Talk:Knaster-Tarski Lemma/Corollary/Power Set/Proof 2

I don't personally care if this is here or if it isn't, but someone who wants to read about sets without knowing what a complete lattice is will probably care. I will work on it in the afternoon (Tuesday). --Dfeuer (talk) 05:29, 2 April 2013 (UTC)

Thinking more on this, it's probably not worth going to the trouble of proving the least and greatest without using lattices, but it is worth the trouble of proving the corollary that there is a fixed point. So should this be renamed Knaster-Tarski Lemma/Power Set/Corollary/Proof 2 or Knaster-Tarski Lemma/Corollary/Power Set/Proof 2? --Dfeuer (talk) 05:38, 2 April 2013 (UTC)

I'm at a complete befuddlement as to the thinking behind the naming of this page. Is there someone able to explain? --prime mover (talk) 17:20, 2 April 2013 (UTC)

Yes.

History: Knaster and Tarski together published Knaster-Tarski Lemma/Power Set. That lemma generalizes trivially to complete lattices, the Knaster-Tarski Lemma. Later, Tarski proved a stronger form of the generalization of the lemma, the Knaster-Tarski Theorem.

Relevance: Smullyan and Fitting present a proof that every increasing mapping from a power set to itself has a fixed point, which they use as a lemma to the Cantor-Bernstein-Schröder Theorem. They claim it to be a specialization of a lattice theorem by Knaster and Tarski (which appears to be a historical error). We should, to support S & F, offer the non-lattice form of this corollary.

Name: I was thinking thus: a trivial corollary of K-T L is that every increasing mapping on a complete lattice has a fixed point. This page is that corollary specialized to the power set. --Dfeuer (talk) 17:35, 2 April 2013 (UTC)

Yep okay fair enough - makes sense now we see the full thread of results. --prime mover (talk) 18:45, 2 April 2013 (UTC)