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Topic: Whitehead & Russell on Reductio A.A. and when Constructivism is mainstream #200; 2nd ed; Correcting Math
Replies: 7   Last Post: Oct 19, 2009 1:18 AM

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Keith Ramsay

Posts: 1,745
Registered: 12/6/04
Re: Whitehead & Russell on Reductio A.A. and when Constructivism is
mainstream #200; 2nd ed; Correcting Math

Posted: Oct 18, 2009 11:05 PM
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On Oct 18, 4:26 pm, Nam Nguyen <namducngu...@shaw.ca> wrote:
|Be careful. Keep answering my question with *if*-this-then-and-*if*-
that-then
|and you might end up agreeing with me there's a meta mathematical
statement
|that has no absolute truth, which would signal the advent of
relativity in
|mathematical reasoning.

If you want to start talking about "absolute truth"
you need to have some kind of grasp of what you
mean by the term.

If I adjoin to Q a new primitive predicate A(n)
and provide no new axioms, then lots of statements
pertaining to A are then independent of Q. Is A(0)
true? Can't tell. Is A(1) true? Can't tell. As
long as I don't supply you with an interpretation
(so that A has a specific meaning) these sentences
don't constitute specific statements any more than
"x=3" does before we say what x represents. This is
a kind of relativity that always exists. It comes
from just not specifying what "A" refers to.

If we consider sentences in the language of Q,
lots of people will say that each one is either true
or false in an absolute way, even if we have no way
of determining which it is. There's no use in trying
to attack this point of view by means of twiddling
around with elements of mathematical logic. You need
to make some kind of philosophical argument about
what is required for a statement to be absolutely
true or absolutely false. (And if you do, plenty of
people won't be persuaded. That's just how
philosophy works.)

|Note that I emphasized on "_syntactically_" to signal that the
question and
|the answer be discussed syntactically, not model-theoretically.

There's no distinctive "model theoretical"
consistency distinct from syntactic consistency.

[...]
|In my recent replies to KR, the key question I have is:
|
| >> Syntactically speaking, would it be possible that a formula F be
| >> undecidable in all consistent extensions of the Robinson
Arithmetic Q
| >> formal system?

As Aatu explained, it's not.

If F were independent of all consistent extensions
of Q, then since Q is consistent, then F would be
independent of Q. Hence there would be no proof of
~F in Q, and Q+F would be consistent. By the
original assumption again, F would be independent
of Q+F. But F is provable in Q+F, a contradiction.
The original assumption, then, that F is independent
of all consistent extensions of Q is impossible.

This proof does not rely on the law of excluded
middle. It goes straight from the assumption that
F is independent of all consistent extensions of Q
to a contradiction without dividing by cases. There
is no "either-or" needed.

|---> The Objective
|
|The objective I have in this question is to establish a meta
mathematical
|statement (about FOL) that would have no absolute truth though in its
form
|it'd appear as if there would be an absolute truth value.

It's futile to try unless and until you get some
basis for claiming that a statement has no
absolute truth value.

I don't think your strategy of resorting to meta-
levels is promising at all. To show that you need
to go to a "meta" level you need already to show
that there's some issue with the original language
that calls for it.

There are cases where we would say that a sentence
in a language doesn't have a fixed meaning. We
determine that not by considering it syntactically,
but by considering what it means in the given
context. If you would prefer not to get off into
some philosophical argument, then as there's no
non-philosophical way of addressing issues of
meaning or ultimate truth, you just have to set
the whole question aside.

Keith Ramsay



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