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Results tagged with ordinal-numbers
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user 7206
An ordinal is the order type of a well-ordered set. The first few ordinals are $0, 1, 2, \dots, \omega, \omega+1, \dots$ where $\omega$ is the order type of $\mathbb{N}$, and $\omega+1$ is the order type of $\mathbb{N}$ together with a maximum element.
1
vote
Is it consistent with ZFC that for all ordinals $\alpha, \beta < \omega$ it holds that $2^{\...
To your edit, note that for every ordinal $\alpha$ it holds that $\alpha\le\aleph_\alpha$. This is because there are $\alpha$ many cardinals below $\aleph_\alpha$.
Since the function $\kappa\mapsto 2 …
- 39.9k
2
votes
Why do ordinal collapsing functions use regular cardinals?
If $\kappa$ is a singular cardinal and we collapse every ordinal below $\kappa$ to be of size $\lambda$, then you might want to say that $\kappa=\lambda^+$. But $\sf ZFC$ proves that $\lambda^+$ has t …
- 39.9k
12
votes
Accepted
Hartogs' Number of the Reals and $\Theta$ without choice
Other than $\aleph_1\leq\aleph(\Bbb R)\leq\aleph^*(\Bbb R)$ and at least one of them is sharp, not much is provable. And of course, both of these inequalities are easy to prove.
We can have $\aleph(\ …
- 39.9k
5
votes
Prime numbers and limit ordinals
You can encode using recursive functions fairly large ordinals. But you are trying a bit too hard, I think, to encode $\omega^2$. Perhaps a simpler method would be as follows:
Let $\pi$ be a bijectio …
- 39.9k
2
votes
Accepted
Good set theory in which to study ordinal-indexed sequences?
Note that the class-length sequence $V_\alpha$ is definable, so given a set $X$, so is the sequence $V_\alpha\setminus X$ going to be definable (via the parameter $X$).
If you intend to use more, per …
- 39.9k
7
votes
Accepted
Elementary countable submodels in Gödel's universe
Very clearly not. Take some countable elementary submodel $M_0$ of $L_{\omega_2}$, and take $M_1$ to be another one, but with $M_1$ a end extension of $M_0$. We can find such models by first finding t …
- 39.9k
14
votes
Accepted
Intuition about ordinal fixed points $\alpha = \aleph_\alpha$
Your intuition is finitary, and therefore wrong. Compare, for example, the two sequences:
$\alpha_n=n$, and
$\beta_n=2^n$.
It is easy to see that $\alpha_n<\beta_n$ for all $n$. We even know from el …
- 39.9k
6
votes
Accepted
What is the order type of $L$ with Godel's well ordering?
The first question can be answered by taking the idea of definable well-orderings which are not set-like. That is, we can consider the formula $\varphi(x,y)$ which states that $x,y$ are distinct ordin …
- 39.9k