It provides for the existence of sets by separating off certain elements of existing sets.
This illustrates the fact that the principle of abstraction implies the existence of sets the elements of which are all objects having a certain property.
This is a restricted version of the principle of abstraction, now known as the principle of comprehension, for it provides for the existence of sets corresponding to formulas.
Thus, the axioms that Zermelo formulated are restrictive insofar as the asserting or implying of the existence of sets is concerned.
It restricts that principle, however, in two ways: (1) Instead of asserting the existence of sets unconditionally, it can be applied only in conjunction with preexisting sets, and (2) only "definite" formulas may be used.
The inventory of clinics data incites to conclude to the probable existence of sets of bifurcations in the determinism of troubles.
Now in ZF one can prove the existence of sets with a non-denumerable number of elements such as the set ℜ of real numbers.
Most of the time, large cardinal principles entail the existence of sets that are larger than any sets which can be guaranteed by ZFC to exist.
Quine granted the existence of sets, in part because they obey the extensionality axiom: sets are identical iff they have the same members.
The objections to the axiom arise from the fact that it asserts the existence of sets that cannot be explicitly defined.
We do not approach it by attempting to justify a principle that implies the existence of sets via definite descriptions which we don't yet know to be well-defined.
