Question about cardinality and CH

[Werg22]

Under the continuum hypothesis, we readily see that that $$|{a < \aleph_1 : \textrm{a is a cardinal}}| = \aleph_0$$. What happens under the negation of CH? Is this equality still true or not? If the latter, always under the negation of CH, are there any infinite cardinals lambda for which the inequality $$|{a < \lambda : \textrm{a is a cardinal}}| = \lambda$$ fails?

 

[micromass]

First, what do you mean with $$|a<\aleph_1 : a \text{ is a cardinal}|$$. Do you simply mean the cardinality of the set $$\{a<\aleph_1 : a \text{ is a cardinal}\}$$??

In that case, it is always true that

$$\{a<\aleph_1 : a \text{ is a cardinal}\}=\mathbb{N}\cup \{\aleph_0\}$$

So wheter CH holds or not, this set is always countable.
In general we have that (by definition almost)

$$\{a<\aleph_\alpha : a \text{ is a cardinal}\}=\mathbb{N}\cup \{\aleph_\beta~\vert~\beta<\alpha\}$$.

For your second question, this is not always true. For example:

$$\{a<\aleph_2 : a \text{ is a cardinal}\}=\mathbb{N}\cup \{\aleph_0,\aleph_1\}$$.

But this is also countable, so $$|a<\aleph_2 : a \text{ is a cardinal}|=\aleph_0$$

Where CH does come into play, is with the set

$$|a<2^{\aleph_0} : a \text{ is a cardinal}|$$

If CH is true, then this is $$\aleph_0$$. But if CH is not true, then it can be any cardinal. E.g. it is consistent with ZFC that

$$|a<2^{\aleph_0} : a \text{ is a cardinal}|=\aleph_{666}$$

and so on if you replace 666 with any ordinal.

I hope this answer was helpful...

 

[CRGreathouse]

and so on if you replace 666 with any ordinal.

I don't think you can replace it with any ordinal. What about, e.g., $$\omega_1$$? Could that really be consistent?

 

[micromass]

But it is consistent with ZFC that $$2^{\aleph_0}=\aleph_{\omega_1}$$. See Easton's theorem...

 

[yossell]

I don't think you can replace it with any ordinal. What about, e.g., $$\omega_1$$? Could that really be consistent?

I believe there are some very weak limits on possible cardinalities of the continuum - I think the cofinality of the continuum is uncountable, and so, for instance, 2^aleph_0 can't be aleph_omega.

 

[Werg22]

Many thanks micromass, you're clarified a few things for me.