Re: An uncountable countable set

Tony Orlow wrote:
MoeBlee wrote:
You ignore instead of asking me for justification. The justification is
in the incompleteness theorem, which is even STRONGER than what I
mentioned. Aside from incompleteness, if you knew even just a little
bit about the subject you'd understand the sense in which you can't get
adequate mathematics (such as, say, enough to do calculus) from just
logical axioms.

Listen, in retrospect, I agree with the statetement, "To get an adequate
amount of mathematics, you have to adopt axioms that are not derivable
from pure logic alone."

Hey, look at that, I DID finally get through to Orlow on at least one
point. This is a point that took nearly A YEAR for him to come to grips
with, but finally I did get through.

However, that does not mean they should not have
ANY justification.

Some folks will agree and others disagree with that.

As I said below, IFR is justified geometrically given
the graph of a function, very intuitively. N=S^L is based on
combinatorics. Is that pure logic? Somewhere in the statement of an
axiom should be included the intent, as should be the case with
governmental laws.

And the axioms of Z set theory have intuitive bases also.

If you say so. What is it, then, in a nutshell? I mean, you can't define
"mathematics", but maybe something a little more restrictive could be a
good place to start. So, why don't you explain induction?

I could, but textbooks such as Enderton's 'A Mathematical Introduction
To Logic' do a much better job than I can do. And I very much prefer
not to explain in a vacuum, without giving other explanations that come
prior in a systematic treatment of the subject. If you ever want to
take it from the top, then let me know.

First order PA doesn't give you real analysis. And the PA axioms are
non-logical axioms.

Like I said, my axioms of infinity are constructive axioms, rather than
deductive. There are rules for creating the system and rules which
follow from those regarding how it behaves.

Since you've not stated a system, only you would know how it works.

MoeBlee

.

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