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No, what the article is talking about is the question whether or not the cardinality of real numbers is the smallest uncountable infinity or some other, larger
by Sharlin 5y ago
No, what the article is talking about is the question whether or not the cardinality of real numbers is the smallest uncountable infinity or some other, larger uncountable infinity. The only countable infinity is aleph-0, the cardinality of natural numbers, and Cantor showed that aleph-0 is too small to hold all reals. So reals must be uncountable, but there is an infinite hierarchy of uncountable infinities, and it is not known which one is the cardinality of reals (although in practice it's suspected to be either aleph-1, which is what the Continuum Hypothesis states, or aleph-2).
- keymone 5y agounrelated: how do we know there are no alephs between 0 and 1?
- Sharlin 5y agoIf you assume Axiom of Choice, you can define cardinalities in terms of ordinal numbers (an extension of naturals that generalizes the notion of "counting", or "indexing"). And ordinal numbers are well-ordered, ie. every element has a unique "successor" element.
- elcomet 5y agoThat's the definition of aleph-1. The continuum hypothesis is that 2^aleph-0 (number of sets of integers or number of reals) = aleph-1.
- orangecat 5y ago(although in practice it's suspected to be either aleph-1, which is what the Continuum Hypothesis states, or aleph-2) There are some arguments that it's "really" much larger, as in greater than aleph-n for any finite n. See https://risingentropy.com/the-continuum-hypothesis-is-false/ https://risingentropy.com/the-continuum-hypothesis-is-false/ (incidentally an excellent blog if you're interested in these sorts of things).