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From: |
Rob Freeman <lists@xxxxxxxxxxxxxxxxxxx> |

Date: |
Tue, 16 Feb 2010 11:28:32 +1300 |

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<7616afbc1002151428s67110ef8j8a8e32cd8b33f82@xxxxxxxxxxxxxx> |

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: > |

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