# 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).

axiom collocations

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 sentences

5) 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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