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The $\aleph$ hierarchy, I think, is a sequence of infinite cardinals. $\aleph_0$ is $|\mathbb{N}|$, possibly by definition. $\aleph_1$ is the next highest cardinal.

A lot has been written about whether $\aleph_1 = |\mathbb{R}|$ or, equivalently $\aleph_1 = \beth_1$ and how the continuum hypothesis is independent of ZFC.

But there definitely isn't a cardinal $\kappa$ such that $\aleph_0 < \kappa < \aleph_1$.

Why do we know that there's a successor cardinal to $\aleph_0$ / least cardinal greater than $\aleph_0$?

What would break if we insist, by fiat, on the existence of a hierarchy of cardinals indexed by the non-negative reals or non-negative rationals (in the way that $\aleph_n$ is indexed by $n \in \mathbb{N_0}$)? If cardinals aren't primitive things and the existence of a successor falls right out of how they're defined, this question might not make sense.

Greg Nisbet
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  • cardinals are ordinals; ordinals are well-ordered – Angina Seng Sep 19 '18 at 06:19
  • @LordSharktheUnknown true only if assuming choice – ℋolo Sep 19 '18 at 06:24
  • @LordSharktheUnknown When I think of cardinality, I think of equivalence classes on sets based on the existence of bijections. I think of a "cardinal" or "cardinality" as a name for one of the classes. That's probably naive. Are cardinals defined as particular ordinals in the ordinal hierarchy? Is the fact that bijections exist between same-sized sets just another result and not fundamentally related to how cardinality is defined? – Greg Nisbet Sep 19 '18 at 06:28
  • @GregoryNisbet do you assume axiom of choice? If not than your definition is the correct way to think about cardinals. But shark is correct with a little change in either case. I'll post an answer about this – ℋolo Sep 19 '18 at 06:31
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    @Holo - Well, if the question is about $\kappa$ such that $\aleph_0 < \kappa < \aleph_1$, then Choice isn't the issue. There is just no such cardinal, since $\kappa < \aleph_1$ implies that $\kappa$ is an $\aleph$ number. What the failure of Choice might do is a bit stranger. – Malice Vidrine Sep 19 '18 at 06:39
  • @MaliceVidrine this is why I said in my reply to OP that a little change will make it correct, you said cardinals are ordinals, instead of aleph numbers are ordinals – ℋolo Sep 19 '18 at 06:42
  • What is the definition of a "discrete hierarchy"? Does the fact that the cardinal $\aleph_0$ or $\aleph_\omega$ has no immediate predecessor, but is the limit of the set of all small ordinals, not conflict with calling the alephs a "discrete hierarchy"? – bof Sep 19 '18 at 07:00
  • @bof, "discrete hierarchy" wasn't meant to be a term of art. I'm asking why there's a smallest cardinal greater than $\aleph_0$ . The question that asked why there's no $\aleph_{1/2}$ said it better. – Greg Nisbet Sep 19 '18 at 07:02

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First: assuming the axiom of choice we have that the class of cardinals is equal to the union of all limit ordinals hence equal to $On$(or $Ord$) thus it is well ordered.

If we do not assume choice we need to go back to the definition of $\aleph_a$, we have $\aleph_0=\omega$, and $\aleph_{a+1}=\aleph_a^+$ and if $a$ is limit ordinal then $\aleph_a=\bigcup\{ \aleph_b\mid b< a\}$, now because ordinals are well founded there exists unique successor cardinal for every aleph number, thus aleph one is well defined

ℋolo
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