# 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.

axiom collocations

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 sentences

5) 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 axiom

15) 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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