# What does it mean: “the closure of the axioms”?

«As we shall see, the logical axioms are so designed that the logical consequences (in the semantic sense, cf. p. 56) of the closure of the axioms of $K$ are precisely the theorems of $K$.» Page 60 “Introduction to Mathematical Logic“ SECOND EDITION by ELLIOTT MENDELSON The same is in fourth edition.

What does it mean: “the closure of the axioms”?

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When you say "the same is in the fourth edition", is it located elsewhere (not p.60) of the fourth edition, I searched on line, found 4th ed., but there is no statement there resembling the statement you're interested in. I suspect more is meant by "the logical consequence...of the closure of the axioms of K are precisely the theorems of K" than closure of a particular formula whose variables are bound by a universal quantifier... –  amWhy May 20 '11 at 23:45
See "universal closure" here en.wikipedia.org/wiki/… –  Bill Dubuque May 30 '11 at 20:53

Maybe: the closure of a formula means universally quantifying over all its free variables? And perhaps this definition is somewhere before page 60?

Edit. Look in the index under: closure, (universal) of a formula

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Sorry for second edition. In fourth edition page 70 and instead of "closure" -- "closures" –  Victor Victorov May 21 '11 at 0:09
GEdgar: "Maybe: the closure of a formula means universally quantifying over all its free variables? And perhaps this definition is somewhere before page 60?" I was looking for the definition and found none. I doubt that it is mean closure of a formula means "universally quantifying over all its free variables" because axioms are logically valid. So it is no difference for closed formulas and not closed. (Meaning both types are valid). –  Victor Victorov May 21 '11 at 0:17
Amy J. M. “I suspect more is meant by "the logical consequence...of the closure of the axioms of K are precisely the theorems of K" than closure of a particular formula whose variables are bound by a universal quantifier...” I suspect the same. One more strange thing that from the 2nd to 4ed edition there was changed from “closure” to “closures”. Also, the word “closure” is gone from Russian translation. But all what I think counts to find definition of “closure of the axioms”. –  Victor Victorov May 21 '11 at 1:33
GEdgar: “Look in the index under: closure, (universal) of a formula” I seriously doubt that this is a case. Axioms are logically valid. So it is no difference for closed formulas and not closed. (Meaning both types are logically valid). –  Victor Victorov May 21 '11 at 2:30
GEdgar! You are right. It is closure of formula. Mendelson changed from “closure of the axioms” in second edition to “closures of the axioms” in fourth edition. So, we are really talking about “closure of each axiom (formula). The case is closed. Thank you. –  Victor Victorov May 21 '11 at 21:10