Strictly Positive Integers have same Cardinality as Natural Numbers

Theorem

Let $\Z_{>0} := \set {1, 2, 3, \ldots}$ denote the set of strictly positive integers.

Let $\N := \set {0, 1, 2, \ldots}$ denote the set of natural numbers.

Then $\Z_{>0}$ has the same cardinality as $\N$.

Proof

Consider the mapping $f: \N \to \Z_{>0}$ defined as:

$\forall x \in \N: \map f x = x + 1$

Then $f$ is trivially seen to be a bijection.

The result follows by definition of cardinality.

Sources

• 2010: Raymond M. Smullyan and Melvin Fitting: Set Theory and the Continuum Problem (revised ed.) ... (previous) ... (next): Chapter $1$: General Background: $\S 5$ The continuum problem