# 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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### List of limit ordinals [closed]

I am trying to understand ordinals and cardinals. Seeing a list of limit ordinals would be helpful. A small question outside of this, are the limit ordinals related to cardinals? In what way? ...
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### Surjective function into Hartogs number of a set

For any set $A$, there is a surjective function $f:\mathcal{P}(A\times A)\longrightarrow\mathcal{H}(A)$. $\mathcal{H}(A)$ is the Hartogs number of $A$. Proof attempt: Suppose $R\subseteq A\times A$. ...
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### Derivatives of ordinal order

This question actually arises from this answer to another question, which contains the sentence A function is smooth is it has derivatives of infinite order. While the author surely didn't ...
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### Lebesgue Premeasure via Transfinite Induction

If $I=[a,b)$ we write $|I|=b-a$ for the length of $I$. Given a theorem of Caratheodory, the tricky part in showing the existence of Lebesgue measure is this: Lemma If $[0,1)$ is the disjoint union of ...
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### How to prove the not-so-long rays are homeomorphic to the reals?

The long ray (half of the long line) is an interesting topological space. It is defined as the order topology on $\omega_1$$\times [0, 1)$ with lexicographic order. Basically, it is an uncountable ...
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### How many limits of limits are there in $\omega_1$?

We know there are $\omega_1$-many limit ordinals less than $\omega_1$. What about ordinals that are limits of limit ordinals? Are there also $\omega_1$-many of these?
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### How to view beth number as ordinal?

The following is the snippet of the paragraph given in my lecture note: (Note: $\beth$ is the beth-number.) (When I introduced $\beth_\alpha$ I told you that it was to be a particular size of ...
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### Height of an ordered field

I'm studying ordered fields, and a specific notion regarding ordered fields that I will denote here by their "height". If $k$ is an ordered field, and $\alpha$ is a non-empty ordinal, a ruler of ...
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### 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 ...
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### The ordinals as a free monoid-like entity

Let $\kappa$ denote a fixed but arbitrary inaccessible cardinal. Call a set $\kappa$-small iff its cardinality is strictly less than $\kappa$. By a $\kappa$-suplattice, I mean a partially ordered set ...
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### formal definition of ordinal addition by recursion

I'm reading Kunen's Set Theory, An Introduction to Independence Proofs (1980). On page 26 he explains how to introduce ordinal addition through recursion. For the sake of convenience i'll give the ...
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### limit and infinite ordinals: same thing? [duplicate]

I'd like to clarify my understanding re: limit and infinite ordinals. This post says: An initial infinite ordinal is a limit ordinal. Is it true the other way around? That is if I have a limit ...
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### Multiplying two ordinals where one has been raised to power of $\omega$. Term order matters?

When multiplying two ordinals that are both raised to some power, my book says that one adds the exponents. But what happens if one of the exponents is $\omega$ ? Does the order of the terms matter ? ...
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### example of multiplication of ordinals with infinite cardinality with larger value on right where we dont' take the max?

I recall reading about a rule for multiplying ordinals where at least one is infinite, and where the cardinality of the multiplier (on right) is larger than the multiplicand (on left). If I recall ...
How can you multiply these ordinal numbers: $(\omega+1)(\omega+1)(\omega2+2)$ I tried and have gotten to this: $(\omega^2+1)(\omega2+2)$ Is that the correct way, or did i made a mistake?
### any example of non-zero ordinals $a,b,c$ for which $a < b$, but $a^c \ge b^c$?
I am working from a book that gives the above problem, but no solution ;^(. That is: Show an example of non-zero ordinals $a,b,c$ for which $a < b$, but $a^c \ge b^c$. Exponentiation here ...