In mathematics, the axiom of choice, or AC, is an axiom of set theory equivalent to the statement that " the product of a collection of non-empty sets is non-empty ".

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.

In Martin-Löf type theory and higher-order Heyting arithmetic, the appropriate statement of the axiom of choice is ( depending on approach ) included as an axiom or provable as a theorem.

The axiom of choice is not the only significant statement which is independent of ZF.

Conjecture is contrasted by hypothesis ( hence theory, axiom, principle ), which is a testable statement based on accepted grounds.

It is therefore possible to adopt this statement, or its negation, as a new axiom in a consistent manner ( much as we can take Euclid's parallel postulate as either true or false ).

Rand defined an axiom as " a statement that identifies the base of knowledge and of any further statement pertaining to that knowledge, a statement necessarily contained in all others whether any particular speaker chooses to identify it or not.

The ninth, final axiom is a second order statement of the principle of mathematical induction over the natural numbers.

* An axiom or postulate is a statement that is accepted without proof and regarded as fundamental to a subject.

The statement that is now called the continuum hypothesis and is known to be independent of the Zermelo – Fraenkel axioms for set theory ( including the axiom of choice ).

* The prisoner will be hanged next week and its date will not be deducible in advance using this statement as an axiom ( B ).

Raeder finally ended his speech with the statement: " Do not think only of the present day or the present war ; think instead of the millennia in which the German nation has already struggled and of the centuries that lie before us and require a wise use of all our resources from this day on ... Remember, therefore, the most important axiom in the thoughts of our Führer and Supreme Commander ; it is not the individual, the family or the clan that count, but the Volk and Volk alone.

To understand this axiom, note that the clause in parentheses in the symbolic statement above simply states that A and B have precisely the same members.

The axiom of extensionality can be used with any statement of the form

Some treatments of axiomatic set theory prefer to do without this, and instead treat the above statement not as an axiom but as a definition of equality.

Negating the Playfair's axiom form, since it is a compound statement (... there exists one and only one ...), can be done in two ways.

To verify a formal proof when the set of axioms is infinite, it must be possible to determine whether a statement that is claimed to be an axiom is actually an axiom.

Each time a new statement is added as an axiom, there are other true statements that still cannot be proved, even with the new axiom.

On the other hand, other foundational descriptions of category theory are considerably stronger, and an identical category-theoretic statement of choice may be stronger than the standard formulation, à la class theory, mentioned above.

Because of this, is usually a more effective choice under these operating systems ; sensitivity for these two combinations can be enhanced by the CONFIG. SYS statement.

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.

This choice of terminology is justified by the fact random oracles support a statement with probability 0 or 1 only.

the choice of which statement is R and which is ¬ R is arbitrary.

then, seen as a statement about cardinal numbers, it is equivalent to the Axiom of choice.

She reiterates the intended outcome through her statement of " choice of an appropriate antecedent genre guides the rhetor toward a response consonant with situational demands " ( Jamieson 414 ).

Within an imperative programming language, a control flow statement is a statement whose execution results in a choice being made as to which of two or more paths should be followed.

and these sets cannot be identified without making such a choice – any statement that identifies them " depends on choice of identification ".

Menander ’ s further statement that her father was a murderer sheds some light on her choice of a way to solve the “ Naboth ” problem in 1 Kings 21.

However, the momentum space in deformed special relativity is curved, which is a statement independent of the choice of coordinates.

No official statement has been made on the reasons behind the choice of F / A-18 fighters as Australia's primary combat commitment, but it is commonly assumed that the obvious alternative of sending a substantial land force instead was considered to involve an unacceptably high risk of casualties, particularly given the possibility of house-to-house fighting in Iraqi cities.

By the early 19th century one's choice of carriage was only in part based on practicality and performance ; it was also a status statement and subject to changing fashions.

David Boaz, head of the libertarian think tank Cato Institute, raised questions with Frontlines choice of subjects in a prepared statement, " Ending Taxpayer Funding for Public Broadcasting ", before the United States Senate, arguing,

In his statement, he claimed that he had " deprived the government of its goal of sentencing me to death ," and that " the fact that I have entered an agreement with the government is purely a tactical choice on my part and in no way legitimates the moral authority of the government to judge this matter or impute my guilt.

Although the play does show a certain structural amateurishness ( there are eleven acts varying in length from twenty-five seconds to an hour and a half ), the statement it makes concerning the ceaseless yearning and searching of youth is profound and worthy of our attention.

