[FOM] Godel's Theorems
Robbie Lindauer
robblin at thetip.org
Thu May 1 17:30:37 EDT 2003
I think the Wittgenstein objection to *) is not that it isn't provable
or that it's not a mathematical statement, just that the word "proves"
here has been given a new meaning ("diagonalization") and that if you
allow that you can change the kinds of things that you call proof, then
you can prove anything.
As LW says, his point is not to disprove any particular mathematical
proposition, only to invite people to realize that the mathematical
paradise isn't what we think it is.
When a mathematician says "prove" s/he means that "if you use these
axioms and transformation rules, you 'get' these results".
LW would add a caveat (I think):
"There is no clarification for the word 'get' here. We can decide to
play be the rules the way it is usually done but that doesn't get us
what was desired out of 'prove'."
Best,
robbie lindauer
>> *) there is a true sentence in the language of PA which is not
>> provable in PA.
>>
>> 1. Conventional wisdom is that this is now a fully established
>> theorem of mathematics (or ordinary mathematics as currently
>> practiced by the overwhelming majority of mathematicians). Is there
>> agreement on this?
>>
>> 2. For those who do not agree, do they believe that *) is not a
>> mathematical statement capable of mathematical proof? E.g., this
>> could be on the grounds that they do not accept the usual
>> mathematical definition of "true sentence in the language of PA".
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