In Zermelo–Fraenkel set theory (ZFC), the axiom of unrestricted comprehension is replaced with a group of axioms that allow construction of sets.
On the other hand, among axiomatic set theories, ZFC is comparatively weak.
Since SCH is a consequence of GCH, which is known to be consistent with ZFC, SCH is consistent with ZFC.
However, this hypothesis can neither be proved nor disproved within the widely accepted ZFC axiomatic set theory, if ZFC is consistent.
The pattern illustrated in the previous sections with Peano arithmetic, ZFC, and ZFC + "there exists an inaccessible cardinal" cannot generally be broken.
The following statements (none of which have been proved false) cannot be proved in ZFC (the Zermelo-Fraenkel set theory plus the axiom of choice) to be independent of ZFC, under the added hypothesis that ZFC is consistent.
independent of ZFC (the canonical axiomatic set theory of contemporary mathematics, consisting of the Zermelo–Fraenkel axioms plus the axiom of choice), assuming that ZFC is consistent.
It is now known that this statement is independent of ZFC (the Zermelo–Fraenkel axioms of set theory, which is the standard axiom system assumed in mathematics).
Most working set theorists believe that the large cardinal axioms that are currently being considered are consistent with ZFC.
Mathematicians have shown there are many statements that are neither provable nor disprovable in Zermelo–Fraenkel set theory with the axiom of choice (ZFC), the standard system of set theory in mathematics (assuming that ZFC is consistent); see List of statements undecidable in ZFC.
Usually when one talks about inner models of a theory, the theory one is discussing is ZFC or some extension of ZFC (like ZFC + [...] a measurable cardinal).
Strong mathematical theories such as ZFC induce propositional proof systems as well: a proof of a tautology [...] in a propositional interpretation of ZFC is a ZFC-proof of a formalized statement ' [...] is a tautology'.
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