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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:
Other approaches do not require free will to be a fundamental constituent of the universe; ordinary randomness is appealed to as supplying the "elbow room" believed to be necessary by libertarians. Free volition is regarded as a particular kind of complex, high-level process with an element of indeterminism.
Bell's 1964 theorem requires the possibility of perfect anti-correlations: the ability to make a completely certain prediction about the result from the second detector, knowing the result from the first. [5] The theorem builds upon the "EPR criterion of reality", a concept introduced in the 1935 paper by Einstein, Podolsky, and Rosen.
The Shannon–McMillan–Breiman theorem, due to Claude Shannon, Brockway McMillan, and Leo Breiman, states that we have convergence in the sense of L1. [2] Chung Kai-lai generalized this to the case where X {\displaystyle X} may take value in a set of countable infinity, provided that the entropy rate is still finite.
The Hawking singularity theorem is based on the Penrose theorem and it is interpreted as a gravitational singularity in the Big Bang situation. Penrose shared half of the Nobel Prize in Physics in 2020 "for the discovery that black hole formation is a robust prediction of the general theory of relativity".
Gödel's theorem, informally stated, asserts that any formal theory sufficient to express elementary arithmetical facts and strong enough for them to be proved is either inconsistent (both a statement and its denial can be derived from its axioms) or incomplete, in the sense that there is a true statement that can't be derived in the formal theory.
The free will theorem says that if we have free will, then particles must have free will. This presumably is counterintuitive. It makes no claim about a world in which we don't have free will (a deterministic world). There's no way to argue for free will on the basis of this theorem - and yet, this is what the section claims, without any ...
The basic theorem is that the resulting homology is an invariant of the manifold (that is, independent of the function and metric) and isomorphic to the singular homology of the manifold; this implies that the Morse and singular Betti numbers agree and gives an immediate proof of the Morse inequalities.