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nLab > Latest Changes: smooth natural numbers

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  1.  
    • CommentRowNumber1.
    • CommentAuthorUrs
    • CommentTimeJan 12th 2010
    • (edited Jan 12th 2010)

    created an entry smooth natural numbers

    I tried to extract there the fundamental mechanism that makes the "nonstandard natural numbers" in Moerdijk-Reyes Models for Smooth Infinitesimal Analysis tick. In their book the basic idea is a bit hidden, but in fact it seems that it is a very elementary mechanism at work. I try to describe that at the entry. Would be grateful for a sanity check from topos experts.

    I find it pretty neat how the sequences of numbers used to represent infinite numbers in nonstandard analysis appears (as far as I understand) as generalized elements of a sheaf in a sheaf topos here.

    • CommentRowNumber2.
    • CommentAuthorMike Shulman
    • CommentTimeJan 12th 2010

    It is neat that unbounded sequences come up, but I don't yet see an actual connection to nonstandard analysis.

    • CommentRowNumber3.
    • CommentAuthorUrs
    • CommentTimeJan 12th 2010
    • (edited Jan 12th 2010)
    This comment is invalid XHTML+MathML+SVG; displaying source. <div> <blockquote> It is neat that unbounded sequences come up, but I don't yet see an actual connection to nonstandard analysis. </blockquote> <p>I am not sure if it has been fully formalized (in the very last paragraphs of the appendix it says that the logical relation ot nsa needs to be better understood) but in the axioms this nonstandard natural numbers object satisfies (chapter VII) and in the concrete constructions done with it, there is at least a lot of similarity with nsa axioms and constructions.</p> <p>Specifically for the sequences: there is what is called the "generic non-standard natural number" (p. 252) which is the generalized element of <img src="/extensions/vLaTeX/cache/latex_e7cb01c8e127f2a15ababfb26f597864.png" title=" N = \ell C^\infty(\mathbb{N})" style="vertical-align: -20%;" class="tex" alt=" N = \ell C^\infty(\mathbb{N})"/> on the <em>domain of definition</em> <img src="/extensions/vLaTeX/cache/latex_b9955b9b422af9860010d42a82f1fd32.png" title=" K = \ell C^\infty(\mathbb{N})/I " style="vertical-align: -20%;" class="tex" alt=" K = \ell C^\infty(\mathbb{N})/I "/>, where <img src="/extensions/vLaTeX/cache/latex_8f5b7b8c3198dcbde0250d812fc1fb30.png" title="I" style="vertical-align: -20%;" class="tex" alt="I"/> is the ideal of sequences that constantly vanish above some integer, given by the canonical injection that is the formal dual of the canonical projection of the ring onto its quotient.</p> <p>Now, the function ring dual to that object <img src="/extensions/vLaTeX/cache/latex_4a344b61963db1926b2fa55ce41f262e.png" title="K" style="vertical-align: -20%;" class="tex" alt="K"/> is the ring of sequences modulo the equivalence relation that identifies two sequences if they are identical above some integer. I have to remind myself of the details of nsa models, but that's the kind of rings used there, isn't it?</p> </div>
    • CommentRowNumber4.
    • CommentAuthorUrs
    • CommentTimeJan 12th 2010
    • (edited Jan 12th 2010)
    This comment is invalid XHTML+MathML+SVG; displaying source. <div> <blockquote> but that's the kind of rings used there, isn't it? </blockquote> <p>So the ring of sequences modulo those with constant tails is not an order, but just a preorder, but it's still pretty similar "in spirit" to sequences that are regarded as equivalent when thy "coincide on an ultrafilter" or whatever the terminology is.</p> <p>I am also still a bit unsure what exactly the right way is to phrase the universal property of the "generic non-standard natural number" in the sense above.</p> </div>
    • CommentRowNumber5.
    • CommentAuthorMike Shulman
    • CommentTimeJan 13th 2010

    I'll buy that it's similar "in spirit," but overall I am wary of this trying to say that nonstandard analysis can be done in a smooth topos. So much of "classical" nonstandard analysis depends on the transfer principle that whatever we can do in this situation, I just don't see it meriting the same name.

