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The difference is that Tarski is interested in the structure of the set of all fixed-points: they form a lattice, you can talk of greatest fixpoint, the sup of all fixpoints is a fixpoint, etc. He is not concerned about reaching fixpoints by iterating a function, starting from some seed, and building
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The above page currently redirects to this one, but I can't see any mention of that exact phrase in the article. As someone who knows nothing about the subject this would be important to me to have it explained somewhere. Could something be inserted, or should we delink the redirect? --
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I found a paper on the who discovered what, called "Fixed point theorems and semantics: a folk tale". I do not have the time to read it, but it should fix the problem of a lack of references. The link is
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The difference is that in the Kleene fixed-point theorem, f has to be Scott-continuous (i.e. has to preserve directed suprema), while in
Knaster-Tarski, it only needs to be monotonic, which is weaker. --
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The ascending Kleene chain is the chain that starts with the bottom element of L and goes up through applications of the monotone function f. I've made the term explicit now. Is that beter?
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The issue between requiring continuity or just monotonicity is not the central difference (with just monotonicity you can have a Kleene fixpoint theorem by iterating in the transfinite).
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I added the proof. If you find any mistakes, or if I'm overly vague at some points, please fix it or write a note here.
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Now I think it's right :-). I'll change the link to 'continuous' to point to the article on Scott continuity instead.
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Knowledge. If you would like to participate, please visit the project page, where you can join
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Did you try just searching for it on google books? There are lots of hits. — Carl
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This article has no references at all, and I could not find any myself. Anyone?
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fixed Point is wrong, since another fixed point might not be of the form
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http://www.sciencedirect.com/science/article/pii/0020019082900655
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I think this is wrong: continuity does not imply monotonicity.
318:{\displaystyle f(\mathbb {M} )=\mathbb {M} \setminus \{\bot \}}
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a converging sequence, which is the point of Kleene's theorem.
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