:I am not much familiar with GST, but since it does not have any axiom of infinity, its natural numbers ( and any other model of Robinson arithmetic, for that matter ) may form a proper class.
32.
He did so however on the basis of an explicit acceptance of Cantor s Axiom of Infinity, which, as Mayberry points out, is best understood as simply a contradiction of Euclid s Common Notion 5 as applied to arithmoi.
33.
But the need of " some " doctrine of types is less doubtful than the precise form the doctrine should take; and in connection with the axiom of infinity it is particularly easy to see the necessity of some such doctrine ".
34.
Second, however, even if ZFC is formulated in so-called free logic, in which it is not provable from logic alone that something exists, the axiom of infinity ( below ) asserts that an " infinite " set exists.
35.
However, Principia Mathematica required, in addition to the basic axioms of type theory, three further axioms that seemed to not be true as mere matters of logic, namely the axiom of infinity, the axiom of choice, and the axiom of reducibility.
36.
The ZF axioms can also be written using a constant symbol representing the empty set; then the axiom of infinity uses this symbol without requiring it to be empty, while the axiom of empty set is needed to state that it is in fact empty.
37.
The axiom of regularity prevents this from happening . ) The minimal set " X " satisfying the axiom of infinity is the von Neumann ordinal ?, which can also be thought of as the set of natural numbers \ mathbb { N }.
38.
In type theory and in outgrowths thereof such as the axiomatic set theory NFU . J . Barkley Rosser showed that the existence of such a type-level ordered pair ( or even a " type-raising by 1 " ordered pair ) implies the axiom of infinity.
39.
A controversy that goes back to the early twentieth century concerns the issue of purely theoretic existence theorems, i . e ., theorems depending on non-constructive foundational material such as the axiom of infinity, the axiom of choice, or the law of excluded middle.
40.
The Formalist movement, following Hilbert s program of saving the mathematical fruits of Cantor s Axiom of Infinity via finitary consistency proofs, likewise, in the very definitions of formal systems and the establishment of their properties, accorded a special status to indefinite iteration and associated definitions by recursion and proofs by induction.
