For example, ZF + DC + BP is consistent, if ZF is.

It is also consistent with ZF + DC that every set of reals is Lebesgue measurable ; however, this consistency result, due to Robert M. Solovay, cannot be proved in ZFC itself, but requires a mild large cardinal assumption ( the existence of an inaccessible cardinal ).

Also Sierpinski proved that ZF + GCH ( the generalized continuum hypothesis ) imply the axiom of choice and hence a well-order of the reals.

It follows from ZF + axiom of determinacy that ω < sub > 1 </ sub > is measurable, and that every subset of ω < sub > 1 </ sub > contains or is disjoint from a closed and unbounded subset.

It is due to this intermediate status between ZF and ZF + AC ( ZFC ) that the Boolean prime ideal theorem is often taken as an axiom of set theory.

ZF + AC < sub > ω </ sub > suffices to prove that the union of countably many countable sets is countable.

Hence, as a consequence of the incompleteness theorems, it is not possible to prove the relative consistency of ZF + AD with respect to ZF.

It was proved by Kurt Gödel that any model of ZF has a least inner model of ZF ( which is also an inner model of ZFC + GCH ), called the constructible universe, or L.

It is implied by the continuum hypothesis, and thus certainly consistent with ZFC, but is also known to be consistent with ZF + ¬ CH.

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Unlike full AC, DC is insufficient to prove ( given ZF ) that there is a nonmeasurable set of reals, or that there is a set of reals without the property of Baire or without the perfect set property.

DC is ( over the theory ZF ) equivalent to the statement that every ( nonempty ) pruned tree has a branch.

Furthermore, AD implies the consistency of Zermelo – Fraenkel set theory ( ZF ).

However, if there are infinitely many Woodin cardinals with a measurable above them all, then L ( R ) is a model of ZF that satisfies AD.

However, that particular case is a theorem of Zermelo – Fraenkel set theory without the axiom of choice ( ZF ); it is easily proved by mathematical induction.

If the method is applied to an infinite sequence ( X < sub > i </ sub >: i ∈ ω ) of nonempty sets, a function is obtained at each finite stage, but there is no stage at which a choice function for the entire family is constructed, and no " limiting " choice function can be constructed, in general, in ZF without the axiom of choice.

The debate is interesting enough, however, that it is considered of note when a theorem in ZFC ( ZF plus AC ) is logically equivalent ( with just the ZF axioms ) to the axiom of choice, and mathematicians look for results that require the axiom of choice to be false, though this type of deduction is less common than the type which requires the axiom of choice to be true.

It is possible to prove many theorems using neither the axiom of choice nor its negation ; such statements will be true in any model of Zermelo – Fraenkel set theory ( ZF ), regardless of the truth or falsity of the axiom of choice in that particular model.

For example, the Banach – Tarski paradox is neither provable nor disprovable from ZF alone: it is impossible to construct the required decomposition of the unit ball in ZF, but also impossible to prove there is no such decomposition.

Similarly, all the statements listed below which require choice or some weaker version thereof for their proof are unprovable in ZF, but since each is provable in ZF plus the axiom of choice, there are models of ZF in which each statement is true.

Assuming ZF is consistent, Kurt Gödel showed that the negation of the axiom of choice is not a theorem of ZF by constructing an inner model ( the constructible universe ) which satisfies ZFC and thus showing that ZFC is consistent.

Assuming ZF is consistent, Paul Cohen employed the technique of forcing, developed for this purpose, to show that the axiom of choice itself is not a theorem of ZF by constructing a much more complex model which satisfies ZF ¬ C ( ZF with the negation of AC added as axiom ) and thus showing that ZF ¬ C is consistent.

The assumption that ZF is consistent is harmless because adding another axiom to an already inconsistent system cannot make the situation worse.

The contributions of Kurt Gödel in 1940 and Paul Cohen in 1963 showed that the hypothesis can neither be disproved nor be proved using the axioms of Zermelo – Fraenkel set theory, the standard foundation of modern mathematics, provided ZF set theory is consistent.

Thus ZF together with " there exists a weakly inaccessible cardinal " implies that ZFC is consistent.

), it is consistent with ZF that a measurable cardinal can be a successor cardinal.

This shows that both propositions are consistent with the basic axioms of set theory, if ZF itself is consistent.

More general and powerful list-building facilities are provided by " list comprehensions " ( previously known as " ZF expressions "), which come in two main forms: an expression applied to a series of terms, e. g.:

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However the situation is different when the language is of arbitrary large cardinality since then, though the completeness and compactness theorems remain provably equivalent to each other in ZF, they are also provably equivalent to a weak form of the axiom of choice known as the ultrafilter lemma.

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However, if an inaccessible cardinal κ is assumed, then the sets of smaller rank form a model of ZF ( a Grothendieck universe ), and its subsets can be thought of as " classes ".

However, κ does not need to be inaccessible, or even a cardinal number, in order for V < sub > κ </ sub > to be a standard model of ZF ( see below ).

The definition of implies ( in ZF, Zermelo – Fraenkel set theory without the axiom of choice ) that no cardinal number is between and.

In 1970, Solovay demonstrated that the existence of a non-measurable set for Lebesgue measure is not provable within the framework of Zermelo – Fraenkel set theory in the absence of the Axiom of Choice, by showing that ( assuming the consistency of an inaccessible cardinal ) there is a model of ZF, called Solovay's model, in which countable choice holds, every set is Lebesgue measurable and in which the full axiom of choice fails.

The restriction to ZF renders any claim that relies on either the axiom of choice or its negation unprovable.

" Such conditional statements are provable in ZF when the original statements are provable from ZF and the axiom of choice.

When one attempts to solve problems in this class, it makes no difference whether ZF or ZFC is employed if the only question is the existence of a proof.

The former is equivalent in ZF to the existence of an ultrafilter containing each given filter, proved by Tarski in 1930.

1920: Thoralf Skolem corrected Löwenheim's proof of what is now called the downward Löwenheim-Skolem theorem, leading to Skolem's paradox discussed in 1922 ( the existence of countable models of ZF, making infinite cardinalities a relative property.

showed that existence of measurable cardinals in ZFC, real valued measurable cardinals in ZFC, and measurable cardinals in ZF, are equiconsistent.