Linked Questions

5
votes
1answer
223 views

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 ...
2
votes
0answers
70 views

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 $\aleph_1$.
0
votes
0answers
29 views

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 ...
24
votes
9answers
3k views

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 ...
13
votes
3answers
1k views

Can the cardinality of continuum exceed all aleph numbers in ZF?

More precisely, is either of the following two statements consistent with ZF: $2^{\aleph_0}\geq\aleph_{\alpha}$ for every ordinal number $\alpha$, $2^{\aleph_0}\leq\aleph_{\alpha}\implies 2^{\aleph_0}...
6
votes
4answers
778 views

Does every ordinal have cardinality no greater than $\aleph_\mathbb{0}$?

My notes say that the ordinals $\omega + 1, \omega + 2, ... , 2 \omega, ... , 3 \omega, ... \omega^2, ... $ are all countable, and hence have cardinality equal to $\omega = \aleph_\mathbb{0}$. So I ...
8
votes
3answers
649 views

Why infinite cardinalities are not “dense”?

What tells us that the structure of the cardinals is "discrete"? I'm not using the words "discrete" and "dense" with their formal meanings. Maybe I have this confusion because I'm using concepts ...
10
votes
1answer
985 views

What is $\aleph_0$ powered to $\aleph_0$?

By definition $\aleph_1 = 2 ^{\aleph_0}$. And since $2 < \aleph_0$, then $2^{\aleph_0} = {\aleph_1} \le \aleph_0 ^ {\aleph_0}$. However, I do not know what exactly $\aleph_0 ^ {\aleph_0}$ is or how ...
5
votes
4answers
1k views

Existence of a well-ordered set with a special element

One of the most mind boggling results in my opinion is, with the axiom of choice/well-ordering principle, there exist such things as uncountable well-ordered sets $(A,\leq)$. With this is mind, does ...
10
votes
1answer
875 views

Is the axiom of choice needed to show that $a^2=a$?

A comment on this answer states that choice is needed for the statement that $a^2=a$ for all infinite cardinals $a$. In Thomas Jech's Set Theory (3rd edition), his theorem 3.5 proves this statement ...
1
vote
3answers
2k views

How to prove that the set of all countable ordinals, $\omega_1$, is uncountable / has the cardinality $\aleph_1$?

The only reasoning I've seen given for this is that it's uncountable because it can't include itself an element. I'm a little unconvinced and was looking for a more proper formal proof demonstrating ...
3
votes
2answers
453 views

True Statement in ZF is true statement in ZFC?

How do we know an $ \aleph_1 $ exists at all? Here, you can see that $\aleph_2≦2^{\aleph_0}$, which is a contradiction to CH in ZFC. So if 'true statements in ZF is true in ZFC' is true, CH must be ...
5
votes
1answer
608 views

Intuition about the size of $\aleph_k$ with $k>1$

Assuming CH for simplicity, I know of some more or less intuitive way to think about difference in sizes of $\aleph_0$ and $\aleph_1$. The most straightforward is the distinction of natural/rational ...
2
votes
1answer
1k views

Why are the countable ordinals a set?

The countable ordinals are themselves either countable or uncountable. They cannot be countable since that would involve a set with itself as an element, so they are uncountable. If they are ...
1
vote
2answers
996 views

Non-aleph infinite cardinals

I'm now confused with a concept of $\aleph$. 1.$\aleph$ is a cardinal number that is well-ordered in ZF.(Defined as an initial ordinal that is equipotent with). Does that mean $\aleph_x$ in ZF may ...

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