Category:Axiom of Infinity

This category contains results about Axiom of Infinity.

There exists a set containing:

$(1): \quad$ a set with no elements

$(2): \quad$ the successor of each of its elements.

That is:

$\exists x: \paren {\paren {\exists y: y \in x \land \forall z: \neg \paren {z \in y} } \land \forall u: u \in x \implies u^+ \in x}$

Let $\omega$ be the class of natural numbers as constructed by the Von Neumann construction:

$\ds 0$

$:=$

$\ds \O$

$\ds 1$

$:=$

$\ds 0 \cup \set 0$

$\ds 2$

$:=$

$\ds 1 \cup \set 1$

$\ds 3$

$:=$

$\ds 2 \cup \set 2$

$\ds $

$\vdots$

$\ds $

$\ds n + 1$

$:=$

$\ds n \cup \set n$

$\ds $

$\vdots$

$\ds $

Then $\omega$ is a set.

Pages in category "Axiom of Infinity"

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