# Zfc

### Some articles on zfc:

Large Countable Ordinal - Beyond Recursive Ordinals - “Unprovable” Ordinals
... For example, if ZFC has a transitive model (a hypothesis stronger than the mere hypothesis of consistency, and implied by the existence of an inaccessible ... Such ordinals are beyond the strength of ZFC in the sense that it cannot (by construction) prove their existence ... is a 1-elementary submodel of L the existence of these ordinals can be proven in ZFC, and they are closely related to the nonprojectible ordinals ...
List Of Statements Undecidable In ZFC
... discussed below are provably undecidable in ZFC (the Zermelo–Fraenkel axioms plus the axiom of choice, the canonical axiomatic set theory of contemporary ...
Equiconsistency - Consistency Strength
... The usual set of axioms of set theory is called ZFC ... Arithmetic in this case) it can be proven that the theories ZFC+A and ZFC+B are equiconsistent ... as the metatheory in question, but even if the metatheory is ZFC or an extension of it, the notion is meaningful ...
The Reflection Principle As A Theorem of ZFC
... One form of the reflection principle in ZFC says that for any finite set of axioms of ZFC we can find a countable transitive model satisfying these ... In particular this proves that ZFC is not finitely axiomatizable, because if it were it would prove the existence of a model of itself, and hence prove its own ... of the reflection principle says that for any finite number of formulas of ZFC we can find a set Vα in the cumulative hierarchy such that all the formulas ...
Morse–Kelley Set Theory - Discussion - Model Theory
... ZFC, NBG, and MK each have models describable in terms of V, the standard model of ZFC and the von Neumann universe ... Then Vκ is an intended model of ZFC Def(Vκ) is an intended model of NBG Vκ+1, the power set of Vκ, is an intended model of MK ...