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Compressibility/Decidability/Genz: there can be no computer program that decides if any amount of data is compressible.
Stronger: there is no way to prove that it is not compressible.
Compressibility: can be proven but not refuted.
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Example number pi: π can be generated by a finite program.
There are numbers that cannot be calculated in principle:
Omega/Chaitin/Genz: this is what Chaitin calls a certain number of which not a single digit can be calculated. It is not accessible to any rule, it is outside mathematics.
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Gregory Chaitin.
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Decidability/calculability/undecidable/non-calculable/Genz: non-calculable numbers are actually the same as non-decidable numbers.
Incalculability/physics/quantum cosmology/Genz: apparent indecidability: the ... of the wave-function of the universe shows apparant indecidability. It deals with the possible geometry of three-dimensional spaces.
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Wave function.
Simplified: e. g. a circle (one dimensional): to calculate the wave function of the universe for the circle as an argument: the wave function can be represented as a sum of summands, where there is a series of handleless cups, one series of cups with a handle, one series of cups with two handles, etc., whereby the handles can be shaped differently in each case. These represent four-dimensional spaces (with time as 4th dimension).
Circle: here time is added as the 2nd dimension. Together they form the two dimensions of the cup surfaces.
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3rd dimension: the 3rd dimension in which the surfaces are embedded, serves only as an illustration. It has no equivalent in reality.
Problem: it is not possible to decide which cups are to be regarded as the same, which cups are to be regarded as different (cups with differently shaped handles have the same topology).
Question: undecidable: whether two cups have the same or different number of handles. (Of course, this is about four, not two dimensions.)
Indecidability/Genz: indecidability occurs here only if a computer is to perform the calculation: to describe a cup, it is covered with a certain number of equal triangles.
Problem: there cannot be a computer program that decides for any number of covering flat triangles whether two (four-dimensional) cups have the same number of handles.
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Theorem: the theorem is rather tame: it now excludes that a program makes a decision for any number of flat triangles, but not for a given number - e. g. one million - flat triangles. This is simply a matter of increasing accuracy.
That would be an example of an unpredictable number.
Wave function of the Universe/Genz: it could be shown that there are calculable representations of it, so that its incalculability (similar to that of > NOPE) suggested by the regulation of the figure does not actually exist.
Definition NOPE/Genz: the smallest number that can only be determined by more than thirteen words minus the smallest number that can only be determined by more than thirteen words
N.B.: the rule is impracticable, but we still know that NOPE = 0!
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Problem/Genz: there cannot be a program that decides in finite time if any program ever stops.
"Stopping problem"/"Non-stopping theorem"/Genz: the "stopping problem" is not a logical but a physical problem. It is impossible to perform infinitely many logical steps in finite time.
Time travel/time reversal/time/decision problem/Genz: if time travel were possible, the stopping problem would only be valid to a limited extent.
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Time, >
Time reversal, >
Time arrow, >
Symmetries.
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Stopping problem/Platonism/Genz: in a platonic world where there are only logical steps instead of time, the non-stopping theorem would also be valid. The point here is the admissibility of evidence rather than its feasibility.
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Proofs, >
Provability.
