this post was submitted on 11 Apr 2024
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Turing Incompleteness is a pathway to many powers the Computer Scientists would consider incalculable.
In fact, there's infinite problems that cannot be solved by Turing machnes!
(There are countably many Turing-computable problems and uncountably many non-Turing-computable problems)
Infinite seems like it's low-balling it, then. 0% of problems can be solved by Turing machines (same way 0% of real numbers are integers)
The subset of integers in the set of reals is non-zero. Sure, I guess you could represent it as arbitrarily small small as a ratio, but it has zero as an asymptote, not as an equivalent value.
The cardinality is obviously non-zero but it has measure zero. Probability is about measures.