[FOM] Question about Set Theory as a formal basis for mathematics

[Harvey Friedman]

On 2/25/06 9:23 AM, "Andrea Proli" <aprol at tin.it> wrote:

> Hello everyone,
> I am a newbie here, I have not a deep knowledge of mathematics because it
> is not the primary subject of my studies. However, my personal interests
> brought me to an effort in understanding the very foundations of
> mathematics, which I assume to be (most say) Set Theory.

Are you a student in philosophy?
> 
> There is a question I would like to ask this mailing list about ZF Set
> Theory, and all other Set Theories in general. The question is: are they
> really stable, formal foundations for mathematics?

I am not sure exactly what you mean by stable, but in any case my answer is
yes.

> I mean: as far as I know, ZF is a first-order theory, and first-order
> theories have a standard denotational, model-theoretic semantics.

ZFC has been the Gold Standard for foundations of mathematics for more than
80 years. At this time, the overwhelming preponderance of mathematics is
equally founded in ZFC without the axioms of foundation and replacement.
(There are very interesting exceptions, and the situation may change
radically in the future).

The remaining axioms are instantly recognized by mathematicians, and form
(with a small quibble about the exact formulation of infinity) a system
already proposed by a famous paper of Zermelo in 1908 which I discussed a
bit in http://www.cs.nyu.edu/pipermail/fom/2006-February/010050.html

>In model  
> theory, symbols are given an interpretation in terms of sets and relations
> (which are also sets). Isn't this a circular definition?

The reasons that ZFC has attained the status of Gold Standard seem to have
no direct connection with the model theoretic semantics of first order
logic. 

The importance of models of (systems in) first order logic for f.o.m. is
more indirect. 

ZFC has been naturally divided into two parts.

1. Axioms and rules of pure logic.
2. Axioms of set theory.

(There are deep questions about the fundamental basis of this division, and
this is a longer story that is ongoing).

Can we know that we are not missing items in 1?

The standard answer is "yes, we know that we are not missing items in 1".
The reason is that 

a. Axioms and rules of pure logic are intended to apply "to any situation".
b. The preceding is appropriately formalized with the notion of validity and
logical consequence, as given by the notion of model of first order logic
(with equality). 
c. There is a theorem to the effect that, in first order logic (with
equality), the valid sentences are exactly those that can be proved using a.
Analogously for "logical consequence".

It is true that b,c are accomplished normally in terms of models as set
theoretic objects, just as you state.

However, it is well known that the models needed in order to carry out b,c
are from a very restricted class that can be properly viewed as "non set
theoretic models". 

More explicitly, there are two ways to address your issue.

A. The amount of set theory needed to carry out b,c is extremely minimal. In
fact, in an appropriate sense, b,c can be carried out within a system of
arithmetic - without any set theory.

B. We can consider only countable models, or even a very special subclass
called arithmetic models - still b,c work. And again, only a very minimal
amount of set theory is needed to do this.

The bottom line is that b,c can be carried out with a very minimal
commitment to set theory - or even (arbuably) none at all.

> The semantics of sets is defined in terms of sets, and this recursive
> definition does not seem to be explicited (kind of a "fixpoint" definition
> would be more comprehensible to me...)

I don't quite know what you mean by "the semantics of sets". If you just
mean that sets have not been appropriately defined in terms of simpler
notions, then you are right. If they were, then these simpler notions would
arguably be better for f.o.m. than set theory.

> This is quite different from a mere axiomatization: I can accept that sets
> are not defined in terms of anything else because they are the
> foundational element of mathematics, but it seems somehow "wrong" to me
> that they are defined in terms of themselves, in such an implicit
> recursion.

In the usual foundation for mathematics, the notion of set remains
undefined. 

I have some idea of what you might be driving at, but I am not clear enough
about what it is for it to make sense for me to address it. I need an
example of what difficulty you are referring to.
> 
> So, the semantics of ZF is given in terms of what ZF itself defines? Or am
> I simply confused?

As I said earlier, the force of ZFC for foundations of mathematics is not
directly related to any "semantics". Perhaps I don't know what you mean here
by "semantics of ZF".

Harvey Friedman