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

Sun Feb 26 11:03:22 EST 2006

Harvey, Aatu, William,
I all thank you for your help. I really appreciate when someone loses part  
of his time to avoid that someone else loses an amount of its own. This is  
a really huge scientific service. Unfortunately, I think I am not learned  
enough to grasp the profound issues embodied in your answers. I am a  
Computer Science PhD student, with a very narrow training in mathematics.  
However, I try to clarify my point.

Harvey, by "stable" I meant "something that is not going to collapse on  
itself", in the sense that if I lean on your shoulders to stay up, and you  
lean on my shoulders to stay up, it is very likely that we both fall down.  
This is the pattern I was seeing in what purports to be the foundation of  
mathematics: on the one hand, Set Theory (say ZF, but I suppose it could  
be any first-order Set Theory) leans on Model Theory because, as a  
first-order theory, its semantics ("what is it about") is given by the  
family of all possible first-order structures ("states of affairs") in  
which their formulas are valid - i.e. the admissible arrangements of the  
world it describes; on the other hand, Model Theory leans on Set Theory  
because such descriptions, i.e. first-order structures (interpretations),  
are made up from a domain of discourse, which is a "set" of individuals,  
 from a "set" of relations, and from a "set" of functions, relations and  
functions being in turn "set"s.

Hoping that my overall ignorance has not prevented me from understanding  
what you say, such "sets" Model Theory refers to are not exactly the same  
"things" ZF is about; instead, they are something described by a more  
"specialized" and restricted theory, thus it is not true that "ZF ---leans  
on---> MT ---leans on---> ZF", because actually "MT ---leans on---> XX"  
(and so the loop is not closed), where XX is not ZF but something less  
general and less powerful or simply something different. Is this a correct  
interpretation of your words? If so, how is XX formalized, what are those  
"sets" Model Theory talks about?

Aatu, thank you too for your explanation. What I ambiguously called the  
"semantics of sets" (I apologize for this ambiguity, Harvey) should  
actually have been "what variables in set theory are allowed to range  
over". Aatu, you say:

"The semantics of the first order language of set theory in which these  
axioms are formulated in is given by saying that the quantifiers range  
over the so called cumulative hierarchy of sets obtained by starting with  
the empty set and iterating the powerset operation "as long as possible".  
This hierarchy, and the concepts necessary to understand its description,  
are not in defined by ZF(C) but must be understood the same way one  
understands what the natural numbers are, what a tree is and what this or  
that mathematical concept means, whatever that way is."

I did not understand what do you mean by "the concepts necessary to  
understand its description, are not in defined by ZF(C) but must be  
understood the same way one understands what the natural numbers are". You  
mean that natural numbers are not formally defined based on set theory, in  
Number Theory for example (I don't know, I'm guessing)? What are the  
involved "concepts" you talk about? Also, I did not understand why  
variables can range only over the "cumulative hierarchy", which I suppose  
to be 0,{0},{{0}},{{{0}}},... if you can obtain it by only applying the  
powerset operator iteratively: I would say that variables can also assume  
value {0, {0}}, for example, which is not generated by only powerset  
operator and empty set. But I am sure I'm missing something.

William, thank you for your reference. I have read the nice  
(unfortunately, the only) review published on Amazon and it looks like the  
subject is of interest to me. However, I actually have a tall stack of  
books on my desktop that I need to read before pushing others on top of  
them :) By the way, I will certainly keep in mind your book for future  
learning.

Thank you all,

Andrea

In data Sat, 25 Feb 2006 18:01:06 +0100, Harvey Friedman  
<friedman at math.ohio-state.edu> ha scritto:

> 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
>
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