natural numbers, and similarly one might think that the number of fractions is greater than the number of integers. However, this is not the case. According to the powerful and beautiful theory of infinite numbers put forward in the late 1800s by the highly original Russian-German mathematician Georg Cantor, the total number of fractions, the total number of integers and the total number of natural numbers are all the same infinite number, denoted ℵ0 (‘aleph nought’). (Remarkably, this kind of idea had been partly anticipated some 250 years before, in the early 1600s, by the great Italian
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I think there's some error in this. Take set theory for example: Natural numbers are countably infinite meaning you can have a bijective relationship between one Natural Integer and the positive integers. Real numbers however are not countably infinite because there is no bijection.
I just left this note to myself as the distinction between countable and uncountable infinite should be noted. I suppose there's no degree to infinites permitting us to define the numbers as the same "aleph nought". Check this out.
