What is the defenition of $\mathcal{c}$ and $\aleph_1$ if we assume ZFC without CH.

Question

I am reading an intro to a chapter of Andreas Blass called "Combinatorial Cardinal Characteristics of the Continuum" and I am getting a bit confused. When I studied "Discrete Mathematics", it was assumed that $\mathcal{c} = \aleph_1$ which are both the cardinality of $[0,1)$. This was the definition.

Now, when we ommit the Continuum Hypothesis, what are the definitions? My guess is that $\mathcal{c}$ is defined as the cardinality of $[0,1)$, which is known to be equal to $2^{\aleph_0}$ which is the cardinality of $P(\mathbb{N})$.

So, what is the definition of $\aleph_1$?

Thank you! Shir

Answer 1

The definition of $\aleph_1$ is the cardinality of the least uncountable ordinal. This is also the cardinality of the countable ordinals themselves.

$2^{\aleph_0}$ is the definition of the cardinality of the power set of the natural numbers, which is also the cardinality of the real numbers.

Note that the above don't even use the axiom of choice. It's just the basic definitions and we can prove all that without the axiom.