Re: Another Reason Why Collatz is Unprovable

Aatu Koskensilta wrote:
Craig Feinstein wrote:
I would define a reasonable axiom system as an axiom system in which
there is no doubt that its axioms are true.

There are no reasonable axiom systems in that sense. The truth of every
formal theory ever put forth in logic has been doubted by someone. In
order to prove a statement of form "P isn't provable in any reasonable
theory" a mathematical characterization of reasonableness is required.

I disagree. There are certain axioms which no one disputes. a=b & b=c
implies a=c is an example. An axiom that I am using in my Collatz paper
could be stated:

"Given a function f, number n, and constant c, in order to prove that
f(n)=c, it is necessary to specify the formula for the function f (when
it is applied to n) in the proof."

For example, if we let f=T (the Collatz function), n=3, c=5, we have to
specify the formula T(n)=(3n+1)/2 (since n=3 is odd), so
T(3)=(3*3+1)/2=5. Just try to prove T(3)=5 without specifying the
formula for T(n) (when it is applied to n) in the proof. It's clearly
impossible.

Anyone who disputes this axiom is fooling himself or herself. A major
criticism of my proof, I think, centers around the rejection of this
axiom or more likely the misunderstanding of this axiom. Of course,
this does not make it an unreasonable axiom - they could reject a=a as
an axiom, but it would still be a reasonable axiom.

Because the axiom of choice is controversial (or at least was
controversial) and there are legitimate reasons for concern as to its
validity, as it is nonconstructive, I would not categorize it as
"reasonable" in the context of this discussion. If you are talking
about axioms like this, then I would agree with you.

Craig


--
Aatu Koskensilta (aatu.koskensilta@xxxxxxxxx)

"Wovon man nicht sprechen kann, daruber muss man schweigen"
- Ludwig Wittgenstein, Tractatus Logico-Philosophicus

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