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Tagged with ordinals reference-request
34
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Can the class of ordinals be extended even further? [duplicate]
Is it possible for anything to come after all ordinals? I don't see why not. For example, one can take a non-ordinal set $S$, and then add in all the ordered pairs $(\alpha, S)$ to $ON$, where $\alpha$...
- 21.5k
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References that give the cofinality of ordinal addition, multiplication and exponentiation
$\newcommand{\cf}{\operatorname{cf}}$
Let $\alpha$, $\beta$ be ordinals. I believe that we have
\begin{align*}
\cf(\alpha+\beta)=\cf(\beta),\quad\beta\neq 0;\quad\cf(\alpha\cdot\beta)=\begin{cases}\cf(...
- 1,923
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Reference request: monoids on ordinal numbers
It is well-known that $(\text{Ord},+,0)$ and $(\text{Ord},\cdot,1)$ are monoids, but I haven't found references on these structures or other simpler ones (like $(\omega_1,+,0)$). For example, it would ...
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Writing infinitely long expressions in set theory
Using just the three symbols {, }, and , we can write any hereditarily finite set as a finite sequence of those three symbols. However, I have thought of something recently. What if we allow ...
- 21.5k
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Classification of compact Hausdorff spaces $X$ of cardinality $2^{\aleph_0}>|X|\geq \aleph_1$?
On my own time, I've been curious/investigating some properties of compact Hausdorff spaces of cardinality $\aleph_{\alpha}$, for $\alpha \geq 0$ a particular ordinal. I am aware of the following ...
- 5,830
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Reference request: order type of subset of limit ordinals
Let $\lambda$ be an ordinal, and let $f(\lambda)$ be the ordinal representing the order type of the limit ordinals strictly less than $\lambda$. Then $f$ is a weakly increasing function, $f(\lambda + ...
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Reference for Analysis book in which natural numbers constructed from sets
Could anyone suggest books on Mathematical/Real Analysis that construct natural numbers through sets not Peano axioms?
I find construction of natural numbers through sets more convenient. So I ...
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Texts on limits of ordinal sequences
Let $\alpha$ be a limit ordinal, and let $f$ be a function from $\alpha$ to $\mathbb{R}$. Has anyone or any book or text defined the notion of limits for general limit ordinals, not just the limit ...
- 21.5k
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Well-founded orders and sums
I am looking for a reference on well-founded ordered sets, that would mention the following notions and results:
$\\$
a) Consider a well-ordered set $(I,<)$ and a family $(X_i,<_i)$ of well-...
- 5,125
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Theory of the von Neumann hierarchy
The sets $V_\alpha$ in the cumulative von Neumann hierarchy, defined by transfinite induction:
$\begin{align}
V_0 &= \varnothing \\
V_{\alpha+1} &= \mathcal{P}(V_\alpha) \\
V_\lambda &= \...
- 1,907
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Countable order-dense subset in a linear order
I ended up posting an answer to my own question, and I believe I knew this answer before I posted this question, but forgot about it. At any rate, here it is for the record.
My question is inspired ...
- 13.5k
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Where is the theorem related to the construction of countable admissible ordinals by Turing machines with oracles?
Wikipedia contains the following information in the article "Admissible ordinal":
By a theorem of Sacks, the countable admissible ordinals are exactly those constructed in a manner similar to the ...
- 922
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1
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What is the name of a theorem which says that if $\alpha$ and $\beta$ are ordinals, then $\alpha\in\beta$, or $\beta\in\alpha$, or $\alpha=\beta$?
This Math.SE question contains the following information:
There is a theorem which says given any two ordinals $\alpha$ and $\beta$, exactly one of the following holds: $\alpha\in\beta$, or $\...
- 922
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Are ordinals greater than $\varepsilon_0$ used outside Ordinal Analysis?
I know of Conway's use of ordinals to exhibit the algebraic closure of $\mathcal{F}_2$. I also read a document about the Cantor Bendixson rank of some family of groups. But I found no applications of ...
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Suggestion for a textbook including Von Neumann definition of ordinal numbers
I'm interested in the generalization of natural numbers and especially ordinals. I have searched through some set theory textbooks but unable to find Von Neumann definition of ordinal numbers.
I have ...
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