In mathematics, the axiom of choice, abbreviated AC or AoC, is an axiom of set theory equivalent to the statement that a Cartesian product of a collection of non-empty sets is non-empty. Informally put, the axiom of choice says that given any collection of sets, each containing at least one element, it is possible to construct a new set by arbitrarily choosing one element from each set, even if the collection is infinite. Formally, it states that for every indexed family of nonempty sets, there exists an indexed set such that for every . The axiom of choice was formulated in 1904 by Ernst Zermelo in order to formalize his proof of the well-ordering theorem.
In mathematics, the Hausdorff maximal principle is an alternate and earlier formulation of Zorn's lemma proved by Felix Hausdorff in 1914. It states that in any partially ordered set, every totally ordered subset is contained in a maximal totally ordered subset.
A joke is a display of humour in which words are used within a specific and well-defined narrative structure to make people laugh and is usually not meant to be interpreted literally. It usually takes the form of a story, often with dialogue, and ends in a punch line, whereby the humorous element of the story is revealed; this can be done using a pun or other type of word play, irony or sarcasm, logical incompatibility, hyperbole, or other means. Linguist Robert Hetzron offers the definition:
A joke is a short humorous piece of oral literature in which the funniness culminates in the final sentence, called the punchline… In fact, the main condition is that the tension should reach its highest level at the very end. No continuation relieving the tension should be added. As for its being "oral," it is true that jokes may appear printed, but when further transferred, there is no obligation to reproduce the text verbatim, as in the case of poetry.
Recursion occurs when the definition of a concept or process depends on a simpler version of itself. Recursion is used in a variety of disciplines ranging from linguistics to logic. The most common application of recursion is in mathematics and computer science, where a function being defined is applied within its own definition. While this apparently defines an infinite number of instances, it is often done in such a way that no infinite loop or infinite chain of references can occur.
In mathematics, the well-ordering theorem, also known as Zermelo's theorem, states that every set can be well-ordered. A set X is well-ordered by a strict total order if every non-empty subset of X has a least element under the ordering. The well-ordering theorem together with Zorn's lemma are the most important mathematical statements that are equivalent to the axiom of choice. Ernst Zermelo introduced the axiom of choice as an "unobjectionable logical principle" to prove the well-ordering theorem. One can conclude from the well-ordering theorem that every set is susceptible to transfinite induction, which is considered by mathematicians to be a powerful technique. One famous consequence of the theorem is the Banach–Tarski paradox.
Zorn's lemma, also known as the Kuratowski–Zorn lemma, is a proposition of set theory. It states that a partially ordered set containing upper bounds for every chain necessarily contains at least one maximal element.
In mathematics, Tychonoff's theorem states that the product of any collection of compact topological spaces is compact with respect to the product topology. The theorem is named after Andrey Nikolayevich Tikhonov, who proved it first in 1930 for powers of the closed unit interval and in 1935 stated the full theorem along with the remark that its proof was the same as for the special case. The earliest known published proof is contained in a 1935 article of Tychonoff, A., "Uber einen Funktionenraum", Mathematical Annals, 111, pp. 762–766 (1935).
Stand-up comedy is a comedic performance to a live audience in which the performer addresses the audience directly from the stage. The performer is known as a comedian, comic, or stand-up. It is usually a rhetorical performance but many comics employ crowd interaction as part of their set or routine.
In set theory, Zermelo–Fraenkel set theory, named after mathematicians Ernst Zermelo and Abraham Fraenkel, is an axiomatic system that was proposed in the early twentieth century in order to formulate a theory of sets free of paradoxes such as Russell's paradox. Today, Zermelo–Fraenkel set theory, with the historically controversial axiom of choice (AC) included, is the standard form of axiomatic set theory and as such is the most common foundation of mathematics. Zermelo–Fraenkel set theory with the axiom of choice included is abbreviated ZFC, where C stands for "choice", and ZF refers to the axioms of Zermelo–Fraenkel set theory with the axiom of choice excluded.
An ethnic joke is a remark aiming at humor relating to an ethnic, racial or cultural group, often referring to an ethnic stereotype of the group in question for its punchline.
In mathematics, the Boolean prime ideal theorem states that ideals in a Boolean algebra can be extended to prime ideals. A variation of this statement for filters on sets is known as the ultrafilter lemma. Other theorems are obtained by considering different mathematical structures with appropriate notions of ideals, for example, rings and prime ideals, or distributive lattices and maximal ideals. This article focuses on prime ideal theorems from order theory.
Reverse mathematics is a program in mathematical logic that seeks to determine which axioms are required to prove theorems of mathematics. Its defining method can briefly be described as "going backwards from the theorems to the axioms", in contrast to the ordinary mathematical practice of deriving theorems from axioms. It can be conceptualized as sculpting out necessary conditions from sufficient ones.
In mathematics, the dimension theorem for vector spaces states that all bases of a vector space have equally many elements. This number of elements may be finite or infinite, and defines the dimension of the vector space.
In mathematics, the axiom of dependent choice, denoted by , is a weak form of the axiom of choice that is still sufficient to develop most of real analysis. It was introduced by Paul Bernays in a 1942 article that explores which set-theoretic axioms are needed to develop analysis.
A mathematical joke is a form of humor which relies on aspects of mathematics or a stereotype of mathematicians. The humor may come from a pun, or from a double meaning of a mathematical term, or from a lay person's misunderstanding of a mathematical concept. Mathematician and author John Allen Paulos in his book Mathematics and Humor described several ways that mathematics, generally considered a dry, formal activity, overlaps with humor, a loose, irreverent activity: both are forms of "intellectual play"; both have "logic, pattern, rules, structure"; and both are "economical and explicit".
Comedic device refers to a kind of device used to make a statement more humorous. In layman's terms, it is what makes things funny.
Anti-humor is a type of indirect and alternative humor that involves the joke-teller's delivering something that is intentionally not funny, or lacking in intrinsic meaning. The practice relies on the expectation on the part of the audience of something humorous, and when this does not happen, the irony itself is of comedic value — humor can be, and often is, about unexpected contrast (see Theories of humor § Incongruous juxtaposition theory). Anti-humor is also the basis of various types of pranks and hoaxes.
Anxiety/Uncertainty Management (AUM) theory is known as the high levels of anxiety one may experience as they come in contact with those of another culture. This concept was first introduced by William B. Gudykunst to further define how humans effectively communicate based on their anxiety and uncertainty in social situations. Gudykunst believed that in order for successful intercultural communication a reduction in anxiety/uncertainty must occur. This is assuming that the individuals within the intercultural encounter are strangers. AUM is a theory based on the Uncertainty Reduction Theory (URT) which was introduced by Berger and Calabrese in 1974. URT provides much of the initial framework for AUM, and much like other theories in the communication field AUM is a constantly developing theory, based on the observations of human behaviour in social situations.
Jerry Lloyd Bona is an American mathematician, known for his work in fluid mechanics, partial differential equations, and computational mathematics, and active in some other branches of pure and applied mathematics.
Equivalents of the Axiom of Choice is a book in mathematics, collecting statements in mathematics that are true if and only if the axiom of choice holds. It was written by Herman Rubin and Jean E. Rubin, and published in 1963 by North-Holland as volume 34 of their Studies in Logic and the Foundations of Mathematics series. An updated edition, Equivalents of the Axiom of Choice, II, was published as volume 116 of the same series in 1985.