If a continuation card follows a statement that does not permit continuation cards, the compiler will generate a NOP and issue an error message.

However, B's example of his uncle does not contradict A's statement, which says nothing about non-Republicans.

Note that " completeness " has a different meaning here than it does in the context of Gödel's first incompleteness theorem, which states that no recursive, consistent set of non-logical axioms of the Theory of Arithmetic is complete, in the sense that there will always exist an arithmetic statement such that neither nor can be proved from the given set of axioms.

Cameron also points out that whether he refers to Plato or to Crantor, the statement does not support conclusions such as Otto Muck's " Crantor came to Sais and saw there in the temple of Neith the column, completely covered with hieroglyphs, on which the history of Atlantis was recorded.

Using the above theorem it is easy to see that the original Borsuk Ulam statement is correct since if we take a map f: S < sup > n </ sup > → ℝ < sup > n </ sup > that does not equalize on any antipodes then we can construct a map g: S < sup > n </ sup > → S < sup > n-1 </ sup > by the formula

Tacitus ' statement that they were " German in their way of life and types of dwelling " implies a sedentary bias, but their close relations with the Sarmatians, who were nomadic, may indicate a more nomadic lifestyle, as does the wide geographical range of their attested inhabitation.

Francis Jennings noted that the statement made by Benjamin Franklin frequently quoted by proponents of the thesis does not support for this idea as it is advocating for a union against these " ignorant savages " and called the idea " absurd ".

In a meeting with 8 Jewish and 8 Protestant Dutch leaders in Israel in May 2011, a statement of cooperation was issued, indicating, for the most part, that the Protestant Church recognizes the issues involved with the Palestinian Christians and that this is sometimes at odds with support for the State of Israel, but standing up for the rights of the Palestinians does not detract from the emphasis on the safety of the State of Israel and vice versa.

The phrase " Cogito ergo sum " ( I think, therefore I am ) is also commonly associated with Descartes ' theory, because in his own methodological doubt, doubting everything he previously knew in order to start from a blank slate, the first thing that he could not logically bring himself to doubt was his own existence: " I do not exist " would be a contradiction in terms ; the act of saying that one does not exist assumes that someone must be making the statement in the first place.

A proposition that says: " The product of the sum and the difference of a and b should give us the difference of the squares of a and b " does express a normative proposition, but this normative statement is based on the theoretical statement "( a + b )( a-b )= a²-b² ".

It made the same statement with regard also to later ordinations by those bishops, saying that, " as for those who have already thus unlawfully received ordination or any who may yet accept ordination from these, whatever may be the validity of the orders ( quidquid sit de ordinum validitate ), the Church does not and will not recognise their ordination ( ipsorum ordinationem ), and will consider them, for all legal effects, as still in the state in which they were before, except that the ... penalties remain until they repent " ( Sacred Congregation for the Doctrine of the Faith, Decree Episcopi qui alios of 17 September 1976-Acta Apostolicae Sedis 1976, page 623 ).

The explicit statement that Jesus was himself the Arche does not come from John's gospel but from the Letter to the Colossians.

The rule does not make any statement about whether any particular reference or comparison to Adolf Hitler or the Nazis might be appropriate, but only asserts that the likelihood of such a reference or comparison arising increases as the discussion progresses, irrespective of whether it is appropriate or not.

But now We join Our voice to that of Our predecessor John XXIII of venerable memory, and We make Our own his words: " No statement of the problem and no solution to it is acceptable which does violence to man's essential dignity ; those who propose such solutions base them on an utterly materialistic conception of man himself and his life.

Impeachment does not necessarily result in removal from office ; it is only a legal statement of charges, parallel to an indictment in criminal law.

For the Jews slew him, although he was a most just man .” However, this statement does not appear in the extant manuscripts of Josephus.

( Köstenberger pages 104-105 ) Köstenberger states that the statement by Josephus that some people recognized Jesus as the Messiah is consistent with the grammar of Josephus elsewhere but does not imply that Josephus himself considered Jesus the Messiah.

His statement, regarded as a logical principle purely and apart from material facts, does not therefore amount to more than that of Aristotle, which deals simply with the significance of negation.

" However, Epimenides ' statement that all Cretans are liars can be resolved as false, given that he knows of at least one other Cretan who does not lie.

A volunteered statement by a person in custody does not implicate Miranda.

Regardless of how it is interpreted, intuitionism does not equate the truth of a mathematical statement with its provability.

* She does not realize that " eating meat is wrong " is a true statement.