# 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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