comprehension axiom in a sentence
1) Second-order semantics eliminates the need for the comprehension axiom, because these semantics imply that every possible set exists.
2) This holds because the set of comprehension axioms is a computable set (i.e., a recursive set).
3) ACA0 is RCA0 plus the comprehension scheme for arithmetical formulas (which is sometimes called the "arithmetical comprehension axiom").
4) For example, one can take the theory ACA given by the Peano postulates together with comprehension axioms for first-order formulas only.
comprehension axiom example sentences5) As a general model, any model of analysis must of course satisfy all of the comprehension axioms; later we will consider weakening this requirement.
6) Moreover, a deductive calculus can be given for second-order logic (adapted from first-order logic and augmented by the comprehension axioms) that will be complete for the general semantics.
7) Thus a sentence is valid in the general semantics iff it is logically implied (in first-order logic) by the set of comprehension axioms.
8) In fact, ZFC actually does circumvent Russell's paradox by restricting the comprehension axiom to already existing sets by the use of subset axioms.
9) The deductive systems considered by Shapiro (1991) and Henkin (1950) add to the augmented first-order deductive scheme both comprehension axioms and choice axioms.
10) The initials "RCA" stand for "recursive comprehension axiom", where "recursive" means "computable", as in recursive function.
11) to be true in M . Call such sentences comprehension axioms .
12) Returning to the case where φ("n") is a formula with a free variable "n" and possibly other free variables, we define the comprehension axiom for φ to be:
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