[FOM] Only one proof
Aatu.Koskensilta at uta.fi
Mon Aug 31 15:41:59 EDT 2009
Quoting William Tait <williamtait at mac.com>:
> Two examples in set theory are Goedel's proof of the consistency of CH
> relative to ZF using inner models and Cohen's proof of the consistency
> of Not-CH using forcing.
Another example: Gödel's proof for the first incompleteness theorem
understood as the result that the set of Pi-1 truths is productive.
I'm not aware of any way of proving this result without going through
essentially the recursion theoretic contortions found in the original
proof. (I may well be just ignorant.)
> There seems to be something different about these example in
> comparison with yours. [Qua examples, they seem less interesting.]
> What is it?
The results you mentioned are both truly fundamental, to advanced set
theory they're a bit like any immediate application of the principle
of mathematical induction, the least number principle, the method of
infinite descent, are to number theory. This is of course a very
boring observation, but with out current logical and conceptual tools
I don't think we can do any better.
Aatu Koskensilta (aatu.koskensilta at uta.fi)
"Wovon man nicht sprechen kann, darüber muss man schweigen"
- Ludwig Wittgenstein, Tractatus Logico-Philosophicus
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