Re: Another Reason Why Collatz is Unprovable


GJ Woeginger wrote:
Craig Feinstein <cafeinst@xxxxxxx> wrote:
#> My proof on arXiv.org rules out a proof by contradiction because it
#> rules out any proof. It is a completely logical proof. No one on usenet
#> has found any flaws in the proof. The only things people have been
#> doing are claiming that my definition of "random" is not rigorous
#> (because it does not specify a formal language, which is really
#> irrelevant in the context of my proof) and giving straw-man arguments
#> which prove false statements and claiming that my proof uses the same
#> type of arguments. If you have faith in logic, then you should have
#> faith in my proof.
#
# My proof never claims that you have to treat every integer
# individually. My proof says: let's pretend that we have a proof of
# Collatz with L bits. It then shows that there is a specific n for which
# any proof that Collatz halts at one with input n requires at least L+1
# bits. From this, I conclude that any proof of Collatz must have at
# least L+1 bits, so we have a contradiction, as our pretend proof only
# has L bits. Therefore, Collatz is unprovable.

In your paper, you do not discuss at all the axiom system that
your are using. All your statements on "proofs" are absolute
statements, that do not even touch the question of the underlying
axiom systems in the slightest way.

Suppose, that I use as axiom system PA plus the axiom that the
Collatz algorithm always reaches 1.
In this axiom system, there is a very short and very trivial
proof that the Collatz algorithm always reaches 1.
This horribly collides with your absolute statement that
"Collatz is unprovable".

Using this line of reasoning, you can make 2+2=5 an axiom and then
prove 2+2=5, contradicting the well-established fact that 2+2=5 is
unprovable, i.e., that it is impossible to prove that 2+2=5. If you
think mathematics is some kind of game, then this is perfectly OK, but
if you believe like me that mathematics is more than just a game, then
your argument is very weak.

The only axiom needed for my proof to work is the axiom that in order
to prove that T^k(n)=1, it is necessary to specify the formula for
T^k(n) in the proof. I don't know anyone who doesn't accept this axiom
as obviously true.

Craig


--Gerhard

___________________________________________________________
Gerhard J. Woeginger http://www.win.tue.nl/~gwoegi/

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Relevant Pages

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  • Re: Another Reason Why Collatz is Unprovable
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  • Re: Another Reason Why Collatz is Unprovable
    ... # any proof that Collatz halts at one with input n requires at least L+1 ... # least L+1 bits, so we have a contradiction, as our pretend proof only ... that I use as axiom system PA plus the axiom that the ... that the Pythagorean Theorem is wrong by claiming that if one were to ...
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