[FOM] From Compactness to Completeness
blumal at mail.biu.ac.il
Thu Dec 23 04:34:09 EST 2010
Is there an intuitive and elementary argument-headlines are enough- which
show how the compactness of fol implies its completeness?
In Wikipedia on "Compactness Theorem" under the heading "Proofs" we have:
"In fact, the compactness theorem is equivalent to G?del's completeness
theorem, and both are equivalent to the Boolean prime ideal theorem, a weak
form of the axiom of choice."
And, in Kurt Godel's 1930 completeness proof he has for Theorem IX one
formulation of the completeness theorem of fol, and then Godel continues "
IX follows immediately from Theorem X. [which is, my own insertion] For a
denumerably infinite set of formulas to be satisfiable it is neccessary and
sufficient that every finite subsystem be satisfiable" p 590 in J. van
Hijenoort's "From Frege to Godel' (1967). J.W. Dawson points this out in his
(1993) 'The compactness of first-order logic:from g?del to lindstr?m',
History and Philosophy of Logic, 14: 1, p17.
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