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Is this question acceptable?: Is this question acceptable: "Examples of Genericity in Mathematics"

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    • CommentAuthordavidac897
    • CommentTimeDec 23rd 2010
     
    Here is the rough text of the topic. It's for community wiki:

    It seems like a lot of results in mathematics amount to showing that something of a certain type exists, then showing that the generic thing actually satisfies that property (or that "most" things satisfy that property). In other words, that "most" things satisfy that property. Often, the result seems obvious, since it seems that there are "enough" things of a certain type A such that at least one of them has to be of another type B. It's then a matter of showing that there's no special reason why everything of type A actually lies outside the side of things of type B, in order to show that something of the desired type actually exists.

    For example, when showing that every vector space over an infinite field is not the union of finitely many proper subspaces, or when showing the primitive element theorem, it's clear that "most elements" work (in the latter case, most elements generate the whole extension - it's a matter of showing that there actually is one!).

    I'm curious to see how many places in mathematics we can find where this basic phenomenon appears, or where it's used to prove something.

    It's a well-defined notion in algebraic geometry, so I imagine there are a number examples from there, but it's a common intuitive idea in mathematics (showing that "most" things should satisfy the conditions you want), so there should be various other examples.
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    I think it's okay. Perhaps you should clarify whether you are more interested in

    • examples where "most" can be made rigorous and used to prove existence,
    • examples where "most" can be made rigorous but it is hard to prove that the rigorous definition actually applies, or
    • examples where "most" doesn't have a known rigorous definition and existence must be proven some other way, or is perhaps an open question.
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    I do wonder what the purpose of that question would be though. Would it provide you with a better understanding of the technique? I doubt it. Would it give you any insight into the various areas? Probably only very superficially. You are saying "I'm curious to see how many places in mathematics we can find...". Do you mind explaining, why you are curious? Should we have similar questions about proofs by induction, proofs using the pigeon hole principle, etc.?

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    Genericity (in the sense of "holds in a comeager set") is a fairly standard technique in descriptive set theory.
    • CommentAuthorYemon Choi
    • CommentTimeDec 24th 2010
     

    The theme has also been seen (from what little I know) a fair bit in classical Fourier/harmonic analysis: there is an article by Kahane discussing certain Banach spaces of functions on the circle, and showing that certain "pathologies" occur outside a meagre subset (in the sense of Baire category). If I recall correctly, several of these example are actually given by random Fourier series.

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    I'm not sure I would vote to close it, but I second Alex's question. What is it you really expect to learn from asking a question like this? I'm rather skeptical of questions of the form "What are a bunch of examples of X?"

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