Re: Cantorian pseudomathematics

imaginatorium@xxxxxxxxxxxxx wrote:
> Tony Orlow wrote:
> > > > Can we consider it an axiom that a^b is finite for finite a and b, but not for
> > > > infinite a or b and nonzero a and b?
> > >
> > > No, that would be a theorem.
> >
> > Based on which axioms(s), specifically?

I didn't catch seeing that last question in the thread, so I'm
supposing that the above is sufficient context. I take it, I hope
correctly, that Tony Orlow is asking which axioms are used to prove
that a^b is finite for finite a and b but not for infinite a or
infinite b (with both a and b nonzero).

In ZF, I think that if we looked down the trail of theorems, then most
likey the axioms of extensionality, replacement, power set, and union
are all used. There might be a way to use fewer axioms, especially by
giving conditional forms of some theorems along the way, but I don't
know. Anyway, this does miss the point. It is not required to keep a
log of the axioms that are required for each theorem (even though such
a log can be generated; cf. Norm Megill's web site), especially since
most texts do not render proofs formally but rather take it for granted
that whatever has been proven from axioms can be used in a step in a
proof of another theorem so that the trail of exactly which axioms are
used is often not consulted This is entirely justified by a provable
transitive property of proofs:

Let A = the set of axioms

Let S be a set of formulas.

Let f be a formula.

If A |- s, for every seS, and S |- f, then A |- f.

MoeBlee

.

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