Re: Godel [was Re: qmail Re: freebsd - Re: recommended Virus Scanner?]
Thus spake Tom (tb.31123.nospam@comcast.net):
...
> I thought it was neither complete (the doesn't capture all truths thing)
> nor consistent (may contain both a statement and its complement)[1].
> But I can look that up.
>
> The Stanford prof told me the Lambda calculus (Lisp-ish stuff) almost
> proved one of the two.
"Almost"? Hardly counts. I also don't understand what he was telling
you. Church's lambda-calculus and Turing's turing machine are
equivalent (as I believe Turing showed as part of his PhD thesis,
which Church supervised) and AFIK neither was trying to disprove
Go"del which, as someone has already pointed out, is well proved and,
the nature of mathematical proof being what it is, it is highly
unlikely that anyone is going to find a logical error almost 75 years
later - Go"del's proof has, after all, been pored over by countless
mathematicins and many of them must have desparately wanted to be able
to refute it.
AIR Go"del says that in any axiomatic system which is at least complex
enough to contain the axioms of arithmetic, then there are statements
which can be made but not proved within that system. It is possible to
add further axioms to prove the statements, but then this richer
axiomatic base will lead to new statements which cannot be proved with
the richer set of axioms.
--
|Deryk Barker, Computer Science Dept. | Music does not have to be understood|
|Camosun College, Victoria, BC, Canada| It has to be listened to. |
|email: dbarker@camosun.bc.ca | |
|phone: +1 250 370 4452 | Hermann Scherchen. |
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