Articles of ordinals

Best known upper and lower bounds for $\omega_1$

What are the best known lower and upper bounds for $\omega_1$? Are there sharper lower bounds than $\varepsilon_0$? And are there any known upper bounds which can be explicitly constructed like $\varepsilon_0$? Since $\varepsilon_0$ is countable, $\varepsilon_0$ must be less than $\omega_1$. But in all of the 45 books on logic and set theory on […]

A limit ordinal $\gamma$ is indecomposable if and only if $\alpha + \gamma = \gamma$ if and only if $\gamma = \omega^{\alpha}$ for some $\alpha$.

This question already has an answer here: Indecomposable limit ordinals 3 answers

Cantor-Bendixson rank of a closed countable subset of the reals, and scattered sets

I am reading the notes in the following link, and I am a bit unclear about the connection between scattered sets, countable sets, and sets for which the Cantor-Bendixson derivative is eventually the empty set. On page 3, the author says: A closed set X (of the reals) is scattered if $X^{\alpha}=\emptyset$, where $\alpha$ […]

A transitive set of ordinals is an ordinal

This is Exercise III.2.20 of Bourbaki’s Set Theory. (Von Neumann ordinals are actually called “pseudo-ordinals” by Bourbaki, but I simply call them ordinals here) Let $X$ be a transitive set, and suppose that each $x\in X$ is an ordinal. Then $X$ is an ordinal (Hint: for each $x\in X$, $x\cup\{x\}$ is an ordinal contained in […]

I read that ordinal numbers relate to length, while cardinal numbers relate to size. How do 'length' and 'size' differ?

I read that ordinal numbers relate to length, while cardinal numbers relate to size. How do ‘length’ and ‘size’ differ? Note : I am an absolute novice, and I’m having a little trouble visualizing ordinal numbers.

Understanding countable ordinals (as trees, step by step)

Even though ordinal numbers – considered as transitive sets – are perfect non–trees, it is worth (and natural) to visualize them as trees, starting from the finite ones which are given as non-branching trees of finite height: There are some crucial steps in the process of “understanding” larger ordinals, resp. when considering them as trees1. […]

Any Set of Ordinals is Well-Ordered

Is this a theorem of ZF or a theorem of ZFC? My suspicion is that it is a theorem of ZFC (the proof I’ve seen in P. L. Clark’s online notes on countable ordinals requires selecting an element from each set of a countable collection of sets). Does the situation change if the ordinals in […]

Ordinal addition is associative

We’ve been asked to teach ourselves a unit on ordinals for our final exam tomorrow, I grasp how to prove that certain ordinals are distinct but I am having trouble figuring out a proof to show ordinal addition is associative. All the proofs I have found online use methods that we have not covered yet. […]

What's the rank of this well founded relation?

Definition A tree is an ordered list of trees. (N.B these are finite objects and there is a very simple computable bijection of them with $\mathbb N$) Examples [] and [[],[],[]] and [[[]],[[],[],[]],[[],[],[[]]]] but it’s much better to think of them like this (can ignore the colors for now) I was wondering what is the […]

Injection from the set of countable ordinals $\Omega$ into $\mathbb{R}$

I’m reading through this and I’d like to define an injective function from the set of countable ordinals $\Omega$ into $\mathbb{R}$ using transfinite induction (or maybe transfinite recursion?). Clearly, $\emptyset \in \Omega$ will be defined to map to zero: $f(\emptyset) := 0$. Next one would probably make a distinction between limit ordinals and successor ordinals. […]