[FOM] post re: H. Friedman post "Concerning First and Second-Order Logics" of 22 May 2016
Gregory Taylor
Gregory.Taylor at baruch.cuny.edu
Thu Mar 16 13:01:07 EDT 2017
Dear Professor Davis,
What appears below is what I would post concerning the H. Friedman post "Concerning 1st and 2nd order "logic"" of May 22, 2016.
Thank you.
R. Gregory Taylor (FOM member)
In his post "Concerning 1st and 2nd order "logic"" of May 22, 2016, H. Friedman wrote
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For validity, we can use sentences in set theory that begin with a universal quantifier over all sets, followed
by any set theoretic formula where the quantifiers are bounded to the power set of a set. In fact, we can use the form
(1) (for all sets x)(phi holds in (POW(x) union x, epsilon).)
where phi is first order.
Then we get a kind of equivalence between validity in 2nd order logic and 1). There is an effective map from sentences in 2nd order logic to sentences in 1) such that, provably in ZFC, the given sentence is valid if and only if its value is
true. And there is an effective map from sentences in 1) to sentences in 2nd order logic such that, provably in ZFC,
the given sentence is true if and only if its value is valid in 2nd order logic.
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Is the following the correct reading of what appears above?
+++++++++++++++++++++++++++++++
In fact, we can use the form
(1) (for all sets x)(phi holds in (POW(x) union x, epsilon))
where phi is a first-order sentence.
Then there is an effective mapping from sentences of 2nd-order logic to first-order sentences such that, provably in ZFC,
the given 2nd-order sentence is valid if and only if (1) holds of its image under the mapping. And there is an effective mapping from first-order sentences to sentences in 2nd-order logic such that, provably in ZFC, the given first-order sentence satisfies (1) if and only if it image under the mapping is valid in second-order logic.
+++++++++++++++++++++++++++++++
Greg Taylor
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