User:IssaRice/Strength of a mathematical statement: Difference between revisions

Revision as of 18:47, 1 October 2018

A puzzle: why do we say P is stronger than Q if P is a subset of Q, but we also say that a theorem is stronger if it is more general (so bigger)?

One reply/intuition uses something like possible world semantics, e.g. see Wei Dai's post on Aumann's agreement theorem. There is just one possible world (a single $ω \in Ω$ ), but our information state is the set of all possible worlds that we cannot distinguish, so the less we know, the more possible worlds we think we could be in.
One visualization is to use a Venn diagram. The stronger the statement, the more our movement is restricted, as we are forced to be in more and more sets.

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+==Strong vs subset==
+A puzzle: why do we say P is stronger than Q if P is a subset of Q, but we also say that a theorem is stronger if it is more general (so bigger)?
+* One reply/intuition uses something like possible world semantics, e.g. see [https://www.lesswrong.com/posts/JdK3kr4ug9kJvKzGy/probability-space-and-aumann-agreement Wei Dai's post on Aumann's agreement theorem]. There is just one possible world (a single <math>\omega \in \Omega</math>), but our information state is the set of all possible worlds that we cannot distinguish, so the less we know, the more possible worlds we think we could be in.
+* One visualization is to use a Venn diagram. The stronger the statement, the more our movement is restricted, as we are forced to be in more and more sets.
 ==External links==