FOM: Re: CH and 2nd-order validity

[Roger Bishop Jones]

In response to John Steel Monday, October 16, 2000 4:46 PM

> Roger Bishop Jones wrote:
>
> > Does CH have a truth value under the semantics of first order set
> theory?
> > If that semantics is exclusively determined by the semantics of first
> order
> > logic and the axioms of first order set theory (as Hilbert might have
> > insisted) then it does not.
>
>     I believe he suffers here from an illusion that lurks in
> a lot of philosophical discussions on mathematics. This is the idea
> that it is the function of formal semantics--1st or 2nd order model
> theory--to assign meaning to our mathematical language. Formal semantics
> models certain aspects of meaning assignments, but if one uses it to
> communicate the meaning of language L, one is essentially translating L
> into the language of set theory. Obviously, one cannot communicate, or
> sharpen, the meaning of the language of set theory this way.

I beg to differ.

Firstly it clearly is the function of formal semantics (--1st or 2nd order
model
 theory--) to assign meaning to 1st or 2nd order languages, and it can
properly be
used for giving the sematics of "non-logical" elements of such languages
(such as the membership assertion).

I don't know what you mean by "our mathematical language" and I have
no opinions about its semantics.
I was speaking of specific formal languages like first order set theory
or second order set theory.

Secondly I have at no point suggested that this can be done entirely
formally.
In fact I have been exploring semantic notions which cannot
be fully formalised, to name but three: 2nd-order validity,
well-foundedness and arithmetic truth.
It is precisely my point that these are important.

So far as the difficulties concerning defining the semantics of
set theory in set theory are concerned, these difficulties apply to
any attempt to make semantics precise.
All such attempts end in circularity, though the cycles may become less
plain when informal notions are used (as they ultimately must be).

It is not in the least obvious that one cannot communicate
or sharpen the meaning of set theory in this way.
A formal semantics is frequently valuable in sharpening our understanding
of a language even if blatently meta-circular.
My remarks however, were primarily about informal semantics, for
which nevertheless, model theoretic language is entirely appropriate.

>     The meaning of the language of set theory is determined by its use, in
> mathematics, science, and everyday life. One learns its meaning by using
> it, just as one learns one's mother tongue. In its use, the language of
> set theory is neither 1st order nor second order; those are properties of
> certain formal mathematical models of possible meaning assignments to its
> syntax. (I believe there was some discussion a while back about whether we
> do or should use a 1st or 2nd order language. I have no idea what the
> difference would be.)

The "meaning is use" dogma is a blight on modern philosophy.
It is trotted out as if it were a truism and then used to derive substantive
conclusions which it could not possibly justify if it were.

Its relevance is in any case exclusively to natural languages not to
artificial formally defined languages.
In relation to natural languages it is normal to distinguish between
correct and incorrect usage.
The meaning of language is "determined" by the correct usage,
not by the incorrect, but clearly we need to know the meaning
before we can make this distinction.

As to set theory, I am astounded to hear it suggested that set theorists
decide on what the concept "set" means by considering the usage of this term
"in mathematics, science, and everyday life".
Even if you think they should I doubt that they do (and I personally
don't think they should).

To chose but one simple aspect of the semantics of the term "set"
there is the question of well-foundedness.
This is semantic, there are of course perfectly coherent concepts
of set which are well-founded and which are not well-founded.
They are different concepts, but equally legitimate.
Surely no-one looks to "mathematics, science, and everyday life"
to decide which of these is the true concept of set?

A set theorist has simply to decide which of these concepts
(kinds of set) he
wants to work with (and of course, he can work with both).
It is up to him to make clear the concepts he is using by defining
his terminology clearly in his publications.
We may be interested to know which is more convenient
for use in science and elsewhere, but this does not "determine meaning",
for artifical languages this is done by fiat.

>     The mistaken idea the meaning of the language of set theory is
> determined by assigning a model to it lies behind a lot of the (excuse my
> French) baloney written on Skolem's paradox.

The meaning of a language is determined by whatever method is chosen
by the person or persons defining the language.
The notion of an "interpretation" in first order logic was invented as
and is exactly what you need to assign meaning to the "non-logical"
aspects of a first order language.
It is therefore virtually impossible to determine the meaning of a
language without directly or indirectly defining its models.

It is of course possible to give poor definitions of languages,
but to suggest that the semantics of a language cannot be defined
by giving its models seems to me incorrect.

The standard account of the semantics of classical set theory is
the informal description of the cumulative heirarchy.
This is nothing but an attempt to describe "the standard model".

Roger Jones
RBJones at RBJones.com