Linked Questions

0
votes
2answers
156 views

Is it true that $|\mathbb{R}|=2^\omega=\omega_1$? [duplicate]

Is it true that $|\mathbb{R}|=2^\omega=\omega_1$? Note that $\omega_1$ is the successor of $\omega$ and $2^\omega$ is |all functions from $\omega$ to 2|.
12
votes
4answers
2k views

How is $\epsilon_0$ countable?

In Wikipedia, it says that any epsilon number with the index that is countable is countable. How is it? Out of all those numbers, I especially want to know why $\epsilon_0$ is countable. Thanks.
10
votes
3answers
2k views

Ordinal exponentiation - $2^{\omega}=\omega$

This is my understanding of ordinal arithmetic - two ordinals are the same as one another if there is an order-preserving bijection between them. So for instance $$1+\omega = \omega$$ because if $...
7
votes
2answers
2k views

How to define countability of $\omega^{\omega}$ and $\omega_1$? in set theory?

How is the ordinal $\omega_1$ defined? I know that it is a supremum of all smaller ordinals, but then $\omega^\omega$ is also a supremum of all smaller ordinals. How can we distinguish these two ...
13
votes
3answers
494 views

Is $2^{\aleph_0} = \aleph_1$?

I was reading a thread on Examples of Common False Beliefs in Mathematics on MathOverflow, in which a user wrote: $$2^{\aleph_0} = \aleph_1$$ This is a pet peeve of mine, I'm always surprised ...
2
votes
2answers
1k views

What is the cardinality of $\omega^\omega$?

I have started reading about Baire Space as a prerequisity for the subject of Borel hierarchies. And if I understand correctly, Baire space is $\omega^\omega$ (The space of all functions from $\omega$ ...
0
votes
2answers
742 views

Ordinal Exponentiation and transfinite induction

Let $a\ne0$ be any ordinal By transfinite recursion theorem, we can define a function f:OR→OR such that i) $f(0)=1$ ii) $f(b^+)=f(b)a$ iii) $f(b)=\sup\{f(r)|r<b\}$ if $b$ is a limit ordinal. Then ...
0
votes
1answer
962 views

First uncountable ordinal number

Why is $\Omega$ the first uncountable ordinal number and not $\omega^\omega$? Isnt the latter a countably infinite product of countably infinite sets and hence uncountable?
0
votes
1answer
249 views

Finding a limit using arithmetic over cardinals

What is the value of: $$\lim_{n \to \infty} \frac{n}{2^n} (n \in \mathbb{N})$$ It seems to me that I can use L'Hopital's rule, but does that rule take into account types of infinity? More precisely, ...
2
votes
1answer
639 views

Exponentiation and Set of Functions Notation.

Given arbitrary sets $A$ and $B$, the notation $A^{B}$ is mostly clear from context to mean $A^{B} = \{f : f : B \rightarrow A\}$. However, when these sets are ordinal or cardinals, especially $\...
3
votes
1answer
215 views

order of infinite countable ordinal numbers

I'm trying to understand ordinal arithmetic. If one had an ordered list of the some subset of countable ordinal numbers, what order would the following 6 countably infinite ordinals be in? If the ...