[FOM] n-th order ZFC

Roger Bishop Jones rbj at rbjones.com
Sat Jul 9 15:04:33 EDT 2011

On Saturday 09 Jul 2011 05:09, meskew at math.uci.edu wrote:

> Can you name any mathematical problem that can be solved
> by appealing to the categoricity of second order set
> theory?  It seems that the power of first order set
> theory actually lies in its ambiguity, because without
> that we would not have the techniques of downward
> Lowheim-Skolem, reflection, ultraproducts, elementary
> embeddings, inner models, forcing.

There has been much previous debate about the merits of 
second order logic on FOM, in which I have participated, and 
which can be seen in the archives.

On this occasion I have not been engaged in advocacy, but 
have simply been providing factual answers to questions 
posted to FOM.
In my answers I have noted my belief that for none of the 
purposes for which I find it convenient to use higher order 
logics are they strictly essential.

If anything I have said is incorrect then I would be pleased 
to be corrected.

Note however that none of the results you mention ceases to 
be true for someone using higher order logic (though of 
course they are not about higher order logic).  If they are 
relevant to the problem they can still be used (and I do 
sometimes use higher order logic for reasoning about first 
order languages).

Roger Jones

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