Re: Mathematician Using Nil With Students

Henrick Hellström [StreamSec] wrote:
Some formal systems are complete and other systems are incomplete.

True, but first order logic is not complete in the sense used in the
incompleteness theorem.

(I must admit I hadn't heard of the Completeness Theorem. According to
Wikipedia it says that First Order Predicate Calculus is Complete, so I
had to look it up in my old text books, but all I could find was a proof
by Henkin from 1949 that non-modal Lower Predicate Calculus is Complete.
Times like this you realize that perhaps you too have read one book too
much and one book too little.)

Gödel proved the completeness of first order logic in his 1929
dissertation. Henkin, in his thesis, gave a novel proof of the
completeness theorem. Virtually every modern exposition of mathematical
logic follows Henkin's proof rather than Gödel's original. As to
"non-modal Lower Predicate Calculus", it's the same thing as first
order logic, which is non-modal in that it does not deal with such
notions as necessity and possibility and "lower" in the sense that
higher order quantifiers, that is quantifiers ranging over relations
and properties, are not allowed.

--
Aatu Koskensilta (aatu.koskensilta@xxxxxxxxx)

"Wovon man nicht sprechen kann, daruber muss man schweigen"
- Ludwig Wittgenstein, Tractatus Logico-Philosophicus

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