# Tagged Questions

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### Prove that ordinal multiplication is left distributive

Suppose $\alpha, \beta$ and $\gamma$ are ordinals. Prove the distributive law $\alpha \cdot ( \beta + \gamma) = \alpha \cdot \beta + \alpha \cdot \gamma$. The following is my proof: Proof: We use ...
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### Ordinal arithmetic and functions

I have two function $G$ and $F$ defined on ordinals and I know that $$G(\alpha +\omega )\subseteq F(\gamma +\alpha+\omega)$$ when $G(\alpha)\subseteq F(\gamma)$ and $\alpha$ is a limit ordinal. I ...
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### Definition of continuity of ordinal function

In the book Introduction to Set Theory' by Hrbacek and Jech, chapter $6$ Ordinal Numbers, section $6$ Normal Form, I don't understand the definition of continuity of ordinal numbers. Ordinal ...
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### How uncountable is the set of countable ordinals?

From the answers (and comments) to this question on the uncountability of countable ordinals I don't get a lucid picture: How can I see that the uncountability of the set of countable ordinals ...
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### The structure of countable ordinals

Consider the recursively defined hyperoperation sequence $\circ_i$ \begin{array}{rcrclclcl} x& \small{+}&(y\ {\small+}1)&:=&x& &&{\small+}&1\\ x& ...
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### A representation of well-orderings?

Is there a well-ordering $P$ of the set of real numbers $\mathbb{R}$ such that there is NO function $f: \mathbb{R}->\mathbb{R}$ satisfying the property: for all $x,y \in \mathbb{R}$, $xPy$ iff ...
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### Showing ordinal addition and ordinal multiplication are defined

I've been reading absoluteless results in Kunens't latest Set Theory text. After talking about $\Delta_0$ formulas and absoluteness, he mentions that certain concepts are absolute, but not $\Delta_0$. ...
Good evening to all. I have two exercises I tried to resolve without a rigorous success: Is it true or false that if $\kappa$ is a non-numerable cardinal number then $\omega^\kappa = \kappa$, where ...