1
vote
2answers
83 views

What does $\alpha+\gamma$ mean when $\alpha$ and $\gamma$ are well-ordered sets?

I was asked to prove the following: let $\gamma$ be a well ordered set with the following property: for any $\alpha$ and $\beta$ well ordered sets, if $\alpha+\gamma=\beta+\gamma$ then ...
7
votes
1answer
91 views

What classification of countable ordinals above $\omega_1^{CK}$ exists?

What classification of countable ordinals above the Church–Kleene ordinal $\omega_1^{CK}$ exists? Are there such things as $\omega_2^{CK}$, $\omega_{\omega_\omega^{CK}}^{CK}$ or ...
0
votes
2answers
91 views

What is a notation for the minimal ordinal of $\mathbb{R}$?

What is a notation for the minimal ordinal of $\mathbb{R}$? I know that $\beth_1$ and $\mathfrak{c}$ designate the cardinality of $\mathbb{R}$, and that $\Omega$ denotes the smallest uncountable ...
3
votes
1answer
115 views

What is this function on ordinals called?

Define a function $f:\mathbf{ON}\to\mathbf{ON}$ as follows: for each ordinal number $\alpha$ let $$f(\alpha)=\operatorname{ord(}\lbrace \beta<\alpha|\text{ }\beta\textrm{ is a limit ...
1
vote
1answer
75 views

Product of ordinals notation

How to denote product of ordinals: $\cdot$ or $\times$? I'm not sure which of these two multiplication symbols to use.
6
votes
2answers
177 views

Least non-arithmetical ordinal

As I understand, there exists the least ordinal $\alpha$ such that there is no well-ordering of $\mathbb{N}$ which is both order isomorphic to $\alpha$ and is an arithmetical set. Is there a ...
0
votes
2answers
374 views

How far do known ordinal notations span?

What is the largest known ordinal number $\alpha$ such that a uniform notation scheme has been developed for all ordinals up to $\alpha$ (there should be no "gaps" in what ordinals are representable), ...