2 ms·
An infinite set can have the same cardinality as a proper subset of itself. Two sets have the same cardinality if all their elements can be placed into a one-t
by _kst_ 4y ago
An infinite set can have the same cardinality as a proper subset of itself.
Two sets have the same cardinality if all their elements can be placed into a one-to-one correspondence with each other. For example, the positive integers can be placed into a one-to-one correspondence with the even positive integers: N <=> N*2
- giomasce 4y agoThat's actually often used as the definition of an infinite set.
- PeterWhittaker 4y agoI remain unconvinced, given that we continue to explore the meaning and possibility of cardinalities beyond aleph 0. See, e.g., [1], and its implications for CH. It strikes me that since we are still learning about cardinalities above aleph 0, we may have made mistakes at 0 itself and that there may be an “aleph -1”, an infinite cardinality less than aleph 0, which could apply, e.g., to the set of primes. This would require proving that the bijection technique is not properly constructivist. That is considered constructivist has long bothered me, but I lack the detailed background in the field to articulate the objection. [1] https://www.quantamagazine.org/how-many-numbers-exist-infinity-proof-moves-math-closer-to-an-answer-20210715/ https://www.quantamagazine.org/how-many-numbers-exist-infini...