FOM: properties

[Vladimir Sazonov]

Randall Holmes wrote:

> Sazonov said:
>
> We have too unsteady ground for discussing 2nd order logic in a
> "pure" form outside ZFC or the like 1st order formalism.  The
> only existing way to make any informal idea or notion (like
> `property') precise and mathematically rigorous consists in an
> explicit presenting some formal rules of reasoning on this
> subject.  We are doing this usually via suitable extension of
> 1st order logic by non-logical axioms.
>
> Holmes answers:
>
> This is a widely held position with which I do not agree.  I do know
> what a property or predicate is -- it is something which is true of
> some things and not of others.  The problem in Sazonov's first example
> is that the "property" he exhibits is not well-defined;

What does it mean "well-defined"? Say, is any property given by 
an arithmetic formula involving unbounded quantifiers (if you do 
not like my first example) well-defined?  Are you sure that such 
a property is either true or false for each natural number? How 
can you check that? Is this an act of belief? Sorry, I am not 
religious, especially in science (and since some time also in 
arithmetic). I can adopt such a belief, but only _temporary_, 
just as a game. 

> I agree with Sazonov that second order logic cannot be effectively
> used without setting up a (necessarily partial) formalization,

"Partial" assumes that there exists something "total" which we 
are trying to approximate. I am even not sure that there exists 
any reasonable "limit" of our approximations/formalizations.

which
> will use first-order logic (and can be construed as a first-order
> theory with nonlogical axioms).
>
> By its nature, a completely "arbitrary" property is something which
> one cannot actually exhibit.  The fact that given any formal scheme
> for enumerating properties of natural numbers one can define a
> property of natural numbers (by diagonalization) which cannot be
> expressed in that scheme strongly suggests that there are "arbitrary"
> properties of natural numbers, though.

Whether it "suggests" depends on a person. It is rather a 
widespread "self-suggestion". My personal experience with "the" 
powersets even of finite sets or with finite binary strings in 
the framework of Bounded Arithmetic (as well as, of course, 
lessons which I got from the independence results of G"odel and 
Cohen) "suggests" me strongly to think in a different direction.

The fact that there are
> members of a kind which I can't become acquainted with does not imply
> to me that I can't be aware of the existence of that kind (or quantify
> over it).

It is good "to be aware", but wherefrom does it follow that 
quantification over "all", even "definite" properties (how could 
we able to understand in _general_ what is "definite" and, 
especially, what is "all"??) makes any sense?  Especially if 
nobody can even check this. It is only an act of belief for 
self-calming if somebody needs such. What can we do practically 
with all of this? 

By the way, what is then "true", standard, unique up to 
isomorphism model for arithmetic if second order quantification 
is so doubtful? 

When we are working with a first-order theory we are normally 
"wearing Platonist's hat" (as Simpson once said).  Is not this 
weak, innocuous Platonism quite enough to our comfortable 
working with any such theory? Why do we need anything more 
in mathematics and its foundations? 


Vladimir Sazonov