Re: Goedel - interesting problem?

From: Aatu Koskensilta (aatu.koskensilta_at_xortec.fi)
Date: 06/03/04 

Date: Thu, 03 Jun 2004 13:27:47 +0300

Acme Diagnostics wrote:

> But in the meantime, I would like to hear
> your substitution for "true in a set of axioms" in less than 26
> characters which is just as *useful* and *reliable* in argumentation
> as further clarified below.

What's wrong with "true arithmetical sentence" or "true sentence about
natural numbers formulated in the language of the formal theory"? Both
of these expression should be easily understood, while I for one can not
fathom what exactly "true in the set of axioms" should mean.

> Due you understand the implications of
> "useful" and "reliable," as the questioner and I have just finished
> examining those, among other things, and that is part of our
> overall context?

Yes. Gödel's theorems can be expressed in an useful, concise and
understandable way without invoking such strange things as "truth in a
set of axioms". For example:

  Gödel's 1st incompleteness theorem says that for any consistent formal
  theory T containing elementary arithmetic we can[1] construct a true
  sentence about natural numbers which is not provable in T.

  Gödel's 2nd incompleteness theorem says that if T is a consistent
  formal theory containing elementary arithmetic and certain simple facts
  about formal provability are provable in T, the statement "T is
  consistent" - formulated as an assertion about natural numbers by means
  of a technical coding mechanism - is not provable.

[1] That is, there is an algorithm for constructing the sentence.

> A mathematical logic textbook would be unuseful in argumentation
> about the Goedel theorem except to mathematical logicians, only
> one profession among many equipped to inference all possible
> implications ("effects") of the Goedel theorem.

But surely one must know Gödel's theorems in sufficient detail in order
to be able to draw any sensible implications from them. 

-- 
Aatu Koskensilta (aatu.koskensilta@xortec.fi)
"Wovon man nicht sprechen kann, daruber muss man schweigen"
  - Ludwig Wittgenstein, Tractatus Logico-Philosophicus

Relevant Pages

  • Re: Goedel - interesting problem?
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