Re:An intutive counterexample to the axiom of choice
by kjaa on Monday January 23, 2023 @09:04AM
(#34488812)
I believe you are confusing rational for real numbers. Rational numbers really do map to the integers, but I believe I can "break" any purported isomorphism from reals to ints, by taking two adjacent ints and asking what number the midpoint between the corresponding reals map to.
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Re:An intutive counterexample to the axiom of choice
by phkahler on Monday January 23, 2023 @09:22AM
(#34489030)
Please be specific. I never said the mapping has nice properties regarding the ordering.
BTW I know rational numbers map to integers, and hence any length tuple can be mapped to integers by pairwise combining them in similar ways. But if any length tuple of integers can map to a single integer, why can't an infinitely long set of integers map to an integer? Like the string of digits in an irrational number represented as a decimal?
Re:An intutive counterexample to the axiom of choice
by Warwolt on Monday January 23, 2023 @10:20AM
(#34489810)
The problem with mapping the reals to the integers is that a list of real numbers is infinite in "two directions". Each real can be potentially described with an infinite number of digits, and you can have an infinite list of real numbers (i.e. infinite sequences of digits). This is, I believe, the basis for Cantor's diagonal argument, which is a quite beautiful proof of there being more reals than integers.