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The mod won ModDB 2010 Mod of the Year awards for Best Original Art, [8] Best Singleplayer Mod [9] and Player's Choice Mod of the Year [10] categories, eventually winning 2nd place at the latter. Operation Black Mesa - An upcoming remake of the Half-Life expansion Opposing Force developed by Tripmine Studio. [11]
Tarski's theorem about choice: For every infinite set A, there is a bijective map between the sets A and A×A. Trichotomy: If two sets are given, then either they have the same cardinality, or one has a smaller cardinality than the other. Given two non-empty sets, one has a surjection to the other. Every surjective function has a right inverse.
In mathematics, Tarski's theorem, proved by Alfred Tarski , states that in ZF the theorem "For every infinite set , there is a bijective map between the sets and " implies the axiom of choice. The opposite direction was already known, thus the theorem and axiom of choice are equivalent.
ZF stands for Zermelo–Fraenkel set theory, and DC for the axiom of dependent choice.. Solovay's theorem is as follows. Assuming the existence of an inaccessible cardinal, there is an inner model of ZF + DC of a suitable forcing extension V[G] such that every set of reals is Lebesgue measurable, has the perfect set property, and has the Baire property.
In mathematical logic, Diaconescu's theorem, or the Goodman–Myhill theorem, states that the full axiom of choice is sufficient to derive the law of the excluded middle or restricted forms of it. The theorem was discovered in 1975 by Radu Diaconescu [ 1 ] and later by Goodman and Myhill . [ 2 ]
Social choice theory is a branch of welfare economics that extends the theory of rational choice to collective decision-making. [1] Social choice studies the behavior of different mathematical procedures (social welfare functions) used to combine individual preferences into a coherent whole.
Unrestricted domain is one of the conditions for Arrow's impossibility theorem. Under that theorem, it is impossible to have a social choice function that satisfies unrestricted domain, Pareto efficiency, independence of irrelevant alternatives, and non-dictatorship. However, the conditions of the theorem can be satisfied if unrestricted domain ...
In mathematics, the axiom of dependent choice, denoted by , is a weak form of the axiom of choice that is still sufficient to develop much of real analysis. It was introduced by Paul Bernays in a 1942 article in reverse mathematics that explores which set-theoretic axioms are needed to develop analysis.