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
You’re on the right track! We can compare the cardinality (size) of two sets by seeing if we can make a one-to-one mapping between the elements of the sets. Like how a farmer might count a herd of sheep– in the morning, get a pile of rocks, and each time a sheep passes, move a rock to a second pile. In the evening you don’t have to count th sheep, but move the rocks back to the original pile. If there is a rock for each sheep, the two sets are the size.
By showing that in ANY possible mapping, there is at least one real number that can’t be mapped to the naturals, you’ve shown that they aren’t the same cardinality.
Btw, this is why sets that can be mapped to the naturals are called “countable” sets– because it’s possible to assign a unique number to each element.
Does this help?
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