FOM: second-order logic is biased

[Stephen G Simpson]

My posting of 24 Mar 1999 17:55:24 (the previous posting, the one
about Platonist/realist assumptions implicit in the idea of absolute
standard semantics) was titled "second-order logic is a myth".  But I
should have titled it "second-order logic is biased".

Randall Holmes 24 Mar 1999 13:48:53 writes:

 > ... mathematics (even arithmetic) cannot be completely captured by
 > any formal system.  This is a well-known fact.

I'm not so sure that's a well-known fact.  Isn't it more of a
well-known philosophical assumption?  Extreme formalists might want to
argue that arithmetic is *nothing but* some particular formal system.
Thus your `well-known fact' appears to be a prejudgement against the
extreme formalists.

Tarski's theorem on undefinability of arithmetical truth may be viewed
as bolstering your `well-known fact'.  But Tarski's theorem itself
assumes (on Platonist/realist grounds?) that there exists a truth
predicate for all arithmetical sentences.  And, in order to conclude
that `arithmetic cannot be completely captured by any formal system',
you probably want to identify `arithmetic' with this truth predicate.
Don't these assumptions amount to a prejudgement against the extreme
formalists?

Could this be another instance of the built-in bias of `second-order
logic with standard semantics' against formalism and in favor of
realism/Platonism?  See also my posting of 24 Mar 1999 17:55:24.

Once again, I am speaking as neither a formalist nor a
realist/Platonist.  I only want all sides to get a fair hearing.

-- Steve