[FOM] first/second order logic

Paul Blain Levy P.B.Levy at cs.bham.ac.uk
Thu Nov 7 12:49:03 EST 2019

> Date: Wed, 6 Nov 2019 10:27:59 -0500
> From: Harvey Friedman <hmflogic at gmail.com>

> [...]
> First order logic has syntax and semantics. The syntax includes most
> importantly the notion of rigorous proof. This is meant as an
> idealized model of mathematical practice. Mathematicians "adhere" to
> rigorous proof in first order logic from ZFC and any rigorous proof in
> ZFC is supposed to represent rigorous mathematical thinking.
> [...]
> So if you care about modeling mathematical thinking then first order
> logic is essentially the only game in town.

Only if we start from the assumption that first and second order logic 
are the only two candidates.

But I would argue (following Mayberry and others) that first order logic 
is too strong for set theory, because its use of unrestricted 
quantification is semantically problematic.

That's why I advocate using a set theory weaker than ZFC that is not a 
first order theory and disallows unrestricted quantification.



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