FOM: the unbearable lightness of classical logic
Neil Tennant
neilt at cohums.ohio-state.edu
Wed Mar 31 08:01:47 EST 1999
There they go again, those darned classical-logician-playwrights!
The Turing-character speaks as follows:
> He [Go"del] did this by constructing a mathematical assertion that said
> --- in effect: 'This assertion cannot be proved.'
> Well --- either it can be proved or it can't. If it can be proved, we have
> a contradiction, and the system is inconsistent. If it cannot be proved,
> then the assertion is true --- but it can't be proved, which means that the
> system is incomplete. Thus mathematics is either inconsistent or it's
> incomplete.
It is of course erroneous to present the proof as proceeding by constructive
dilemma. He should have given the following *constructive* reasoning:
Well --- suppose the system is consistent. If the assertion can
be proved, then we have a contradiction, contrary to the assumption
of consistency. So it can't be proved. But then it is true. So the
system is incomplete. Thus if mathematics is consistent then it's
incomplete.
In the context, of course, it would have been too much to ask the playwright
to reveal that the `But then it is true' calls for omega consistency.
Still, not bad for a playwright!--and thanks to Jeremy Seligman for the
excerpt.
Neil Tennant
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