Re: does sqrt(2) exist in CM?

tchow_at_lsa.umich.edu
Date: 02/15/05 

Date: 15 Feb 2005 14:42:14 GMT

In article <1108344383.188462.163160@c13g2000cwb.googlegroups.com>,
 <examachine@gmail.com> wrote:
>In elementary number theory, one can construct problems which are not
>in general solvable by any finite axiomatic system. This is an informal
>statement. I leave it to you as an exercise to read and understand the
>relevant incompleteness theorems, and work out the construction of the
>diophantine equation(s) which I refer to.

Torkel Franzen understands these results; he is simply trying, in his
inimitable gadfly style, to get *you* to state them precisely, instead
of garbling them. The "exercise" you set for him is clearly one that
you haven't worked out in detail yourself.

Let me try to get you started. First of all, finite axiomatic systems
don't "solve problems"; they prove theorems. Is this a quibble? Well,
if we simply restate your claim as: "One can state theorems which are
not provable in any finite axiomatic system," then this is simply false,
even if we restrict to consistent axiomatic systems. Any true statement
about the integers (expressible in the first-order language of arithmetic)
is a theorem in the finite axiomatic system whose sole axiom is precisely
that true statement. This is a trivial fact---omega or no omega, Chaitin
or no Chaitin.

So I suggest you try the following exercise: State Chaitin's result
precisely, and explain why it doesn't contradict the "trivial fact"
I just stated. (And never mind the question of whether it "extends"
Goedel's results; that's a distraction.) 

-- 
Tim Chow       tchow-at-alum-dot-mit-dot-edu
The range of our projectiles---even ... the artillery---however great, will
never exceed four of those miles of which as many thousand separate us from
the center of the earth.  ---Galileo, Dialogues Concerning Two New Sciences

Relevant Pages

  • Re: does sqrt(2) exist in CM?
    ... one can construct problems which are not ... >in general solvable by any finite axiomatic system. ... I leave it to you as an exercise to read and understand the ... don't "solve problems"; they prove theorems. ...
    (sci.math)
  • Re: does sqrt(2) exist in CM?
    ... one can construct problems which are not ... >in general solvable by any finite axiomatic system. ... I leave it to you as an exercise to read and understand the ... don't "solve problems"; they prove theorems. ...
    (comp.theory)