    • CommentRowNumber6.
    • CommentAuthorUrs
    • CommentTimeJan 13th 2010
    This comment is invalid XHTML+MathML+SVG; displaying source. <div> <blockquote> So much of "classical" nonstandard analysis depends on the transfer principle that whatever we can do in this situation, I just don't see it meriting the same name. </blockquote> <p>There is, by the way, also a transfer principle for the sdg-setup, chapter VII 4 of the book. It establishes a way to deduce statements in the smooth topos <img src="/extensions/vLaTeX/cache/latex_12a5935c99b249ce30e1a7c334e648da.png" title=" \mathcal{B} " style="vertical-align: -20%;" class="tex" alt=" \mathcal{B} "/> (that has nonstandard numbers) from statements in the smooth topos <img src="/extensions/vLaTeX/cache/latex_b0090da802fea5e66b27cb64daaf7a4a.png" title=" \mathcal{G} " style="vertical-align: -20%;" class="tex" alt=" \mathcal{G} "/> (that has not). The central statement is the Transfer Theorem on p. 342.</p> <p>Okay, please let me know explicitly what you are opting for doing with the section on sdg in the entry on nsa, I am not sure what your comments mean to imply. Currently the section starts with the sentence</p> <blockquote> A few aspects of nonstandard analysis can be realized in some models of synthetic differential geometry. </blockquote> <p>and hence tries to make clear that the claim is not that nsa is realized in the smooth topos.</p> <p>I felt it was still useful to have this as a comment there. Do you feel we should remove that section (and maybe move it to the entry on sdg itself)?</p> </div>
    • CommentRowNumber7.
    • CommentAuthorUrs
    • CommentTimeJan 13th 2010

    The transfer principle between the smooth toposes \mathcal{G} and \mathcal{B} applies to all coherent formulas. I created that page by just copying in the definition I see in the book, so it's mighty stubby.

    • CommentRowNumber8.
    • CommentAuthorMike Shulman
    • CommentTimeJan 15th 2010

    Don't take this the wrong way, but have you actually read any "classical" nonstandard analysis? The feel there is totally different from what I see in any smooth toposes. In particular, the transfer principle is used all over the place, usually for very non-coherent formulae. For instance, one of the central facts is that a function f\colon \mathbb{R}\to\mathbb{R} is continuous at x iff for any y\approxeq x we have f(y)\approxeq f(x). This is proven by applying the transfer principle to the ?-? definition of continuity, which is of the form \forall\exists(\Rightarrow), very non-coherent.

    By contrast, in smooth toposes everything is continuous (in fact, smooth, of course), the cited "transfer principle" applies only to a restricted class of coherent formulae, and moreover it relates two different smooth topoi, rather than relating the topos of sets to some nonstandard model. So what I'm saying is that I don't understand why it is useful to draw any connection between the two systems at all; rather I think it is misleading as to the true nature of both. Just the fact that both can have invertible infinitesimals and involve a "transfer" property between two models doesn't make them similar, or mean that intuition from one is at all helpful in thinking about the other.

    The surreal numbers also include invertible infinitesimals, and have a sort of "transfer" property since they are a real-closed field and hence elementarily equivalent to \mathbb{R}. They even contain all possible hyperreal number fields, since they are the universally embedding ordered field. But surreal analysis is very different from nonstandard analysis.

    I'm willing (and would be happy!) to be convinced otherwise, that there is some deep relationship between the two (or three) approaches to infinitesimals and that ideas from one can be fruitfully applied in another. But I haven't seen any evidence of it yet.

    • CommentRowNumber9.
    • CommentAuthorUrs
    • CommentTimeJan 15th 2010
    • (edited Jan 15th 2010)

    Okay, Mike, no problem, if it wasn't clear yet:no, I haven't looked much into standard nonstandard analysis (and I don't plan to). I am interested in smooth toposes and and find it interesting that

    A few aspects of nonstandard analysis can be realized in some models of synthetic differential geometry.

    as the entry says, following the message given in a textbook. What I find interesting I find worth recording on the Lab, so therefore this material.

    And I think it would be a pity if the nLab entry on nonstandard analyis wouldn't mention that many aspects of nonstandard analysis, like existence of invertible infinitesimals, flavors of transfer principles, distributions as plain functions may be modeled in certain smooth toposes. Also that nonstandard infinite numbers find an incarnation as unbounded sequences of ordinary numbers in models for both frameworks I find noteworthy. I find this noteworthy information for anyone who is wondering about how to best implement tools for handling invertible infinitesimals into his or her work and in this context looks into nonstandard analysis.

