
Re: Whitehead & Russell on Reductio A.A. and when Constructivism is mainstream #200; 2nd ed; Correcting Math
Posted:
Oct 18, 2009 11:05 PM


On Oct 18, 4:26 pm, Nam Nguyen <namducngu...@shaw.ca> wrote: Be careful. Keep answering my question with *if*thisthenand*if* thatthen 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 modeltheoretically.
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 "eitheror" 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 nonphilosophical way of addressing issues of meaning or ultimate truth, you just have to set the whole question aside.
Keith Ramsay

