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.

If one rejects that axiom, the situation is more complicated, with additional infinite cardinals that are not alephs.

If the axiom of choice is assumed, as will be the case in the rest of this article, then the two definitions are equivalent.

If the axiom of choice holds, the following conditions on a cardinal are equivalent:

If there was an intuitive element, it was to be isolated and represented separately as an axiom: from there on, the proof was to be purely logical and without gaps.

If A is a nonempty set containing E, then we can form the intersection using the axiom schema of specification as

If an axiom is ever added that makes the system complete, it does so at the cost of making the system inconsistent.

If the axiom of choice holds, the next higher cardinal number is aleph-one,.

* If Martin's axiom and the negation of the continuum hypothesis both hold, then there is a non-free Whitehead group.

If sheaf values are in the category of sets, applying the local identity axiom to the empty family shows that over the empty set, there is at most one section, and applying the gluing axiom to the empty family shows that there is at least one section.

* Veblen's axiom: If a, b, c, d are distinct points and the lines through ab and cd meet, then so do the lines through ac and bd.

If the axiom set is not categorical ( so that there is more than one model ) one has the geometry / geometries debate to settle.

If the axiom of countable choice ( a weaker version of the axiom of choice ) holds, then is smaller than any other infinite cardinal.

If the axiom of choice ( AC ) is used, it can be further proved that the class of cardinal numbers is totally ordered, and thus is the second-smallest infinite cardinal number.

( If the axiom of choice holds, this is the next larger cardinal.

If the universal set existed and the axiom of separation applied to it, then Russell's paradox would arise from

If the second axiom, that of idempotence, is relaxed, then the axioms define a preclosure operator.

If the axiom of choice is not assumed we need to do something different.

If the axiom of choice holds ( so that any cardinal number can be well-ordered ), an infinite cardinal is regular if and only if it cannot be expressed as the cardinal sum of a set of cardinality less than, the elements of which are cardinals less than.

If the axiom of choice holds, then every successor cardinal is regular.

If one defines a homology theory axiomatically ( via the Eilenberg-Steenrod axioms ), and then relaxes one of the axioms ( the dimension axiom ), one obtains a generalized theory, called an extraordinary homology theory.

If the set X above is restricted to be the set of all real numbers, the resulting axiom is called DC < sub > R </ sub >.

If you were a man '' --

If she, Pamela, were being held responsible for his crimes, then hers must be the final act of expiation.

If someone were to drop a match in here, this place would go up like a haystack ''.

If the other pilots were worried, they did not show it.

If it were the enemy, tactically his position was correct.

If it were not for an old professor who made me read the classics I would have been stymied on what to do, and now I understand why they are classics ; ;

If it were not that I knew who it was I could have mistaken it for my Aunt so well did her clothes fit him.

If an automobile were approaching him, he would know what was required of him, even though he might not be able to act quickly enough.

If there were only darkness, all would be clear.

If there were only the mess, all would be clear ; ;

If these people were denied a voice ( do they have a moral right to a voice??

If we were creating a wholly new society, we could insist that our social, political, economic and philosophic institutions foster rather than hamper man ; ;

If living Jews were unavailable for study, the Bible was at hand.

If his scholarship and formal musicianship were not all they might have been, Mercer demonstrated at an early age that he was gifted with a remarkable ear for rhythm and dialect.

If she were not at home, Mama would see to it that a fresh white rose was there.

If the Stearns were not there, grace would be omitted.

If anything, the conservative Democrats were more opposed to Hearst than the Republicans.

If the hardships of the winter at Valley Forge were trying for healthy men, they were, of course, much more so for those not in good health.

If his circumspection in regard to Philip's sensibilities went so far that he even refused to grant a dispensation for the marriage of Amadee's daughter, Agnes, to the son of the dauphin of Vienne -- a truly peacemaking move according to thirteenth-century ideas, for Savoy and Dauphine were as usual fighting on opposite sides -- for fear that he might seem to be favoring the anti-French coalition, he would certainly never take the far more drastic step of ordering the return of Gascony to Edward, even though, as he admitted to the English ambassadors, he had been advised that the original cession was invalid.

If there were no West Berlin problem, imperialist quarters would have invented an excuse for stepping up the armaments race to try to solve the internal and external problems besetting the United States and its NATO partners.

If the new Soviet series has followed the general pattern of previous Russian tests, the shots were roughly half fission and half fusion, meaning a fission yield of 30 to 40 megatons thus far.

If the Hessian troops sent here willy-nilly by the Hessian Government to fight for England in the 1770's were mercenaries, what shall we call the UN troops sent to the Congo willy-nilly by their governments to fight for the United Nations??

If the UN troops are not mercenaries then the Hessians were not mercenaries either.

If he were to go with White, he would be out there two days, not just listening in the dark at some point between here and Papa-san, but moving ever deeper into enemy land -- behind Papa-san -- itself.