    I am still not sure what you are asking me to change about the entry, if this is what you are doingl You say you haven't seen any evidence for a deep relationship, but also the entry doesn't seem to speak of a deep relationship, but indicates just some similarities.

    But probably -- certainly -- the wording can be improved. I see you had already added some warning remarks. Maybe we should not make the section "In SDG" a subsection of the section titled "Models"? Maybe it should be a subsection of a new section "Related concepts"? Would that help?

    Or maybe the phrase "a few aspects may be realized in..." should be reworded to something like "a few aspects also have analogs in..." or the like?

    • CommentRowNumber10.
    • CommentAuthorzskoda
    • CommentTimeJan 15th 2010
    • (edited Jan 15th 2010)

    I would agree with Mike, if he meant that unless the transfer principle is satisfied it is not nonstandard analysis. Having something alternative to Cauchy epsilon delta is one thing and having certain possibility of inverting (the order of) quantifiers is another. It is more essentially about the logic than about infinitesimals.

    You could have a separate entry, say, notions of infinitesimals having all kinds of math related to intuition of infinitesimals. Topoi may be useful in several of them. They are also very useful (specially study of Boolean topoi) in nonstandard analysis, but this is likely not what you meant (and I am surprised you are not interested by calling it "classical").

    • CommentRowNumber11.
    • CommentAuthorUrs
    • CommentTimeJan 15th 2010
    • (edited Jan 15th 2010)

    unless the transfer principle is satisfied it is not nonstandard analysis.

    Sure. But it seems that the entry doesn't claim the contrary, no?

    The entry says

    A few aspects of nonstandard analysis can be realized in some models of synthetic differential geometry.

    This seems to me to be a piece of information which is true and useful to know. How would you rather say it?

    • CommentRowNumber12.
    • CommentAuthorTobyBartels
    • CommentTimeJan 15th 2010

    Re #10:

    I have started infinitesimal number.

    • CommentRowNumber13.
    • CommentAuthorTobyBartels
    • CommentTimeJan 15th 2010
    • (edited Jan 15th 2010)

    I did what I could at infinitesimal number. There is room for expansion.

    • CommentRowNumber14.
    • CommentAuthorzskoda
    • CommentTimeJan 15th 2010

    A few aspects of nonstandard analysis can be realized in some models of synthetic differential geometry.

    This may mislead the reader, in my experience. We had some time ago discussion on this when I asked you if the nonstandard analysis is there and was not familiar with Moerdijk-Reyes book that it is in a particular chapter and that did not reflect my expectations. If you have people who know nonstandard analysis and do not know synthetic geometry you can mislead them this way. It is much better to have a separate entry, then to pile up impressions in such a delicate thing. You see before Cauchy people worked all kind of infinitesimals informally, and this was not a theory, and in particular that was NOT a nonstandard analysis, it was just nonrigorous calculus. Its advantage and its peril was the existence of not-consistently treated differentials. It took few hundreds years to make a correct repair via the genius of Robinson and logical way of solving the problem. For the ultrafilters one has an existence, not a construction. Spirit of these ideas may be compromised by making simply ring-theoretic constructions with limited resemblance.

    • CommentRowNumber15.
    • CommentAuthorUrs
    • CommentTimeJan 16th 2010
    • (edited Jan 16th 2010)

    So what do we do? I am confused!

    I seem to have a different perception of the statements. To me they look like noteworthy supplementary information, to you, and possibly to Mike, they look misleading.

    Please, Mike, and Zoran, do edit the material in question the way you deem appropriate. If you think it has to be removed from the entry on nonstandard analysis, I suppose it could be saved at infinitesimal number? (Even though it is really about infinite numbers!)

    I am a bit baffled that a nice category theoretic perspective on a subject encounters such resistance, but I guess now that if I spent a few days reading original literature on nonstandard analysis I would be transformed into a different mindset and understand what you are trying to tell me.

    • CommentRowNumber16.
    • CommentAuthorMike Shulman
    • CommentTimeJan 16th 2010

    I am a bit baffled that a nice category theoretic perspective on a subject encounters such resistance

    I am all for nice category-theoretic perspectives on subjects. But I think it's important that such perspectives be true (in the sense of capturing something important about the subject) and useful, and I'm not convinced yet that the NSA-SDG connection has either of those properties. I did write a bit here about a different category-theoretic perspective on NSA, which I do think is valuable.

