From: kramsay@aol.commangled (Keith Ramsay)
Subject: Re: Goedel's theorem
Date: 04 Jun 2000 15:48:04 GMT
Newsgroups: sci.math
Summary: [missing]

In article <3934ada7.8111290@news.mindspring.com>,
Postin@thegroup (Dismas) writes:
|>  >The effort to ensure consistency has led to more and more complicated
|>  >restrictions on the structure of models and systems, until the notion
|>  >of "true" in mathematical logic hardly has any relation to the
|>  >thoughts of the naive student seeking philosophical truth.
|>
|>  I have no idea what you're talking about. What are these "more and
|>more complicated restrictions on the structure of models and systems"?
|
|I had in mind Russel's paradox.

Russell's paradox did ruin Frege's system, yes, but we should
distinguish between Frege's set theory and Cantor's set theory.
Cantor's set theory wasn't ruined at all. He had already described
the one basic "restriction" required to avoid the paradoxes, and as
I recall had even warned Frege about it. Cantor recognized a
distinction between sets and what he called "absolutely infinite"
collections-- what we would now call "proper classes", such as
"everything" or "all ordinals", which he regarded as being "too big"
to be collected together as unities.

The formalization most commonly associated with Cantorian set theory
is ZFC. The axioms of ZFC were chosen from prior mathematical work.
Nobody has found a need to restrict ZFC further, and it appears that
mathematical proofs ordinarily can be formalized in it.

So the process of making "restrictions" was not progressive as you
describe it. The "restrictions" didn't get more complicated. I find
the logicians' notion of "truth" natural enough; do you have some
qualms with it for some reason?

|It is discussed in the last several
|paragraphs at
|
|http://www-history.mcs.st-andrews.ac.uk/history/HistTopics/Beginnings_of_
set_theory.html
|
|the comment on "restrictions" is echoed in the sentence beginning
|"Much of the ..." just before the heading HISTORY in
|
|index/03-XX.html#INTRO

It's good of you to provide references, but this last cited discussion
isn't about restrictions, but further developments in logic which were
spurred on by these foundational questions.

|The notion of "first order systems" is just such a restriction.
|Concerning the first point, note the distinction the author makes at
|the top of the page between "semantic" reasoning and "syntactic"
|reasoning.  

People don't consider first order axiom systems because they think
they have to limit themselves to them. Well, not ordinarily.

|>  >It seems blatantly obvious that every non-empty collection of natural
|>  >numbers has a smallest element.  Is it tue?  It can't even be said in
|>  >a first order system, so what does the question mean?
|>
|>  This is a fine example of how exposure to logic can cause people
|>who are otherwise no doubt perfectly sensible to say very strange
|>things. 

Very true!

|I gather that you consider the well-ordering principle for the natural
|numbers to be a first-order property.  Or perhaps that the natural
|numbers have a "platonic" existence.  Very strange.
|
|As I said, this is all elementary, no doubt disposed of in the first
|twenty minutes of a logic course.  

During the first class in a logic course, they may well expose you to
the concept of a formal language, which may or may not be supplied
with an interpretation. When you refer to "what can be said in a
system", you are referring to some kind of semantics for the system.
For instance, I might say "you can't say `there exists a measurable
cardinal' in PA", and it would mean that there doesn't exist a
statement in that language whose associated meaning is that a
measurable cardinal exists. By contrast, you can "say" that there are
infinitely many twin primes in PA because there is a sentence
"(n)(Em) (m>n)&..." whose associated interpretation is that there are
infinitely many twin primes.

Some students appear at some point to get the idea that the above
procedure of assigning an interpretation to a first-order language
isn't possible after all, and that it isn't actually possible to say
all those things that we had previously thought we were saying. This
is not of course something that a logician will ordinarily teach you.

In some cases, the thinking seems to be that unless we have a list of
first-order sentences which are modelled by the intended model, and
only by the intended model, we cannot specify the model. It is this
view, however, which is "very strange". In the midst of an analysis
of first-order language, which has dependended heavily upon our being
able to talk coherently about arbitrary sets of elements (models),
strings of symbols (indexed by integers) and the like, we decide
abruptly that it's all stuff that cannot really be said. But then
we continue to say it.

If you're going to decide that discussions of natural numbers aren't
meaningful after all, you're back at square one. Without natural
numbers, you don't have "first order" languages as usually defined,
and you have to find something else to talk about.

|I was formerly uncertain as to whether you were trying to help via
|leading questions, or were simply seeking an outlet for snide remarks.
|I am no longer uncertain and see no reason for further comment.

Actually, I think it was all meant helpfully. You appear to think that
you've disposed of Torkel Franzen's point of view rather easily, and
it's a pity that this sort of misperception of the situation puts you
beyond the kind of help he can offer.

Keith Ramsay