[FOM] Re: On the Nature of Mathematical Objects

Dmytro Taranovsky dmytro at mit.edu
Fri Aug 13 17:45:21 EDT 2004

Vladimir Sazonov writes:
>Could not you imagine, at least in principle, that some quite 
>mathematical formalism will be found which cannot be grasped
>by set theory in its contemporary form?

One must distinguish between set theory as the study of sets, the first order
language of membership relation, and ZFC.  It is likely that every mathematical
statement can be rephrased as a statement about sets.  Statements about proper
classes are well-defined (i.e. are either true or false) only if they can be
interpreted as statements about sets. One cannot prove universality of the set
theoretical universe, and I oppose defining mathematics as set theory because
of the possibility that there is something beyond.  However, just as with the
Church-Turing thesis (which is also unprovable), every particular example
presented supports the thesis of universality of set theoretical universe.

>It seems much more
>safe to say that mathematics deals with various formalisms
>( + corresponding intuitions, of course) some of which have
>relatively (it is even tempting to say - absolutely - but this
>is definitely not true) universal character, like ZFC.

It is the language of set theory, which has "relatively (it is even tempting to
say - absolutely - but this is definitely not true) universal character." 
Ordinaly, only a small fraction of the expressivenes of that language is
needed.  ZFC is just the default formalism, the formalism you implicitly claim
you use when you announce a theorem.  In a sense, all sufficiently powerful
formal systems are equivalent:  ZFC proves phi iff primitive recursive
arithmetic proves "ZFC proves phi".  Semantically, ZFC captures much of our
basic set theoretical intuition, but our intuition about sets clearly extends
beyond ZFC.

>Say, consider the (well-known paradoxical)
>least natural number which cannot be described with less
>than one hundred symbols.

The issue here is ambiguity of the language.  Technically, English language does
not have a well-defined semantics.  The number that an English sentence refers
to may depend on the time of day or on the number of people in the room. 
However, once one specifies an unambiguous language (such as (in my opinion)
the language of set theory), that statement because well-defined, and one can
rely on intuition and abstract reasoning (and, possibly, physical experiments)
to discover more and more powerful formal systems for that language and
(perhaps) eventually decide what that natural number is.  

Best Wishes,
Dmytro Taranovsky

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