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 Originally Posted by oskar
Is it fair to say it's one of those cases where the technical term simply means something else than the colloquial? Just because the R infinity isn't listable using the N infinity doesn't make the N one countable.
No, N is countable in literally the exact meaning of the word. Whereas with sets like R it's literally imposisble to count all the elements.
The words that probably aren't the best are words like big. In the case of R and N it makes some sense because N is contained in R but the irrationals are also contained in R but by this measure they are the same. I think this is somewhat down to a countable infinity being so insignificant compared to uncountable ones that even though R = R/Q & Q, Q is basically nothing but that's very hand wavy nonsense.
It's all to do with mapping each element to each other.
so {a, b, c, d} has cardinality 4 which is the same as {1, 2, 3, 4} and each element of one set can be mapped to the other i.e. a to 1, b to 2 etc.
N has a smaller cardinality than R and R/Q
R has the same cardinality as R/Q because you can still (don't ask me how) map every element of R to one in R/Q.
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