Timeline for Proving boundedness of continuous images of [0,1] in WKL0
Current License: CC BY-SA 2.5
9 events
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| Oct 9, 2010 at 19:54 | vote | accept | CommunityBot | ||
| Oct 9, 2010 at 13:22 | comment | added | Carl Mummert | I can't vote up comments on an iphone, apparently, but Bjorn has the right point. In reverse mathematics a continuous function is accompanied by its coded representstion. That coded representation is key to defining sets relative to the continuous function. | |
| Oct 8, 2010 at 23:19 | comment | added | Bjørn Kjos-Hanssen | "Continuous function" has a special definition in reverse mathematics which does not seem to mentioned in Wikipedia yet, but which I allude to in my answer. For in-depth study of this subject one should get hold of Simpson's book Subsystems of Second Order Arithmetic. | |
| Oct 8, 2010 at 22:35 | comment | added | user5810 | The definition of continuity gives "for all x in [0,1], there exists an open interval with rational endpoints such that x is in the interval and for all z in [0,1], if z is in the interval then -n < f(z) < n". I don't see how that can be used to show that "there exists a sequence of open intervals with rational endpoints such that for all x in [0,1], if -n < f(x) < n, then x is in one of the intervals". | |
| Oct 8, 2010 at 21:40 | comment | added | Bjørn Kjos-Hanssen | I think that's basically the definition of continuity. Since every object in reverse math that belongs to the model must be represented by an element or a subset of $\mathbb N$, one cannot just use the set-theoretical definition of a function. You're right, a set being $\Sigma_1$ definable in the model does not imply the set is in the model, just that there is a $\Sigma_1$ formula with parameters from the model that defines the set. | |
| Oct 8, 2010 at 20:43 | comment | added | user5810 | How are the definition of continuity show that the sets $f^{-1}([-n,n)]$ are represented as the union of $\Sigma_1$ collections of open intervals with rational endpoints? (I'm also not sure how it's provable that those collections are sets, but that's not needed.) | |
| Oct 8, 2010 at 19:44 | history | edited | Bjørn Kjos-Hanssen | CC BY-SA 2.5 |
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| Oct 8, 2010 at 19:36 | history | edited | Bjørn Kjos-Hanssen | CC BY-SA 2.5 |
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| Oct 8, 2010 at 19:16 | history | answered | Bjørn Kjos-Hanssen | CC BY-SA 2.5 |