[FOM] From Compactness to Completeness
Stephen G Simpson
simpson at math.psu.edu
Wed Dec 29 15:32:11 EST 2010
G. Aldo Antonelli writes:
> The question (which at least I took myself to be presenting to the
> list) is whether the *weaker* form of completeness also requires
> König's lemma.
> By the weaker version of completeness I meant the claim that every
> valid first-order formula is provable, the proof of which usually
> proceeds by extracting a counter-model from a non-terminating proof
This weaker version of completeness "requires" Weak K"onig's Lemma, in
the sense that it implies it over RCA_0. (In fact, the two statements
are equivalent over RCA_0.) However, it does not "require" the full
K"onig's Lemma. See Sections IV.3 and III.7 of my book, Subsystems of
Second Order Arithmetic.
Note: Weak K"onig's Lemma says: every infinite subtree of the full
binary tree has an infinite path. K"onig's Lemma says: every
infinite, finitely branching tree has an infinite path.
> Is this true or (which comes to same thing) mentioned in SSOA?
I strove to make SOSOA as comprehensive as possible, but ultimately I
ran up against G"odel's Incompleteness Theorem.
Stephen G. Simpson
Professor of Mathematics
Pennsylvania State University
foundations of mathematics, recursion theory, mathematical logic
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