FOM: Intuitionism (Tait)
Torkel Franzen
torkel at sm.luth.se
Tue Jun 18 21:24:16 EDT 2002
Sean C Stidd says:
>(1) If Godel's
>'metamathematical demonstration' cited above is one we ought to accept as
>valid, what are the philosophical and/or mathematical consequences, if
>any, beyond those of the incompleteness theorem of doing so? (2) If this
>argument is not one we ought to accept, what is its principled dismissal?
Godel's observation that "Thus, the proposition that is
undecidable in the system PM still was decided by metamathematical
considerations" is a demonstration that the Godel sentence for PM
is true only when coupled with a demonstration that PM is consistent.
Whether you accept the truth of the Godel sentence just boils down
to whether or not you accept that PM is consistent. There is nothing
in this that is any more difficult to formalize than anything else
in (non-trivial) mathematics. Asking whether we "ought to accept"
anything in this context is no different from asking whether we
"ought to accept" anything else in mathematics.
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