| Revision as of 12:23, 14 September 2008 editKatzmik (talk | contribs)3,355 edits →non-standard calculus← Previous edit | Revision as of 09:05, 15 September 2008 edit undoKatzmik (talk | contribs)3,355 edits →your comment: new sectionNext edit → | ||
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| Thanks for your comment at ]. I added a couple of paragraphs to the lead at ]. Please give it a professional edit. I still feel that the thrust of this material goes contrary to the remarks in the first section, as I tried to explain at the talk page. ] (]) 12:23, 14 September 2008 (UTC) | Thanks for your comment at ]. I added a couple of paragraphs to the lead at ]. Please give it a professional edit. I still feel that the thrust of this material goes contrary to the remarks in the first section, as I tried to explain at the talk page. ] (]) 12:23, 14 September 2008 (UTC) | ||
| == your comment == | |||
| Hi, | |||
| You made the following comment at ]: | |||
| Reply to comment of User talk:Katzmik) posted 14:14, 9 September 2008 (UTC). Sorry to take so long to respond. You are correct that quantification over sets is required, but this doesn't make it a higher order theory. For example, there are no type distinctions between sets of integers and integers. In ZFC all variables range over the entire set-theoretic universe. If one had a weaker no-standarad analysis, with limits on the range of quantification, the resulting theory would be less interesting. In fact, you can make the transcendental extension 'R into an ordered field in which the indeterminate t is infinite and 1/t is a non-zero infinitesimal. But this is pretty much useless for a development of calculus. I don't know if I've addressed any of your concerns.--CSTAR (talk) 14:48, 11 September 2008 (UTC) | |||
| :I have thought about your comment for a while and I do not understand it fully, surely this is due to my lack of training in logic. At any rate, I am not sure what you mean when you say | |||
| "there are no type distinctions between sets of integers and integers"; why aren't there? Also, I understand the assertion "In ZFC all variables range over the entire set-theoretic universe" but I am not sure I understand what you are driving at when you say this. Certainly in NAS one needs to interpret statements as referring to internal sets only; I see this not as a weakness of the theory but rather the main tool in the realisation of Robinson's goal. When you say "you can make the transcendental extension 'R into an ordered field in which the indeterminate t is infinite and 1/t is a non-zero infinitesimal. But this is pretty much useless for a development of calculus", are you referring to the absence of a transfer principle in such a naive approach to NAS? ] (]) 09:05, 15 September 2008 (UTC) | |||
Revision as of 09:05, 15 September 2008
Mathematical validity
You reversed my edit at non-standard analysis with the justification that the content of the section is contained in the last sentence of the previous paragraph. The sentence you are referring to mentions vaguely that there is no argument about the mathematical validity of non-standard analysis. I don't think this is sufficiently precise. Namely, even a system containing additional axioms could also be mathematically valid, so long as nobody has found an internal contradiction in such a system. The specific point that non-standard analysis is "conservative" in the sense that it does not go beyond ZFC deserves to be mentioned explicitly. If you disagree please raise the issue at WP math rather than using deletions. For the time being I will revert my edits. Katzmik (talk) 08:11, 31 August 2008 (UTC)
Please respond to my comments at the talk page of non-standard analysis. Katzmik (talk) 13:24, 31 August 2008 (UTC)
non-standard calculus
There is a dispute regarding the proof of the intermediate value theorem, please comment. Katzmik (talk) 12:27, 2 September 2008 (UTC)
Please respond to my comment at talk:transfer principle. Katzmik (talk) 12:43, 10 September 2008 (UTC) and again Katzmik (talk) 14:19, 11 September 2008 (UTC)
Thanks for your comment at talk:transfer principle. I added a couple of paragraphs to the lead at transfer principle. Please give it a professional edit. I still feel that the thrust of this material goes contrary to the remarks in the first section, as I tried to explain at the talk page. Katzmik (talk) 12:23, 14 September 2008 (UTC)
your comment
Hi, You made the following comment at talk:transfer principle:
Reply to comment of User talk:Katzmik) posted 14:14, 9 September 2008 (UTC). Sorry to take so long to respond. You are correct that quantification over sets is required, but this doesn't make it a higher order theory. For example, there are no type distinctions between sets of integers and integers. In ZFC all variables range over the entire set-theoretic universe. If one had a weaker no-standarad analysis, with limits on the range of quantification, the resulting theory would be less interesting. In fact, you can make the transcendental extension 'R into an ordered field in which the indeterminate t is infinite and 1/t is a non-zero infinitesimal. But this is pretty much useless for a development of calculus. I don't know if I've addressed any of your concerns.--CSTAR (talk) 14:48, 11 September 2008 (UTC)
- I have thought about your comment for a while and I do not understand it fully, surely this is due to my lack of training in logic. At any rate, I am not sure what you mean when you say
"there are no type distinctions between sets of integers and integers"; why aren't there? Also, I understand the assertion "In ZFC all variables range over the entire set-theoretic universe" but I am not sure I understand what you are driving at when you say this. Certainly in NAS one needs to interpret statements as referring to internal sets only; I see this not as a weakness of the theory but rather the main tool in the realisation of Robinson's goal. When you say "you can make the transcendental extension 'R into an ordered field in which the indeterminate t is infinite and 1/t is a non-zero infinitesimal. But this is pretty much useless for a development of calculus", are you referring to the absence of a transfer principle in such a naive approach to NAS? Katzmik (talk) 09:05, 15 September 2008 (UTC)