False statementConstruction: AI-adaptedVerification: AI-adaptedprecheck passverified 2026-08-02 (claude-opus-5)

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FALSE: successor-closure alone forces a set to be all of $N$

Statement

False statement. If a set $S \subseteq N$ is nonempty and closed under the successor ( $n \in S \Rightarrow σ (n) \in S$ ), then $S = N$ . (That is, the induction principle would hold without its base case $0 \in S$ .)

Facts & Assumptions

Given: the claim above.

[L1]

The induction principle requires $0 \in S$ (The principle of mathematical induction).

[L2]

$σ (n) \neq 0$ for all $n$ (P1) (The von Neumann naturals form a Peano system).

Refutation

technique · direct

1.1

Take $S = N ∖ {0} = {n \in N : n \neq 0}$ , the set of nonzero naturals; it is nonempty (for instance $1 = σ (0) \in S$ ).

given

1.2

$S$ is closed under $σ$ : for any $n$ , $σ (n) \neq 0$ by P1 [L2], so $σ (n) \in S$ .

2.1

But $0 \notin S$ , so $S \neq N$ ; the nonempty successor-closed set $S$ is not all of $N$ , refuting the claim.

step 1.1step 1.2

3.1

The base case $0 \in S$ is therefore indispensable in the induction principle [L1]; successor-closure and nonemptiness do not suffice.

step 2.1L1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

9 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Mathematical induction (Wikipedia) (standard reference, not scraped)
Peano axioms (Wikipedia) (standard reference, not scraped)
W. Aitken, MATH 378 Ch. 1: The Peano Axioms (CSU San Marcos) (standard reference, not scraped)

FALSE: successor-closure alone forces a set to be all of NN