The existence of the set " N " follows from the axiom of infinity in ZF set theory.
12.
The sets and classes of " V " ? satisfy all the axioms of NBG except the axiom of infinity.
13.
Furthermore, the axiom of infinity guarantees the existence of at least one infinite set, namely a set containing the natural numbers.
14.
Concerning GST : Since it does not have any axiom of infinity, it cannot prove that there is a model of arithmetic.
15.
:In set theory, there is the axiom of infinity, which arises because the other axioms don't define an infinite set.
16.
The hereditarily finite sets, V ?, satisfy the axiom of regularity ( and all other axioms of ZFC except the axiom of infinity ).
17.
Modern constructive set theory includes the axiom of infinity from ZFC ( or a revised version of this axiom ) and the set of natural numbers.
18.
Indeed, using the Von Neumann universe, we can make a model of the axioms where the axiom of infinity is replaced by its negation.
19.
PM define addition, multiplication and exponentiation of cardinals, and compare different definitions of finite and infinite cardinals . 8'120.03 is the Axiom of infinity.
20.
A common objection to Cantor's theory of infinite number involves the axiom of infinity ( which is, indeed, an axiom and not a logical truth ).
