objects to get
infinitesimals, but again, this is not the same as cardinality.

Moreover, you must be very careful to distinguish between the proof of
the existence of certain models and a RECURSIVE axiomatization for a
theory of which the model is a model of.

MoeBlee

.

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