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The free will theorem states: Given the axioms, if the choice about what measurement to take is not a function of the information accessible to the experimenters (free will assumption), then the results of the measurements cannot be determined by anything previous to the experiments. That is an "outcome open" theorem:
2 √ 2 ≈ 2.665 144 142 690 225 188 650 297 249 8731... which was proved to be a transcendental number by Rodion Kuzmin in 1930. [2] In 1934, Aleksandr Gelfond and Theodor Schneider independently proved the more general Gelfond–Schneider theorem, [3] which solved the part of Hilbert's seventh problem described below.
In the 1980s, John Stewart Bell discussed superdeterminism in a BBC interview: [7] [8] There is a way to escape the inference of superluminal speeds and spooky action at a distance. But it involves absolute determinism in the universe, the complete absence of free will. Suppose the world is super-deterministic, with not just inanimate nature ...
The modern proof of the strong law is more complex than that of the weak law, and relies on passing to an appropriate subsequence. [17] The strong law of large numbers can itself be seen as a special case of the pointwise ergodic theorem. This view justifies the intuitive interpretation of the expected value (for Lebesgue integration only) of a ...
The length of unusually long proofs has increased with time. As a rough rule of thumb, 100 pages in 1900, or 200 pages in 1950, or 500 pages in 2000 is unusually long for a proof. 1799 The Abel–Ruffini theorem was nearly proved by Paolo Ruffini , but his proof, spanning 500 pages, was mostly ignored and later, in 1824, Niels Henrik Abel ...
The problem of free will has been identified in ancient Greek philosophical literature. The notion of compatibilist free will has been attributed to both Aristotle (4th century BCE) and Epictetus (1st century CE): "it was the fact that nothing hindered us from doing or choosing something that made us have control over them".
[23] [24] [25] The weak conjecture is implied by the strong conjecture, as if n − 3 is a sum of two primes, then n is a sum of three primes. However, the converse implication and thus the strong Goldbach conjecture would remain unproven if Helfgott's proof is correct.
Renewal theory is the branch of probability theory that generalizes the Poisson process for arbitrary holding times. Instead of exponentially distributed holding times, a renewal process may have any independent and identically distributed (IID) holding times that have finite mean. A renewal-reward process additionally has a random sequence of ...