Re: Orlow cardinality question

In article <MPG.1d1036471116eeed9896ae@xxxxxxxxxxxxxxxxxxxxxxxxx>,
Tony Orlow <aeo6@xxxxxxxxxxx> wrote:

> My position is that there are infinite whole
> numbers in the infinite set, because an infinite set of naturals MUST
> have infinite members. I have given two solid proofs, which you should
> know by now, Randy.

Those "proofs" all require assumptions outside of mathematics which are,
in fact contradicted by the axioms and definitions for standard
arithmentic.

In particular there is no such distinctions in the set of naturals
between finite and infinite naturals.


The set of naturals up to and including any natural can, however, be
proven finite by induction. I.e., for each n in N, {k e N: k <= n} is
finite according to Cantor's definition, since no such set allows
injectins into any proper subset.
> >
> > The definition and the membership of N are static: you can
> > determine a priori whether a given object n is an
> > element of N. Membership is a thing which doesn't
> > change in time. When we ask "how big is set N" we
> > aren't asking "after we run a construction algorithm
> > for a while, how big has the set gotten to be?"
> > We are asking "how many different conceivable objects
> > fit the definition of a member of set N?"


> The definition is inductive, and as such, process-based. The nth natural
> is not defined, according to Peano, until its predecessor is defined.

But they are all defined simultaneously by indiction itself.
.

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