Re: Continuum hypothesis

On Sep 10, 7:37 pm, Alan Smaill <sma...@xxxxxxxxxxxxxxxx> wrote:
"R. Srinivasan" <sradh...@xxxxxxxxxx> writes:
On Sep 7, 9:45 pm, Alan Smaill <sma...@xxxxxxxxxxxxxxxx> wrote:
"R. Srinivasan" <sradh...@xxxxxxxxxx> writes:
On Sep 6, 12:05 am, MoeBlee <jazzm...@xxxxxxxxxxx> wrote:
On Sep 5, 11:11 am, "R. Srinivasan" <sradh...@xxxxxxxxxx> wrote:
..
Look at this FOM post by Torkel Franzen:

http://cs.nyu.edu/pipermail/fom/1998-March/001447.html

Also look at the thread titled "What is the standard model for PA"
that follows this post. TF defends an informal notion of natural
numbers and the people who raised objections are Vladimir Sazonov and
Vaughan Pratt.

TF insists that the consistency of PA is to accepted as an absolute
truth, and you, George, objected vehemently in the following sci.logic
thread titled "Godel statements other than G"

http://groups.google.com/group/sci.logic/browse_frm/thread/222c6e5248...

Again, I find nothing there makes it clear to me that Franzen posted
anything
that can be fairly paraphrased as "Classical logic only makes sense
if
there is a notion of truth that is independent of the human mind and
axiomatizations." And you've added the word "absolute" now too. So,
again, I'm not aware of anything that Franzen wrote (though, of
course, my knowledge is not definitve on this matter) that admits of
being so glibly paraphrased as an assertion of "absolute truth".

I take truths that are independent of (FOL) axiomatizations or
theories to be "absolute" truths.

Can children not be aware of arithmetical facts, simply because they
have not seen FOL axiomatisations of them?

That seems to be what you are saying.

If by arithmetical facts you mean assertions that only involve
finitely many numbers, then yes, children can be aware of them. But
these facts are easiy provable within arithmetic. That is not what I
had in mind. A statement like "Given any natural number n, there is a
natural number m such that m > n" is about infinitely many natural
numbers. Although children may get an appreciation of this assertion,
they do not have any proof of it.

I was thinking of statements such as the associativity of addition,
the commutativity of multiplication.

In fact it will fail for a
ultrafinitist. So the truth of ths assertion is really dependent on
the theory that one has in mind.

Of coures, the provability depends on the theory.
Do you mean by "truth" anything other than provability?

Well, I must confess that I do identify truth with provability as is
done in NAFL. In my (strict) view any other notion of truth is
"Platonic", but you obviously disagree.

But if someone were to insist that
this statement is "true", period, without any reference to the theory
that proves it, that someone would be asserting as "fact" that there
are indeed infinitely many natural numbers.

What about commutativity of addition, say?

In so much as commutativity of addition is intended to apply over a
domain of infintiely many natural numbers, then yes, truth of such an
assertion must be with respect to theories. One could again assert
that commutativity of addition must necessarily hold by the very
definition of natural numbers, but the fact is that the definition
takes for granted the existence of infinitely many naturals, and that
is an axiomatic postulation rather than any fixed truth.

Where "are" they?

Do we need to answer this before understanding commutativity?
Apparently you are happy that 1+1=2 is true;
where "is" 1?

Right there, before your eyes. You postulate the existence of "1" and
maybe provide a construction for it (say, in terms of sets). As long
as you are dealing with finitely many such constructions, they are
there right before your eyes. That is not the case with infinitely
many such constructions and one needs an axiomatic framework (at least
in my view) to deal with these.

It seems
to me that an assertion of absolute truth is being made and the person
concerned is appealing to a Platonic universe of infinitely many
natural numbers.

It is equally consistent with a view that says we are dealing with
mental constructs, which let us envisage going from any given natural
number to its successor.

Or else one could argue that the individual concerned is really making
an axiomatic assertion. I.e., the individual is either asserting an
axiom or else a theorem of some theory T which (s)he has in mind. This
is the NAFL positon, at least for propostions that are formalizable
within NAFL theories.

Which would mean that almost no-one ever makes
(universally quantified) arithmetical assertions.

I say they *should not* do so, without reference to the theory that
proves such propositions.

Regards, RS

.

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