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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 ...
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 ...
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 ...
For infinite dimensional manifolds, this is sometimes taken to be the definition of an immersion. [4] An injectively immersed submanifold that is not an embedding. If M is compact, an injective immersion is an embedding, but if M is not compact then injective immersions need not be embeddings; compare to continuous bijections versus homeomorphisms.
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 ...
Stuart Kauffman favoured self-organisation, the idea that complex structure emerges holistically and spontaneously from the dynamic interaction of all parts of an organism. Michael Denton argued for laws of form by which Platonic universals or "Types" are self-organised.
Currently, solid-state NMR is widely used in the field of structural biology to determine the structure and dynamic nature of proteins (protein NMR). [ 15 ] In 1990, Richard Henderson produced the first three-dimensional, high resolution image of bacteriorhodopsin using cryogenic electron microscopy (cryo-EM). [ 16 ]
Combinatorics is an area of mathematics primarily concerned with counting, both as a means and as an end to obtaining results, and certain properties of finite structures.It is closely related to many other areas of mathematics and has many applications ranging from logic to statistical physics and from evolutionary biology to computer science.