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Thus, the intuitive interpretation of the third axiom schema is: given a set a and a
condition on x, P(x), those elements of a for which the condition holds form a set.
It provides for the existence of sets by separating off certain elements of existing ...
An application of the ninth axiom schema then yields the ordinal that follows all of
these in the same sense in which to follows the finite ordinals; using notation from
ordinal arithmetic, it is to2. At this point the iteration process can be repeated.
a finite number of axioms. On the other hand, since the ninth axiom or schema of
ZF provides an axiom for each formula, ZF has infinitely many axioms — which is
unavoidable because it is known that no finite subset yields the full system of ...
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