Re: Goedel - interesting problem?

From: Spike (ismy_at_friend.arf)
Date: 06/14/04 

Date: Mon, 14 Jun 2004 14:44:53 GMT

"Acme Diagnostics" <LFinezapthis@partpostmark.net> wrote in message
news:<40c9a21a$0$92008$45beb828@newscene.com> the following;
>
>> I counted only two criticisms still standing:
>>
>> 1) "What does it mean for a statement to be true in a set of axioms?"
>> This is also the second critic's outstanding question.
>> 2) Explain "true" instead of the Gödel's "provable." See Chris's
>> 200 line post on Sunday.
>>
>> Why not just answer these two questions?
>
>Because so far only Goedel experts have asked them, apparently only
>those unable to inference the entire first sentence in context (happens
>to me too when I'm the expert). They only have the 600X lens whereas
>the assumed reader only has the 50X lens. I repeat, when you look through
>a telescope at 50X you see *completely different things* than you do at
>600X. Saturn becomes a mere pixel in the larger picture. The first
>sentence reads:
>
>"...any set of axioms at least as rich as the axioms of arithmetic has
>statements which are true in that set of axioms, but cannot be proved
>by using that set of axioms."

Just replace the second "set of axioms" with "axiomatic system," and
perhaps again in the second paragraph. The iteration is preserved.

>For the verification of terms, see the second group of appended google
>references. The entire sentence would be inferenced in the top level of
>explanatory description by most educated laybpersons as (for one
>example among myriad variations that say about the same thing:):
>
>"Arithmetic or math, but not exactly, includes true statements that cannot
>be proved by it's axioms."

True, but now it survives examination at a higher power. The last
paragraph fits better. That paragraph is optional, as it does not
belong to the explanation. I believe you added it yourself from a
second message.

I would like to see something about the larger implications of the
theorem at the end. I think it is worth the length. This also serves
your stated purpose.

Relevant Pages

  • Re: Goedel - interesting problem?
    ... Let's refer to the "Masterpiece of explanatory text" aka "Effect of the ... Goedel Theorem" as "the article" for convenience. ... >say exactly what particular axioms of arithmetic one has in mind. ... distinction changes lots of editing criteria for the article. ...
    (sci.logic)
  • Re: Goedel - interesting problem?
    ... "...any set of axioms at least as rich as the axioms of arithmetic has ... statements which are true in that axiomatic math system, ... in the 2nd paragraph for essentially no change in length. ... Goedel proved that any set of axioms at least as rich as the axioms ...
    (sci.logic)
  • Re: What is the 1st order formal system known as PA?
    ... can be deduced in the predicate calculus from the Peano axioms. ... the very paragraph right above! ... It hasn't got a definition of "PA-ish" theory in it. ...
    (sci.logic)
  • Re: Goedel - interesting problem?
    ... >Goedel proved that any set of axioms at least as rich as the axioms ... Do you want to confine it to math? ... trouble spot in this paragraph whatever you do. ...
    (sci.logic)
  • Re: Deep Thoughts # 17: Liar Paradox is a Formal Metamathematical Theorem
    ... >> set of axioms. ... >> follow from the previous part of the paragraph is if we ... David C. Ullrich ...
    (sci.math)