# Definition:Basis for the Induction

## Terminology of Mathematical Induction

Consider a **proof by mathematical induction**:

**Mathematical induction** is a proof technique which works in two steps as follows:

- $(1): \quad$ A statement $Q$ is established as being true for some distinguished element $w_0$ of a well-ordered set $W$.

- $(2): \quad$ A proof is generated demonstrating that if $Q$ is true for an arbitrary element $w_p$ of $W$, then it is also true for its immediate successor $w_{p^+}$.

The conclusion is drawn that $Q$ is true for all elements of $W$ which are successors of $w_0$.

The step that establishes the truth of $Q$ for $w_0$ is called the **basis for the induction**.

Expressed in the various contexts of mathematical induction:

### First Principle of Finite Induction

The step that shows that the integer $n_0$ is an element of $S$ is called the **basis for the induction**.

### First Principle of Mathematical Induction

The step that shows that the proposition $\map P {n_0}$ is true for the first value $n_0$ is called the **basis for the induction**.

### Second Principle of Finite Induction

The step that shows that the integer $n_0$ is an element of $S$ is called the **basis for the induction**.

### Second Principle of Mathematical Induction

The step that shows that the proposition $\map P {n_0}$ is true for the first value $n_0$ is called the **basis for the induction**.

### Principle of General Induction

The step that shows that the propositional function $P$ holds for $\O$ is called the **basis for the induction**.

### Principle of General Induction for Minimally Closed Class

The step that shows that the propositional function $P$ holds for the distinguished $b \in M$ is called the **basis for the induction**.

### Principle of Superinduction

The step that shows that the propositional function $P$ holds for $\O$ is called the **basis for the induction**.

## Also known as

The **basis for the induction** is often informally referred to as the **base case**.

## Also see

## Sources

- 1980: David M. Burton:
*Elementary Number Theory*(revised ed.) ... (previous) ... (next): Chapter $1$: Some Preliminary Considerations: $1.1$ Mathematical Induction