How many real numbers exist? New proof moves closer to an answer

Re:How many real numbers exist? New proof moves closer to an... by quisnam on Friday July 16, 2021 @06:31AM (#27854775)

Many entry-level real analysis courses will cover cardinality after introducing sets and functions. There's also a short article on Wikipedia. [0]

Your intuition about there being more rational numbers might be based on viewing the rationals as a proper superset of the natural numbers. You might similarly consider that there are more natural numbers than even natural numbers. However, by "renaming" every even number, and that shouldn't change how many there are, to half its value, we obtain the natural numbers. Formally, there exists a bijection between N and 2N, as between N and Q, and this is what mathematicians mean when they say that sets have the same size, or cardinality.

[0]: https://en.wikipedia.org/wiki/Cardinality

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Re:How many real numbers exist? New proof moves closer to an... by Someone on Friday July 16, 2021 @02:37PM (#27859617)

To add, for hutzlibu and others like them: as soon as infinite sets (or sequences. See https://plus.maths.org/content/when-things-get-weird-infinit...) are involved math gets counter-intuitive.

In this case, at most one of these two a priori quite reasonable statements can be true:

- if set B is a strict superset of A, B is larger than A.

- if you can map the times in A to the items in B in 1:1 fashion, A and B have the same size.

Giving up the first is deemed less problematic than giving up the second (probably because that means giving up comparing sizes of infinite sets at all, but I’m not familiar enough with that to be sure about that)

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