FOM: Re: first order and second order logic: once more
V. Sazonov
V.Sazonov at doc.mmu.ac.uk
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
interpretations.
Vladimir Sazonov
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