27 questions linked to/from How do we know an $\aleph_1$ exists at all?
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### How do we know that $\omega_1$ exists in ZF? [duplicate]

In ZF, not all "collections of objects" are sets. For example, there is no set of all sets, and there is no set of all ordinals. So, how do we know that there is a set of all countable ordinals? In ...
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### Proof that $\aleph_1$ exists. [duplicate]

Clearly, uncountable cardinals exist. How do we know that $\aleph_1$ is the smallest one? Is there a proof that there is no cardinal strictly between $\aleph_0$ and $\aleph_1$.
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### Proving $\gamma$ is a cardinal such that $\omega < \gamma$ [duplicate]

Denote $W=\lbrace R \mid R \hspace{0.1cm} \text{is a well-order on} \hspace{0.1cm} \omega \rbrace$ and $S=\lbrace \text{Ord}(R)\mid R \in W \rbrace$. The ZF-axioms guarantuee us that $W$ and $S$ are ...
### A simple example of an uncountable set that is not $\mathbb{R}$
Let's suppose that I have only defined $\mathbb{N}$ and then I define the terms finite and infinite set, and also countable and uncountable set. I can think of some examples of finite, infinite and ...