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 first-countability axiom were imposed on the topological spaces in question, the two above conditions would be 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 this choice is less exciting than New York Democrats may wish, it nevertheless must be made.

If Russian pupils have to take these languages, how come American students have a choice whether or not to take a language, but have to face so many exceptions??

If we fail to develop the means to hunt down and destroy the enemy's military force with extreme care and precision, and if war comes in spite of our most ardent desires for peace, our choice of alternatives will be truly frightening.

If the content of faith is to be presented today in a form that can be `` understanded of the people '' -- and this, it must not be forgotten, is one of the goals of the perennial theological task -- there is no other choice but to abandon completely a mythological manner of representation.

If to be a Christian means to say yes where I otherwise say no, or where I do not have the right to say anything at all, then my only choice is to refuse to be a Christian.

`` If Philip Toynbee is claiming that the choice lies between capitulation and the risk of nuclear war, I think he is right.

If we try to choose an element from each set, then, because X is infinite, our choice procedure will never come to an end, and consequently, we will never be able to produce a choice function for all of X.

If the next data item is not available when it is needed, the codec has no choice but to produce silence or guess — and if the data is late, it is useless, because the time period when it should have been converted to a signal has already passed.

If a government is in the habit of appointing and replacing the governor frequently, it clearly has the capacity to micro-manage the central bank through its choice of governors.

If the meet is a 5 optional meet, then the divers will perform 1 optional from each category ( Front, Back, Inward, Reverse, and Twister ) and then 1 voluntary from the category of their choice.

If there is more than one allograph of a unit of writing, and the choice between them depends on context or on the preference of the author, they now have to be treated as separate glyphs, because mechanical arrangements have to be available to differentiate between them and to print whichever of them is required.

If history has to single out one person from among the Mountain Men to receive the admiration of later generations, Carson is the best choice.

If " this statement is false " is denoted by A and its truth value is being sought, it is necessary to find a condition that restricts the choice of possible truth values of A.

If further copies are desired at this point, there is no choice but to make up another stencil from scratch.

Rand summarizes: If chooses to live, a rational ethics will tell him what principles of action are required to implement his choice.

If the case has been submitted to arbitration rather than a national court, say because of a forum selection clause, an arbitrator may decide not to apply local mandatory policies in the face of a choice of law by the parties if this would defeat their commercial objectives.

If you were forced to make a choice.

If either the 2nd player chooses to move to the square diagonal to the first player or the first player's 2nd move is not to this square given the choice, then a starting position before flipping moves commence that differs from the standard Othello position arises.

If a tumor is causing syringomyelia, removal of the tumor is the treatment of choice and almost always eliminates the syrinx.

( If it is the first move of the game, playing on a corner gives " O " more opportunities to make a mistake and may therefore be the better choice ; however, it makes no difference between perfect players.