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In set theory, a branch of mathematics, Kunen's inconsistency theorem, proved by Kenneth Kunen , shows that several plausible large cardinal axioms are inconsistent with the axiom of choice. Some consequences of Kunen's theorem (or its proof) are: There is no non-trivial elementary embedding of the universe V into itself.
A similar example can be constructed using MA. On the other hand, the consistency of the strong Fubini theorem was first shown by Friedman. [17] It can also be deduced from a variant of Freiling's axiom of symmetry. [18]
The system + =, + = has exactly one solution: x = 1, y = 2 The nonlinear system + =, + = has the two solutions (x, y) = (1, 0) and (x, y) = (0, 1), while + + =, + + =, + + = has an infinite number of solutions because the third equation is the first equation plus twice the second one and hence contains no independent information; thus any value of z can be chosen and values of x and y can be ...
The hockey stick identity confirms, for example: for n=6, r=2: 1+3+6+10+15=35. In combinatorics , the hockey-stick identity , [ 1 ] Christmas stocking identity , [ 2 ] boomerang identity , Fermat's identity or Chu's Theorem , [ 3 ] states that if n ≥ r ≥ 0 {\displaystyle n\geq r\geq 0} are integers, then
Identity theorem (complex analysis) Identity theorem for Riemann surfaces (Riemann surfaces) Immerman–Szelepcsényi theorem (computational complexity theory) Implicit function theorem (vector calculus) Impossibility of angle trisection ; Increment theorem (mathematical analysis) Independence of the axiom of choice (mathematical logic)
As an example, consider a proof of the theorem α → α. In lambda calculus, this is the type of the identity function I = λx.x and in combinatory logic, the identity function is obtained by applying S = λfgx.fx(gx) twice to K = λxy.x. That is, I = ((S K) K).
The proofs of the Kronecker–Weber theorem by Kronecker (1853) and Weber (1886) both had gaps. The first complete proof was given by Hilbert in 1896. In 1879, Alfred Kempe published a purported proof of the four color theorem, whose validity as a proof was accepted for eleven years before it was refuted by Percy Heawood.
In real analysis and complex analysis, branches of mathematics, the identity theorem for analytic functions states: given functions f and g analytic on a domain D (open and connected subset of or ), if f = g on some , where has an accumulation point in D, then f = g on D.