There are alternative set theories, e . g . " general set theory " ( GST ), Kripke Platek set theory, and pocket set theory ( PST ), that deliberately omit the axiom of power set and the axiom of infinity and do not allow the definition of the infinite hierarchy of infinites proposed by Cantor.
42.
Russell in his 1920 " Introduction to Mathematical Philosophy " devotes an entire chapter to " The axiom of Infinity and logical types " wherein he states his concerns : " Now the theory of types emphatically does not belong to the finished and certain part of our subject : much of this theory is still inchoate, confused, and obscure.
43.
For example, I believe that if you do it for the Kripke Platek set theory with the axiom of infinity, you'll get the admissible set L _ { \ omega _ 1 ^ { CK } } ( which is, indeed, a model of V = L ) . J . 12 : 59, 2 December 2010 ( UTC)
44.
The Axiom of Infinity of NFU can be expressed as V \ not \ in Fin : this is enough to establish that each natural number has a nonempty successor ( the successor of | A | being | A \ cup \ { x \ } | for any x \ not \ in A ) which is the hard part of showing that the Peano axioms of arithmetic are satisfied.
45.
The Church-Kleene ordinal is again related to Kripke-Platek set theory, but now in a different way : whereas the Bachmann-Howard ordinal ( described G�del universe, " L ", up to stage ?, yields a model L _ \ alpha of KP . Such ordinals are called "'admissible "', thus \ omega _ 1 ^ { \ mathrm { CK } } is the smallest admissible ordinal ( beyond ? in case the axiom of infinity is not included in KP ).
46.
Hilbert states that, with regard to this system, i . e . " Russell and Whitehead's theory of foundations [, ] . . . the foundation that it provides for mathematics rests, first, upon the axiom of infinity and, then upon what is called the axiom of reducibility, and both of these axioms are genuine contentual assumptions that are not supported by a consistency proof; they are assumptions whose validity in fact remains dubious and that, in any case, my theory does not require . . . reducibility is not presupposed in my theory . . . the execution of the reduction would be required only in case a proof of a contradiction were given, and then, according to my proof theory, this reduction would always be bound to succeed ."
