ZFC axiom in a sentence
1) There are many equivalent formulations of the ZFC axioms.
2) Metamath version of the ZFC axioms – A concise and nonredundant axiomatization.
3) It is independent from the ZFC axioms whether "V" = "L".
4) Most of the ZFC axioms state the existence of particular sets defined from other sets.
ZFC axiom example sentences5) For any given finite set of ZFC axioms there is ZFC proof that this set of axioms has countable transitive model.
6) This adding (justified by historical opportunity reasons) is theoretically useless since the concept of Hilbert space can be designed with the standard ZFC axioms.
7) In ZFC, one proves that these notions all generate or apply to sets via the ZFC axioms of "union", "separation", and "power set".
8) If the scope of all quantified variables in the above axioms is restricted to sets, all axioms except III and the schema IV are ZFC axioms.
9) These ontological restrictions are required for ZFC to avoid Russell's paradox, but critics argue these restrictions make the ZFC axioms fail to capture the informal concept of "set".
10) Because there are non-well-founded models that satisfy each axiom of ZFC except the axiom of regularity, that axiom is independent of the other ZFC axioms.
11) One motivation for the ZFC axioms is the cumulative hierarchy of sets introduced by John von Neumann (Shoenfield 1977, sec. 2).
12) There are many equivalent formulations of the ZFC axioms; for a rich but somewhat dated discussion of this fact, see Fraenkel "et al.
13) Landmark results in this area established the independence of the continuum hypothesis from ZFC, and of the axiom of choice from the remaining ZFC axioms.
14) A goal of the ZFC axioms is that each axiom should be true if interpreted as a statement about the collection of all sets in the von Neumann universe (also known as the cumulative hierarchy).
example sentences with axiom15) One has to construct a ZFC proof of Con("T" + "H") for any given finite set "T" of ZFC axioms (by ZFC instruments of course).
16) One verification project, Metamath, includes derivations of more than 10,000 theorems starting from the ZFC axioms and using first order logic.
17) Using the standard ZFC axioms for set theory, every Dedekind-finite set is also finite, but this requires at least the axiom of countable choice.
18) One have to prove that there is finite set T" of ZFC axioms such that if countable transitive model M satisfies T" then "M"["G"] satisfies considered hypothesis "H".
19) For any given finite set T of ZFC axioms there is finite set T' of ZFC axioms such that ZFC proves that if countable transitive model "M" satisfies T' then "M"["G"] satisfies "T".
20) For any given finite set T of ZFC axioms there is finite set T' of ZFC axioms such that ZFC proves that if countable transitive model "M" satisfies T' then "M"["G"] satisfies "T".
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