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| ``Re: Update: "Trans-inductive Posets"''
by smw on 2006-02-19 21:40:07 |
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| The key mechanism which makes induction work, as is well-known, is "well-foundedness." I think this is equivalent to what you are proposing.
A (binary) relation < on a set X is said to be well-founded if every nonempty subset S of X has a (not necessarily unique) <-minimal element. A <-minimal element of S is an element a of S such that there is no element b in S (other than perhaps a itself) with the property that b < a. (Note that this formulation does not need to invoke the idea of a chain, and moreover < is not even assumed to be a partial order.)
The most general form of induction would therefore be "well-founded induction." Transfinite induction is nothing more than well-founded induction on a well-ordered *class.* (e.g. the class of all ordinals.)
Please look at
http://planetmath.org/encyclopedia/WellFoundedInduction.html
(NOTE: In this article, the author seems to restrict the definition of well-foundedness to posets (this is a little different from the definition I just gave), but it is easy to verify that the principle of well-founded induction only uses the property that "every nonempty subset has a minimal element.")
Good luck with your research. |
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