Concept
How do the cardinalities of the natural and real numbers differ?
Stephen Davies, Ph.D. Version 2.2.2 Through Discrete Mathematics A Cool Brisk Walk / Chapter 1
"The cardinality of all these sets is infinity, although as I alluded to previously, | R | is in some sense “greater than” | N | . For the curious, we say that N is a countably infinite set, whereas | R | is uncountably infinite . Speaking very loosely, this can be thought of this way: if we start counting up all the natural numbers 0, 1, 2, 3, 4, …, we will never get to the end of them. But at least we can start counting . With the real numbers, we can’t even get off the ground. Where do you begin? Starting with 0 is fine, but then what’s the “next” real number? Choosing anything for your second number inevitably skips a lot in between."
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