[FOM] First Order Logic

MartDowd at aol.com MartDowd at aol.com
Wed Aug 28 12:06:02 EDT 2013

Expanding on this point, the difference between first-order and  
second-order logic is that in second-order logic set variables are required to  range 
over subsets of the domain.  Second-order logic is not usable in  practice 
because there is no recursively enumerable set of axioms (theorem 41C  of 
Enderton's logic book).
Second-order set theories (such as NBG) are actually first-order theories  
because they are given as a set of axioms, and proofs must use the rules of  
(many-sorted) first-order logic.  In both category theory and small large  
cardinal theory, this is a very useful system, and is a conservative 
extension  of ZFC   I think logicians will always be interested in whether  
theorems of mathematics can be proved in ZFC.  We already know from Boolean  
relation theory that this is not always the case.
- Martin Dowd
In a message dated 8/26/2013 7:56:59 A.M. Pacific Daylight Time,  
hmflogic at gmail.com writes:

Imagine the situation if one proposed a second order version of ZFC, in  
the serious sense of second order here (not the fake notion used in so called  
second order arithmetic, where it is simply a two sorted first order 
theory).  There would be widespread disagreement about whether a purported proof 
within  second order ZFC is correct or not. This directly stems from the fact 
that it  is not a first order formulation. 

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