I’m by no means a mathematician so this is a layman’s confusion after watching Youtube videos.
I understand why the (new) real number couldn’t be at any position (i.e. if it were, its [integer index] digit would be different, so it contradicts the assumption). But how does this **prove** that the set of real numbers is larger? This number indeed couldn’t be at any position, but it can **still** be mapped to an integer, because there are infinite integers – neither set can be larger because neither is finite.
Or are just “larger” or there being “more real numbers than integers” wrong terms to describe cardinality?
Pardon my ignorance.
Thank you.
In: Mathematics
> This number indeed couldn’t be at any position, but it can still be mapped to an integer, because there are infinite integers – neither set can be larger because neither is finite.
The original list of real numbers is, by definition, mapped to every existing integer. There are none remaining to place the new real number. But by the assumptions of the diagonal argument, the list contains **every** real number. Since the assumption (every real number is in the list) and the conclusion (this real number is not) are in contradiction, one of the assumptions made in the construction must be false – specifically, the assumption that you could create a countable list of all real numbers.
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