FOM: Re: first order and second order logic: once more

V. Sazonov V.Sazonov at
Sat Sep 2 10:48:25 EDT 2000

I know only one "scientific" way to define what is 
a standard model (for arithmetic or for second order logic). 
Fix a reasonable version of formal set theory like ZFC and 
define this concept as well as syntactic notions and validity 
of second order formulas there as usually. It is in such 
framework (in which else?) in which undecidability of SOL and 
other related issues may be discussed. On the other hand, 
such a definition of standard model(s) is nothing else than 
a reducing of SOL to First Order Logic based set theory ZFC. 
Deducing in ZFC of so translated SOL-formulas will give 
SOL-"tautologies" relativized to ZFC, and analogously for 
any other formal set theory. There was suggested no other 
way to consider SOL-tautologies. 

These considerations are known to everybody here. Then, what 
is the point for the discussion (if not some related technical 
questions)? I guess, it is assumed here some non-relativized 
(as above), but "absolute" (Platinistic or so called 
"Realistic") concept of standard model(s) existing independently 
on and prior to any formalisms. It seems to me impossible to 
get anything reasonable in this "theological" way. Anyway, 
something mathematically interesting may be done only in a 
formalized approach, because, essentially,  

        mathematics = formalizing intuitions. 

Also note that during formalization the intuition usually 
evolves from amoebae or embryo like state to something higher 
organized and developed, and eventually cannot exist alone, 
as non-formalized, "skeleton-free". For example, our 
intuition on standard model(s) can exists only in the 
framework of ZFC or the like. That is, it is essentially 
relativized and non-absolute. 

There is a good (and of course, not complete) analogy with the 
views on absolute vs. relativistic space-time. Formal systems 
are like coordinate systems with the help of which only we 
can work out (mathematically or rigorously) some approaches 
to the nature or our ideas and intuitions, and between which 
there is a possibility of formal translations / relative 

Vladimir Sazonov

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