[FOM] second-order logic once again
Timothy Y. Chow
tchow at alum.mit.edu
Tue Sep 4 16:37:16 EDT 2012
Robert Black wrote:
> Those of us who, like myself, are card-carrying second-orderists can say
> what this means: the set of second-order validities, *a perfectly
> well-determined set of formulae*, is not r.e. (indeed not even remotely:
> it's not definable in nth-order arithmetic for any n and so on and so
> But suppose you're not a card-carrying second-orderist, so you don't
> think there's a perfectly determinate 'set of second-order validities'.
> (Perhaps you balk at the consequence that CH must have a truth-value.)
> How do you state the incompleteness theorem?
Is there anything special about second-order logic in your question?
It seems to me that you could ask an analogous question about any
mathematical statement that some people think is "meaningful" and others
don't---CH, for example. Some people think that either there is an
uncountable subset of the reals that is not equipollent to the reals, or
there isn't. Others don't think that CH is meaningful in this way.
Those in the latter camp don't *state* CH any differently. They state it
the same way, but just treat it as a formal sentence. If you say, "No,
they must state it differently, because there is definitely something
meaningful to be stated here, and the skeptics don't think the standard
formulation makes a meaningful statement, so they must have some other way
of asserting the meaningful-something here," then I think the skeptics
will disagree that there is a meaningful-something to be stated, beyond
the fact that the standard statement is a formal theorem of a formal
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