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This is incorrect, and a common misconception. Aleph_1 refers to the second-smallest infinite cardinal, whether that's equal to 2^(aleph_0) or not. (Aleph_1 i
by Sniffnoy 5y ago
This is incorrect, and a common misconception.
Aleph_1 refers to the second-smallest infinite cardinal, whether that's equal to 2^(aleph_0) or not. (Aleph_1 is also equal, as has been mentioned, to the number of countable ordinals.)
The continuum hypothesis, in stating that there are no infinite cardinals inbetween aleph_0 and 2^(aleph_0), is equivalent to the statement that 2^(aleph_0) = aleph_1.
The series you refer to is denoted by the Hebrew letter bet, not aleph. So bet_0 = aleph_0, bet_1 = 2^(bet_0), bet_2 = 2^(bet_1), etc. But that is not how the aleph numbers work.
- klyrs 5y agoSome of us prefer to assume the continuum hypothesis just for the sake of this notation... but it turns out that the continuum hypothesis is still pretty hot. 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...
- Sniffnoy 5y ago"Assuming the continuum hypothesis for the sake of this notation" is being deliberately obfuscatory, encouraging confusion, and requiring other mathematicians to do extra work to translate your results to the more general context. There's perfectly good notation for the bet series, and no need to use aleph as a substitute when that simply isn't what it means.
- moomin 5y agoChecks Yes, you're right.