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ultrafilters
editOthers, such as Wim Luxemburg, showed that the same results could be achieved using ultrafilters, which made Robinson's work more accessible to mathematicians who lacked training in formal logic. - Well ... it was already well understood that ultraproduct constructions can replace any single use of the compactness theorem. Various people tried packaging the subject in a variety of ways (Machover, Nelson in particular). There is some similar point that could well be made here but at the moment it is not being made. The underlying difficulty was not so much the compactness theorem as the need to distinguish internal and external sets, and to be aware of the degree to which concepts which might appear to be elementary had set theoretic content. The underlying idea was to treat 2nd order logic as a 1st order notion, which is never a very faithful representation, but to turn the weakness of this interpretation into an advantage - giving one a sort of higher order logic with no new first order consequences. The germ of the idea, without a technical apparatus to support it, is in Leibniz. On the other hand, if you look at the work of Ax-Kochen, then one can really express the results directly in terms of ultraproducts, and one then only needs a tiny bit of logic to decode them usefully. Such is not the case in nonstandard analysis, and if one wants to use the technique one needs some specific training in formal systems, or at least in one particular formal system. (Unless one just wants a specific application, e.g. Loeb measure.) Abu Amaal 03:24, 26 August 2006 (UTC)
Nationality
editIt states at Robinson (name) that Robinson was an "American mathematician". There is nothing in this article about his nationality (although one assumes he was born with German citizenship).
- 1. If he took American nationality it would be good to include this information here.
- 2. What are the guidelines on nationality of the deceased? I can't seem to find any. I assume he would only have taken American nationality when he was living in the US. Then he could only have been an American citizen for at most the last 12 years of his 58 year life. In this case I think he is best described as a German mathematician.