Re: abundance of irrationals!)

Randy Poe said:
>
> aeo6 Tony Orlow wrote:
> > Virgil said:
> > > Those who, like mathematicians, are used to dealing with axiom
> systems
> > > in their mental world, require actual internal conflicts in those
> axiom
> > > systems before rejecting them.
> > This is exactly my point. What is "internal"?
>
> Within the framework composed of the axioms and propositions
> derivable from those axioms.
>
> > Is math a field in itself,
>
> Yes.
>
> > and if so, should it contradict itself?
>
> If axiom system 1 contradicts axiom system 2, that is not
> math "contradicting itself". Each axiom system is a
> laboratory. There's no more requirement that they be
> compatible, than there is that two different chemical
> compounds in different jars in a chemistry lab be
> compatible.
No, it is equivalent to two physical theories predicting two results from the
same experiment. Only one of them is ultimately right. If you want to draw
conclusions about infinities that contradict well-established facts from
related areas of mathematics, go ahead and isolate yourself in an obviously
invalid system. Not my problem.
>
> > Many different kinds of math have developed
> > from two sides at once independently, and this is a beautiful thing,
> when we
> > discover two apparently different systems are actually equivalent in
> some
> > manner, when they agree.
>
> If they're equivalent, then they are actually one system.
>
> Sometimes that happens and yes it's beautiful to discover
> new rich connections.
>
> And sometimes it doesn't. C'est la vie.
>
> > What agrees with cardinality, within the world of
> > math?
>
> As usual, you can only come up with "problems" by adding
> new requirements. Nothing contradicts cardinality. It's the
> only notion of set size we have right now.
Bull. We have a variety of ways of dealing with infinities. Cantor didn't
invent the concept of infinity.
>
> > Does it actually contradict other areas of math and logic?
>
> No. The results you don't like about cardinality are a
> direct result of insisting on the rules of logic. They
> are what unbiased deduction leads you to from the
> axioms. We don't get to say "I don't like these conclusions".
> They are what they are, if we respect logic.
If we understand logic, then we understand that correct deduction can yield
incorrect results when the starting assertions are incorrect. The axioms are
wrong, and the interpretations of proofs are confused. The whole idea of
discussing infinity by counting is by definition futile, and the only way
results can be forced out of such a system is by subtle contradictions. That's
what we have here.
>
> What they contradict is the "common sense" of many
> laymen. But obeying "common sense" is not a requirement.
It's a good start. try it sometime.
>
> - Randy
>
>

--
Smiles,

Tony
.

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