Comments on: Functions with Very Low Symmetry and the Continuum Hypothesis http://xorshammer.com/2009/07/19/functions-with-very-low-symmetry-and-the-continuum-hypothesis/ Some things in mathematical logic that I find interesting Mon, 06 Sep 2010 21:42:48 +0000 hourly 1 http://wordpress.com/ By: dick lipton http://xorshammer.com/2009/07/19/functions-with-very-low-symmetry-and-the-continuum-hypothesis/#comment-300 dick lipton Sun, 11 Jul 2010 15:44:08 +0000 http://xorshammer.com/?p=353#comment-300 This is a very cool post. thanks This is a very cool post.

thanks

]]> By: Kenny http://xorshammer.com/2009/07/19/functions-with-very-low-symmetry-and-the-continuum-hypothesis/#comment-203 Kenny Tue, 21 Jul 2009 02:13:56 +0000 http://xorshammer.com/?p=353#comment-203 That notion of symmetric continuity sounds fun too, though I don't know what a non-logician would make of it. Anyway, it still sounds like fun stuff, even if mainly recreational. That notion of symmetric continuity sounds fun too, though I don’t know what a non-logician would make of it. Anyway, it still sounds like fun stuff, even if mainly recreational.

]]> By: mkoconnor http://xorshammer.com/2009/07/19/functions-with-very-low-symmetry-and-the-continuum-hypothesis/#comment-202 mkoconnor Mon, 20 Jul 2009 23:42:37 +0000 http://xorshammer.com/?p=353#comment-202 The couple of places that I've seen these functions come up have been in the context of the study of symmetrically continuous functions, which is defined to be true of $latex f\colon\mathbb{R}\rightarrow\mathbb{R}$ at $latex x$ if $latex \lim_{h\to 0} (f(x-h) - f(x+h)) = 0$, but I've only seen those discussed in the context of the set theory of the real line. However, that probably says more about my lack of breadth of knowledge than it does about the applications of the context. I've seen the book "Symmetric Properties of real functions" referenced as a analytical basis for studying these types of things, and you can see a preview at: http://books.google.com/books?id=BMWk0X8rl_YC&lpg=PP1&ots=SF0LuDF6P8&dq=thomson%20symmetric%20properties%20of%20real%20functions&pg=PR11 In any case, I could easily be wrong, but as far as I know, this result is only entertainment; I don't know that it tells anything fundamentally new either about cardinalities or real functions. The couple of places that I’ve seen these functions come up have been in the context of the study of symmetrically continuous functions, which is defined to be true of f\colon\mathbb{R}\rightarrow\mathbb{R} at x if \lim_{h\to 0} (f(x-h) - f(x+h)) = 0, but I’ve only seen those discussed in the context of the set theory of the real line.

However, that probably says more about my lack of breadth of knowledge than it does about the applications of the context. I’ve seen the book “Symmetric Properties of real functions” referenced as a analytical basis for studying these types of things, and you can see a preview at:
http://books.google.com/books?id=BMWk0X8rl_YC&lpg=PP1&ots=SF0LuDF6P8&dq=thomson%20symmetric%20properties%20of%20real%20functions&pg=PR11

In any case, I could easily be wrong, but as far as I know, this result is only entertainment; I don’t know that it tells anything fundamentally new either about cardinalities or real functions.

]]> By: Kenny http://xorshammer.com/2009/07/19/functions-with-very-low-symmetry-and-the-continuum-hypothesis/#comment-200 Kenny Mon, 20 Jul 2009 12:19:25 +0000 http://xorshammer.com/?p=353#comment-200 That's a fun construction! It's always neat to have a result where the cardinality $latex \aleph_n$ actually means something about something in some way n dimensional. But not having looked at the Komjáth and Shelah paper, does this tell us anything interesting and new about these cardinalities? These strongly non-even functions seem sort of silly as a sort of non-symmetry, since they just are the injections, while being non-even of order n feels a bit more meaningful, but perhaps too complicated to be intuitively interesting. Is there any motivation for considering these non-symmetric functions beyond the connection to the continuum? That’s a fun construction! It’s always neat to have a result where the cardinality \aleph_n actually means something about something in some way n dimensional.

But not having looked at the Komjáth and Shelah paper, does this tell us anything interesting and new about these cardinalities? These strongly non-even functions seem sort of silly as a sort of non-symmetry, since they just are the injections, while being non-even of order n feels a bit more meaningful, but perhaps too complicated to be intuitively interesting. Is there any motivation for considering these non-symmetric functions beyond the connection to the continuum?

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