Natural Numbers are Infinite

Theorem

The set $\N$ of natural numbers is infinite.


Proof

Let the mapping $s: \N \to \N$ be defined as:

$\forall n \in \N: \map s n = n + 1$

$s$ is clearly an injection.

Aiming for a contradiction, suppose $\N$ were finite.

By Equivalence of Mappings between Finite Sets of Same Cardinality it follows that $s$ is a surjection.

But:

$\forall n \in \N: \map s n \ge 0 + 1 > 0$

So:

$0 \notin \Img s$

and $s$ is not a surjection.

From this contradiction it is seen that $\N$ cannot be finite.

So, by definition, $\N$ is infinite.

$\blacksquare$


Sources

  • 1960: Paul R. Halmos: Naive Set Theory ... (previous) ... (next): $\S 13$: Arithmetic
  • 1964: W.E. Deskins: Abstract Algebra ... (previous) ... (next): $\S 2.4$
  • 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text {III}$: The Natural Numbers: $\S 17$: Finite Sets: Theorem $17.8$
  • 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): infinite set
  • 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): infinite set
  • 2010: Raymond M. Smullyan and Melvin Fitting: Set Theory and the Continuum Problem (revised ed.) ... (previous) ... (next): Chapter $1$: General Background: $\S 1$ What is infinity?
  • 2000: James R. Munkres: Topology (2nd ed.) ... (previous) ... (next): $1$: Set Theory and Logic: $\S 6$: Finite Sets: Corollary $6.4$
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