For most purposes it is simply used as convenient; when considered more carefully it is incorporated, or not, according to whether the axiom of infinity is included.
22.
Thus the axiom of infinity is sometimes regarded as the first " large cardinal axiom ", and conversely large cardinal axioms are sometimes called stronger axioms of infinity.
23.
Thus the axiom of infinity is sometimes regarded as the first " large cardinal axiom ", and conversely large cardinal axioms are sometimes called stronger axioms of infinity.
24.
The hereditarily countable sets form a model of Kripke Platek set theory with the axiom of infinity ( KPI ), if the axiom of countable choice is assumed in the metatheory.
25.
Even if one does not accept the axiom of infinity and therefore cannot accept that the set of all natural numbers exists, it is still possible to define any one of these sets.
26.
The main reason one accepts the axiom of infinity is probably that we feel it absurd to think that the process of adding only one set at a time can exhaust the entire universe.
27.
Mainstream mathematicians consider strict finitism too confining, but acknowledge its relative consistency : the universe of hereditarily finite sets constitutes a model of Zermelo Fraenkel set theory with the axiom of infinity replaced by its negation.
28.
The existence of a set with at least two elements is assured by either the axiom of infinity, or by the axiom schema of specification and the axiom of the power set applied twice to any set.
29.
One can add to this base theory strong axioms of infinity familiar from the ZFC context, such as " there exists an inaccessible cardinal, " but it is more natural to consider assertions about Cantorian and strongly Cantorian sets.
30.
The technical details here are not the main point, which is that reasonable and natural ( in the context of NFU ) assertions turn out to be equivalent in power to very strong axioms of infinity in the ZFC context.
