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  2. Axiom of choice - Wikipedia

    en.wikipedia.org/wiki/Axiom_of_choice

    Many theorems provable using choice are of an elegant general character: the cardinalities of any two sets are comparable, every nontrivial ring with unity has a maximal ideal, every vector space has a basis, every connected graph has a spanning tree, and every product of compact spaces is compact, among many others. Frequently, the axiom of ...

  3. List of statements independent of ZFC - Wikipedia

    en.wikipedia.org/wiki/List_of_statements...

    The mathematical statements discussed below are provably independent of ZFC (the canonical axiomatic set theory of contemporary mathematics, consisting of the Zermelo–Fraenkel axioms plus the axiom of choice), assuming that ZFC is consistent. A statement is independent of ZFC (sometimes phrased "undecidable in ZFC") if it can neither be ...

  4. Group structure and the axiom of choice - Wikipedia

    en.wikipedia.org/wiki/Group_Structure_and_the...

    Using the axiom of choice, one can show that for any family S of sets | ⋃S | ≤ | S | × sup { |s| : s ∈ S} (A). [5] Moreover, by Tarski's theorem on choice, another equivalent of the axiom of choice, | X | n = | X | for all finite n (B). Let X be an infinite set and let F denote the set of all finite subsets of X. There is a natural ...

  5. Category:Biological theorems - Wikipedia

    en.wikipedia.org/wiki/Category:Biological_theorems

    Biology portal; Pages in category "Biological theorems" The following 6 pages are in this category, out of 6 total. This list may not reflect recent changes. B. Bet ...

  6. Immersion (mathematics) - Wikipedia

    en.wikipedia.org/wiki/Immersion_(mathematics)

    Hassler Whitney initiated the systematic study of immersions and regular homotopies in the 1940s, proving that for 2m < n + 1 every map f : M m → N n of an m-dimensional manifold to an n-dimensional manifold is homotopic to an immersion, and in fact to an embedding for 2m < n; these are the Whitney immersion theorem and Whitney embedding theorem.

  7. Discrete mathematics - Wikipedia

    en.wikipedia.org/wiki/Discrete_mathematics

    Discrete mathematics is the study of mathematical structures that can be considered "discrete" (in a way analogous to discrete variables, having a bijection with the set of natural numbers) rather than "continuous" (analogously to continuous functions). Objects studied in discrete mathematics include integers, graphs, and statements in logic.

  8. Mathematical structure - Wikipedia

    en.wikipedia.org/wiki/Mathematical_structure

    In Mathematics, a structure on a set (or on some sets) refers to providing it (or them) with certain additional features (e.g. an operation, relation, metric, or topology). Τhe additional features are attached or related to the set (or to the sets), so as to provide it (or them) with some additional meaning or significance.

  9. Occam's razor - Wikipedia

    en.wikipedia.org/wiki/Occam's_razor

    The choice of the "shortest tree" relative to a not-so-short tree under any optimality criterion (smallest distance, fewest steps, or maximum likelihood) is always based on parsimony. [61] Francis Crick has commented on potential limitations of Occam's razor in biology. He advances the argument that because biological systems are the products ...