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, however, that there is a shorter proof of a theorem from ZFC than from ZF.

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 ).

Basically da Costa and Doria define a formal sentence = NP ' which is the same as P = NP in the standard model for arithmetic ; however, because = NP ' by its very definition includes a disjunct that is not refutable in ZFC, = NP ' is not refutable in ZFC, so ZFC + = NP ' is consistent ( assuming that ZFC is ).

Given that the relevant cardinals exist, it is consistent with ZFC either that the first measurable cardinal is strongly compact, or that the first strongly compact cardinal is supercompact ; these cannot both be true, however.

Although originally controversial, the axiom of choice is now used without reservation by most mathematicians, and it is included in ZFC, the standard form of axiomatic set theory.

As discussed above, in ZFC, the axiom of choice is able to provide " nonconstructive proofs " in which the existence of an object is proved although no explicit example is constructed.

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.

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.

For example, the generalized continuum hypothesis ( GCH ) is not only independent of ZF, but also independent of ZFC.

Basic theories, such as arithmetic, real analysis and complex analysis are often introduced non-axiomatically, but implicitly or explicitly there is generally an assumption that the axioms being used are the axioms of Zermelo – Fraenkel set theory with choice, abbreviated ZFC, or some very similar system of axiomatic set theory like Von Neumann – Bernays – Gödel set theory, a conservative extension of ZFC.

With PCF theory, he showed that in spite of the undecidability of the most basic questions of cardinal arithmetic ( such as the continuum hypothesis ), there are still highly nontrivial ZFC theorems about cardinal exponentiation.

A similar phenomenon occurs in formalized theories that are able to refer to their own syntax, such as Zermelo – Fraenkel set theory ( ZFC ).

The proof sketched in the previous paragraph that the consistency of ZFC + " there is an inaccessible cardinal " implies the consistency of ZFC + " there is not an inaccessible cardinal " can be formalized in ZFC.

However, no proof that the consistency of ZFC implies the consistency of ZFC + " there is an inaccessible cardinal " can be formalized in ZFC.

There are arguments for the existence of inaccessible cardinals that cannot be formalized in ZFC.

This collection, which is formalized by Zermelo – Fraenkel set theory ( ZFC ), is often used to provide an interpretation or motivation of the axioms of ZFC.

Thus, IST is an enrichment of ZFC: all axioms of ZFC are satisfied for all classical predicates, while the new unary predicate " standard " satisfies three additional axioms I, S, and T. In particular, suitable non-standard elements within the set of real numbers can be shown to have properties that correspond to the properties of infinitesimal and unlimited elements.

In the foundations of mathematics, classical mathematics refers generally to the mainstream approach to mathematics, which is based on classical logic and ZFC set theory.

Formally, ZFC is a one-sorted theory in first-order logic.

Isabelle is generic: it provides a meta-logic ( a weak type theory ), which is used to encode object logics like First-order logic ( FOL ), Higher-order logic ( HOL ) or Zermelo – Fraenkel set theory ( ZFC ).

MK can be confused with second-order ZFC, ZFC with second-order logic ( representing second-order objects in set rather than predicate language ) as its background logic.

* ZFC – Zermelo – Fraenkel set theory, extended to include the Axiom of Choice.

The proof of the independence result also shows that a wide class of mathematical statements, including all statements that can be phrased in the language of Peano arithmetic, are provable in ZF if and only if they are provable in ZFC.

Equivalently, these statements are true in all models of ZFC but false in some models of ZF.

Sometimes slightly stronger theories such as Morse-Kelley set theory or set theory with a strongly inaccessible cardinal allowing the use of a Grothendieck universe are used, but in fact most mathematicians can actually prove all they need in systems weaker than ZFC, such as second-order arithmetic.

To avoid illegal set formation, one must only use predicates of ZFC to define subsets.

From the ZFC axioms of set theory ( including the axiom of choice ) one can show that there is a well-order of the reals.

Nonetheless, it is possible to show that the ZFC + GCH axioms alone are not sufficient to prove the existence of a definable ( by a formula ) well-order of the reals.

However it is consistent with ZFC that a definable well-ordering of the reals exists — for example, it is consistent with ZFC that V = L, and it follows from ZFC + V = L that a particular formula well-orders the reals, or indeed any set.

Modifications to this axiomatic theory proposed in the 1920s by Abraham Fraenkel, Thoralf Skolem, and by Zermelo himself resulted in the axiomatic set theory called ZFC.

This theory became widely accepted once Zermelo's axiom of choice ceased to be controversial, and ZFC has remained the canonical axiomatic set theory down to the present day.

ZFC does not assume that, for every property, there is a set of all things satisfying that property.

The enormous changes in world politics have, however, thrown it into confusion, so much so that it is safe to say that all international law is now in need of reexamination and clarification in light of the social conditions of the present era.

Of greater importance, however, is the content of those programs, which have had and are having enormous consequences for the American people.

The content is not the same, however: rather than individual security, it is the security and continuing existence of an `` ideological group '' -- those in the `` free world '' -- that is basic.

Historically, however, the concept is one that has been of marked benefit to the people of the Western civilizational group.

It is interesting, however, that despite this strong upsurge in Southern writing, almost none of the writers has forsaken the firmly entrenched concept of the white-suited big-daddy colonel sipping a mint julep as he silently recounts the revenue from the season's cotton and tobacco crops ; ;

He is still concerned, however, with a personal event.

I think it is essential, however, to pinpoint here the difference between the two concepts of sovereignty that went to war in 1861 -- if only to see better how imperative is our need today to clarify completely our far worse confusion on this subject.

It appears that the dominant tendency of Mann's early tales, however pictorial or even picturesque the surface, is already toward the symbolic, the emblematic, the expressionistic.

More profound and more disturbing, however, is the moral isolation of Raymond Chandler's Philip Marlowe.

This is, however, symptomatic of our national malaise.

We are reminded, however, that freedom of thought and discussion, the unfettered exchange of ideas, is basic under our form of government.

This, however, cannot be done by a community whose very experience of truth is confused and incoherent: it has no absolute standard, and consequently cannot distinguish the absolute from the contingent.

But however we come, finally, to explain and account for the present, the truth we are trying to expose, right now, is that the makers of constitutions and the designers of institutions find it difficult if not impossible to anticipate the behavior of the host of all their enterprises.

The reality of the situation, however, is described by Mr. Lyford: ``

Circular motion, however, since it is eternal and perfectly continuous, lacks termini.

It is a mistake, however, to imagine that Sandburg uses the guitar as a prop.

It is, however, a disarming disguise, or perhaps a shield, for not only has Mercer proved himself to be one of the few great lyricists over the years, but also one who can function remarkably under pressure.

Undoubtedly, however, the significance of the volume is greater than the foregoing paragraphs suggest.

For what we propose, however, a psychoanalyst is not necessary, even though one aim is to enable the reader to get beneath his own defenses -- his defenses of himself to himself.

What is not so well known, however, and what is quite important for understanding the issues of this early quarrel, is the kind of attack on literature that Sidney was answering.