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My understanding is that this is true because there are infinite decimals between every decimal, infinitely. For example, there is infinity between 0.1 and 0.2
by warent 3y ago
My understanding is that this is true because there are infinite decimals between every decimal, infinitely.
For example, there is infinity between 0.1 and 0.2, and infinity between 0.1 and 0.11, etc. i.e. infinite sets of infinity rather than one set of infinity.
In the end it's all infinity, but their sets have higher cardinality described in Aleph terms ... (or something)
https://en.m.wikipedia.org/wiki/Aleph_number https://en.m.wikipedia.org/wiki/Aleph_number
- daef 3y agoYou can uniquely map all rationals onto the natural numbers, thus they are of the same quantity. That doesn't work for all real numbers thou.
- warent 3y agoOh right this is only true for irrational and transcendental numbers
- aimor 3y agoMaybe this is misguided cheat, but couldn't you map any real number (between 0 and 1) to a natural number by mirroring the decimal digits across the decimal point. So 0.123 -> 321, but also sqrt(2)/2 -> ?601707 where ? is the rest of the decimal representation. This creates infinitely large numbers, but it's still a 1-to-1 mapping.
- dangond 3y agoUnfortunately, numbers with infinitely many digits are not natural numbers. You cannot count to ?601707, even with an infinite amount of time.
- rcme 3y agoIt’s not because there are infinite decimals between every two decimal numbers. That applies to the rational numbers too, e.g. there are infinite rational numbers between 1/2 and 3/4. Rather, the real numbers are more dense in a way that makes them fundamentally larger than the integers / rational numbers. “Larger” means not being able to pair up the two sets one by one so that each element of both sets is the member of a pair. No matter how you pair up the integers to the reals, you can prove that some real numbers will be unpaired.