Avril, (01)
I very much prefer constructive methods in mathematics. In fact,
nearly every mathematician would prefer a construction as a proof
in preference to a proof by contradiction. Whitehead (among many
others) pointed out that the only thing a proof by contradiction
shows is that somewhere there is one or more flaws in the proof.
Deciding which of many possible assumptions and methods of reasoning
caused the contradiction is not always easy. (02)
All proofs for anything beyond a countable infinity depend on
the diagonal method for showing that any proposed 1-to-1 map
of integers to reals would create a contradiction. I don't
blame anyone for feeling uneasy about building all the
hierarchies of hierarchies of infinities on such a slender
thread. (03)
On the other hand, I would not recommend the following approach
for number theory: (04)
> Axiom 9. can be maintained, but the meaning of 'every' has to be
> interpreted to denote a totality of something around 10**120,
> or Ackermann(5 5), or something finite that can be written down
> or understood in some way. (05)
The traditional view before Cantor was that infinity means the
absence of a fixed boundary, and that any reference to infinity
was a way of talking about a process that would exceed any fixed
boundary. Cantor changed the "language game" of mathematics by
talking about completed sets that had infinitely many members. (06)
Many 19th century mathematicians strongly objected to that way
of talking, and I sympathize with them. But those mathematicians
would *never* agree to a fixed upper bound on the integers, such
as 10**120, Ackermann(5 5), or any other finite integer. (07)
In fact, 10**120 is an absurdly small integer for today's computers.
It can be stored in less than 60 bytes, and you can download
computational packages that do arithmetic on multi-word integers
that are much larger than that. (08)
And you can be sure that for any upper limit you might care to name
today, somebody will find a use for integers that are much, much
larger. (09)
John (010)
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