# induction axiom in a sentence

1) This sentence is called the second-order induction axiom.

2) induction axioms are also implicit, and are needed only to prove program properties.

axiom collocations
3) In Peano's original formulation, the induction axiom is a second-order axiom.

4) As axioms, we can take the usual Peano postulates, including the second-order induction axiom.

## induction axiom example sentences

5) All of the Peano axioms except the ninth axiom (the induction axiom) are statements in first-order logic.

6) This makes no difference when we study only ω-models, which automatically satisfy every instance of the induction axiom.

7) The subscript 0 in __FORMULA__ indicates that we have not included every instance of the induction axiom in this subsystem.

8) In the presence of the unrestricted comprehension scheme, the single second-order induction axiom implies each instance of the full induction scheme.

9) This follows from the fact that the axioms of Peano arithmetic with the second-order induction axiom have only one model under second-order semantics.

10) Dedekind proved in his 1888 book, "What are numbers and what should they be" () that any two models of the Peano axioms (including the second-order induction axiom) are isomorphic.

11) This form of the induction axiom is a simple consequence of the standard formulation, but is often better suited for reasoning about the ≤ order.

12) The induction axiom is sometimes stated in the following form:

13) Observe that "in fact" Peano's axioms are "9" in number and axiom "9" is the recursion/ induction axiom.

14) If φ("n") is a formula of second-order arithmetic with a free number variable "n" and possible other free number or set variables (written "m"• and "X"•), the induction axiom for φ is the axiom:

### example sentences with axiom

15) The first-order induction schema includes every instance of the first-order induction axiom, that is, it includes the induction axiom for every formula φ.

16) The first-order induction schema includes every instance of the first-order induction axiom, that is, it includes the induction axiom for every formula φ.

17) For each formula φ("x","y"1...,"y""k") in the language of Peano arithmetic, the first-order induction axiom for φ is the sentence

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