# Tagged Questions

In the ZF set theory ordinals are transitive sets which are well-ordered by $\in$. They are canonical representatives for well-orderings under order-isomorphism. In addition to the intriguing ordinal arithmetics, ordinals give a sturdy backbone to models of ZF and operate as a direct extension of ...

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### First uncountable ordinal

I am a beginner of ordinals and I don't know any powerful techniques in it. I come across with a problem about the first uncountable ordinal like this. Let $X$ be a set of uncountable cardinality. ...
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### Order-preserving map of regressive functions on $\omega_1$

I posted the following question a year ago on MO. It did receive some attention, but the answer there remains incomplete (as discussed in some detail in its comments). Meanwhile I got the impression ...
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### An upper bound of a countable set of countable ordinals

Let $X$ be a countable set of countable ordinals. Is there any upper bound of this set under $\omega_1$? In other words, can we find an ordinal $\alpha<\omega_1$ which is larger than any elements ...
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### If $f\colon\kappa\rightarrow\kappa$, then the set of all $\alpha < \kappa$ such that $f(\xi) < \alpha$ for all $\xi < \alpha$ is closed and unbounded.

If $f\colon\kappa\rightarrow \kappa$, then the set of all $\alpha < \kappa$ such that $f(\xi) < \alpha$ for all $\xi < \alpha$ is closed and unbounded. This is from Jech's book (page 103) so ...
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### Ordinals that are not cardinals [duplicate]

I am reading Jech's set theory and he defines a cardinal number as an ordinal $\alpha$ (a cardinal) if $|\alpha| \neq |\beta|$ for all $\beta < \alpha$, and he says that all infinite cardinals are ...
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### A question about well-ordered subsets of partially ordered sets

Let $S$ be an infinite partially ordered set. Is there a name for the smallest ordinal number which is not ordinally similar to any subset of any chain of $S$? Assume that we are working in ZF. If the ...
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### Does every countable sequence of ordinals converge?

Let $A=\{ \alpha_n | n \in \omega \}$ a monotonic increasing sequence. Can we say that there exists an $\alpha \in ON$ such that $\alpha = \lim_{n \in \omega} \alpha_n$? i am asking this because I am ...
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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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### Can set containing not only all ordinal numbers exist?

I've always seen the same proof of non-existence of set of all ordianal numbers. It was based on a fact that such a set would fit into definition of ordinal number, and the set itself would be an ...
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### What is an undefinable ordinal? How large are the least undefinable ordinal and supremum of all undefinable ordinals?

What does it mean to say that a particular ordinal/cardinal number is "definable"? Where and how do we define this "definability"? If there are undefinable ordinals/cardinals, how large are the least ...
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### Open neighborhoods in ordinals with the order topology

While learning about ordinals, my teacher made some remarks about any ordinal α being equipped with the order topology, and some facts, one of which was basically that the finite ordinals and ω are ...
### Is every countable ordinal homeomorphic to a subspace of $\mathbb R$?
I know that every countable ordinal is isomorphic to some subset of $\mathbb R$ as ordered sets. Is it also the case that every countable ordinal (with the order topology) is homeomorphic to some ...