Re: The IDS, the EDS and the DBMS
From: Jan Hidders <jan.hidders_at_REMOVETHIS.pandora.be>
Date: Thu, 16 Sep 2004 22:09:17 GMT
Message-ID: <pan.2004.09.16.22.12.08.821435_at_REMOVETHIS.pandora.be>
>
> Actually, that extract was pasted from
>
> http://www.nationmaster.com/encyclopedia/real-number
>
> where in the next paragraph we read:
>
> "It is not possible to characterize the reals with first-order logic
> alone: the Löwenheim-Skolem theorem implies that there exists a
> countable dense subset of the real numbers satisfying exactly the same
> sentences in first order logic as the real numbers themselves."
>
> Ah, I remember this reference! But, it doesn't seem to have "real"
> mentioned anywhere:-(
Date: Thu, 16 Sep 2004 22:09:17 GMT
Message-ID: <pan.2004.09.16.22.12.08.821435_at_REMOVETHIS.pandora.be>
On Thu, 16 Sep 2004 14:36:08 -0700, Mikito Harakiri wrote:
> "Jan Hidders" <jan.hidders_at_REMOVETHIS.pandora.be> wrote in message
> news:pan.2004.09.16.19.30.51.642372_at_REMOVETHIS.pandora.be...
>> On Wed, 15 Sep 2004 09:51:35 -0700, Mikito Harakiri wrote: >> > "Jan Hidders" <jan.hidders_at_REMOVETHIS.pandora.be> wrote in message >> > news:pan.2004.09.15.16.09.04.217597_at_REMOVETHIS.pandora.be... >> >> >> The question about the complexity of normalization is also >> >> >> interesting. From Tarksi we know that the reals are axiomatisable >> >> > >> >> > Finitely axiomatisable or not? >> >> >> >> Yes, the first order theory of reals is finitely axiomatisable and >> >> in fact decidable. Ten points if you know why this not contradicts >> >> Goedel's incompleteness theorems. ;-) >> > >> > First order theory of reals would be hardly interested to any real >> > matematician (pun intended: real matematician as opposed to >> > logician:-) since the center pillar axiom of reals -- the supremum >> > axiom -- refers to subsets of the reals and is therefore a >> > second-order logical statement. >> >> Hmm. That makes me wonder. Can the finite axiomatization of a >> first-order theory not contain second-order axioms?
>
> Actually, that extract was pasted from
>
> http://www.nationmaster.com/encyclopedia/real-number
>
> where in the next paragraph we read:
>
> "It is not possible to characterize the reals with first-order logic
> alone: the Löwenheim-Skolem theorem implies that there exists a
> countable dense subset of the real numbers satisfying exactly the same
> sentences in first order logic as the real numbers themselves."
That doesn't really answer my question. Besides, characterization is not the most interesting question in the context of database theory, completeness is. And your axioms may characterise the reals without being complete, and on the other hand they can be complete without giving an exact characterization.
>> > As for the ten points, I was unable to google any references to >> > Tarski's work on finite axiomatization of reals. Can you please help? >> >> Jan Van den Bussche refers to it in his paper on the relationship >> between Tarski's work and database theory: >> >> http://citeseer.ist.psu.edu/vandenbussche01applications.html >> >> The original Tarki paper is reference [52]. Look in Section 5 for a >> (partial) explanation and some more recent references. Nothing on line >> as far as I can see, sorry.
>
> Ah, I remember this reference! But, it doesn't seem to have "real"
> mentioned anywhere:-(
Don't trust the PDF search, it will tell you that even "Tarski" isn't mentioned in the article. :-) The PDF was generated from postscript which often results in ugly PDF that is not really searchable.
- Jan Hidders