Re: Yet another inane amateur Godel question
- From: Arturo Magidin <magidin@xxxxxxxxxxxxxx>
- Date: Wed, 17 Jun 2009 17:03:25 -0700 (PDT)
On Jun 17, 6:48 pm, Ask me about System Design <grpad...@xxxxxxxxx>
wrote:
On Jun 17, 4:31 pm, Arturo Magidin <magi...@xxxxxxxxxxxxxx> wrote:
[ context snipped to save bandwidth ]
"Essentially undecidable"
sounds more like philosophy, in which case I might be tempted to
direct you to the great, late and lamented, Torkel Franzen's book on
not abusing Goedel.
"Essentially undecidable" is also a technical term. In mathematical
logic a (consistent first-order) theory T is essentially undecidable
if any consistent extension S is also undecidable.
Thanks! Note, however, that Edward talked about "essentially
undecidable problems", not essentially undecidable theories, so it
would seem that the technical meaning is not what he had in mind.
So a consistent
set of sentences T is essentially undecidable because we cannot add
any set of new sentences (in the same language I believe) to make a
set S containing T such that S is both consistent and decidable.
Do a Web search to find examples of theories T which are and
which are not essentially undecidable.
In the context of the original post, perhaps "problem" is being
confused with "theory" ?
Or perhaps the technical term is being used philosphically, much like
"normal" has a very precise technical meaning (several, in fact,
depending on which particular subfield of math you are in), but none
of them have anything to do with what a philosopher or a psychiatrist
might have in mind when they ask "is what you are doing normal?"
(-:
--
Arturo Magidin
.
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