    I think infinitesimal number would be a good place to talk about comparisons between the various approaches. (Surely wherever there are increasing sequences of reals that represent infinite numbers, there are also decreasing sequences of them that represent infinitesimals?) There should, of course, be pointers to there, and even references to specific aspects of it, on the individual pages. I'll try and do a bit of rearranging and see if it can make us all happy.

    • CommentRowNumber17.
    • CommentAuthorMike Shulman
    • CommentTimeJan 16th 2010

    By the way, is there a principle distinguishing the content that goes on synthetic differential geometry from the content that goes on smooth topos? There seems to be a fair bit of overlap.

    • CommentRowNumber18.
    • CommentAuthorMike Shulman
    • CommentTimeJan 16th 2010

    BTW, I'm not sure if this is contributing to the confusion, but just the fact that the word "non-standard" is used somewhere doesn't necessarily imply any connection to NSA. People do use that just to mean "not standard". (-:

    • CommentRowNumber19.
    • CommentAuthorMike Shulman
    • CommentTimeJan 16th 2010

    I did some rearrangement at infinitesimal number, nonstandard analysis, and smooth natural numbers; let me know what you think.

    • CommentRowNumber20.
    • CommentAuthorTobyBartels
    • CommentTimeJan 17th 2010

    @ Mike #17:

    Based on the titles, I would think that synthetic differential geometry is about the subject in general, including its motivation, main ideas, and so forth; while smooth topos is about that particular concept, including its definition, principle properties, and the like. Compare topology and topological space.

    This seems to be what is happening there now.

    • CommentRowNumber21.
    • CommentAuthorUrs
    • CommentTimeJan 17th 2010

    okay, sounds good. I'll have a look a little later, when I find the time

    • CommentRowNumber22.
    • CommentAuthorzskoda
    • CommentTimeJan 17th 2010
    • (edited Jan 17th 2010)

    It seems Urs, that you are not getting the message that there are genuine and very general topos theoretic frameworks for nonstandard analysis (if somebody resists general nonsense here it is you -- by calling non-sdg-directed approaches "STANDARD nonstandard analysis") and that I and Mike (I assume) are NOT against those. Now you are trying to make a shortcut and replace those by something what is natural for SDG purposes. As I mentioned to you in September I do suspect that there is a common framework which gives both, but it is apparanetly not easy to make such a framework and it seems that it is not quite in Moerdijk-Reyes book.

    • CommentRowNumber23.
    • CommentAuthorUrs
    • CommentTimeJan 18th 2010
    • (edited Jan 18th 2010)

    you are not getting the message that there are genuine and very general topos theoretic frameworks for nonstandard analysis

    Sure. But there are also toposes that come close to the idea of Robinson-NSA but have the additional advantage that they have great use apart from that. Why not mention this on an entry on NSA? Robinson was not infallible. People interested in abstract nonsense formulations of the infinitesimal and the infinite might appreciate being alerted that there are other formalizations that achieve the kind of thing that Robinson wanted to achieve, but differ in detail from the formalization he found.

    But we spent already too much time on discussing this. As I said in my previous comments, I'll go with whatever you and Mike make of this.

    • CommentRowNumber24.
    • CommentAuthorMike Shulman
    • CommentTimeJan 18th 2010

    I agree we've spent too much time on this, but I just want to say again that what Zoran and I are saying is that the toposes of SDG do not "come close to the idea of Robinson-NSA". They do formalize "the" intuitive idea of infinitesimals, but in a way that is not really close to NSA at all, and makes clear (I think) that there really wasn't a unique "idea of infinitesimals" to begin with.

    • CommentRowNumber25.
    • CommentAuthorUrs
    • CommentTimeJan 18th 2010

    Okay. I learn from this that we think very differently of when a formal system comes close to capturing an idea. But that's fine.

    • CommentRowNumber26.
    • CommentAuthorzskoda
    • CommentTimeJan 18th 2010
    • (edited Jan 18th 2010)

    I would not say that Mike objects to the claim that what you propose does come close to some (or the if you really like) idea of infinitesimals, but not that it comes to the idea of nonstandard analysis which is in one sense more specific, but in another wider and more powerful than the study and modelling of infinite and infinitesimal quantities (regarding that the transfer principle leads to host of extended notions not having to do with infinitesimals or real analysis at all, but say on set theory, forcing, probability, on Heyting algebras, spaces of functional analysis etc.).

    • CommentRowNumber27.
    • CommentAuthorUrs
    • CommentTimeJan 18th 2010

    Okay! :-)

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