Pat, (01)
What is clear is that: (02)
1) You keep repeating your belief that a single complete
axiomatization (with useful coverage?) is possible. (03)
2) You admit you don't know how to prove it is possible. (04)
3) You ignore/don't understand that other people have proven it is impossible. (05)
"A complete and consistent logic complex enough to include arithmetic
was shown by Kurt Goedel to be impossible, rendering unrealizable not
only the complete axiomatization of mathematics, but the greatest
hopes of an ideal language philosophy as well." - From modernism to
postmodernism: an anthology By Lawrence E. Cahoone, p.g. 5. (06)
I'm also finding references to a guy named Thoralf Skolem. Anyone else
heard of him? (07)
"In the 1922 lecture the Löwenheim-Skolem theorem was applied to a
formalization of set theory. The result was a relativization of the
notion of set, later known as the Skolem paradox: If the axiomatic
system (e.g. as presented by Zermelo) is consistent, i.e. if it is at
all satisfiable, then it must be satisfiable within a countable
``Denkbereich'' (domain). But does this not contradict Cantor's
theorem of the uncountable, the existence of a never-ending sequence
of transfinite powers? The ``paradox'' of Skolem is no contradiction.
Roughly speaking it asserts that there is no complete axiomatization
of mathematics, and that certain concepts must be interpreted relative
to a given axiomatization and its models and thus have no ``absolute''
meaning." (08)
"Towards the end of the 1929 paper Skolem expressed some doubts about
the complete axiomatizability of mathematical concepts. His scepticism
was based on the set-theoretic relativism which follows from the
Löwenheim-Skolem theorem. In 1929 he could give only some partial
results, but in a paper from 1934 (and a previous one from 1933)
``Über die Nichtcharacterisierbarkeit der Zahlenreihe mittels endlich
oder abzählbar unendlich vieler Aussagen mit ausschliesslich
Zahlenvariablen'' he could prove that there is no finite or countably
infinite set of sentences in the language of Peano arithmetic which
characterizes the natural numbers. Today, this follows as a simple
consequence of Gödel's completeness theorem. The technique used by
Skolem was a more direct model-theoretic construction. And this
technique, suitably refined to the so-called ``ultraproduct''
construction, has been an important tool in recent work on model
theory." (09)
http://www.hf.uio.no/ifikk/filosofi/njpl/vol1no2/skobio/node1.html (010)
By the way, my alternative is to work directly with observations of
different kinds, perhaps indexed by labels, and implement
interoperability based on overlaps between sets of these, as the task
demands. (011)
John's alternative seems to be to have a number of different
formalizations, and devise special rules to select/map between them. I
think devising special mapping rules is much more difficult and less
flexible than working directly with overlaps between sets of
observations. But at least it addresses the core problem that a single
complete axiomatization cannot be found. (012)
-Rob (013)
On Tue, Feb 16, 2010 at 5:40 AM, Patrick Cassidy <pat@xxxxxxxxx> wrote:
> ...
>
> ... I know of no way to prove mathematically that no one will
> ever think up a new concept whose meaning cannot be specified... (014)
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