[FOM] second-order logic once again

[Timothy Y. Chow]

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 
> on).
>
> 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 
system.

Tim