19
votes
1answer
617 views

Why isn't this a computable description of the ordinal of ZF?

In a previous MO question, I was told by several commenters that (a) it's known that there exists a computable ordinal $\alpha_{ZF}$ that "encodes the strength of ZF set theory" (i.e., a least ...
26
votes
1answer
792 views

Is there a computable ordinal encoding the proof strength of ZF? Is it knowable?

In comments on Quora (see, for example, here, here, here), Ron Maimon has repeatedly expressed the strong opinion that Hilbert's program was not killed by Gödel's results in the way typically ...
6
votes
2answers
429 views

The Theory of Transfinite Diophantine Equations [closed]

The theory of Diophantine equations is one of the main stream research areas in number theory. There are many known results and unknown conjectures about the existence of non-trivial solutions for ...
4
votes
4answers
625 views

Prime numbers and limit ordinals

As a set, i.e. as a von Neumann ordinal, the $\omega$-th limit ordinal $\omega^2$ is fairly complex and not so easy to visualize (for the novice). But as an explicit well-ordering of $\mathbb{N}$, ...
10
votes
2answers
420 views

Ways to define “definability”

The notion of a definable set is not expressible in the language of set theory: there is no formula $\delta(x)$ that is equivalent with there being a formula $\phi(y)$ such that $x = \lbrace y : ...
2
votes
1answer
189 views

A question about additively indecomposable ordinals

I'm studying some topics about topological games of length $\alpha\geq\omega$, where I came across the following statement about additively indecomposable ordinals (recall that $\alpha$ is additively ...
2
votes
1answer
207 views

What is the order type of $L$ with Godel's well ordering?

In some sense $Ord$ is a "proper class" ordinal. Unfortunately the notion of a proper class ordinal is not a straight forward generalization of the notion of "set" ordinals because the proper classes ...
7
votes
1answer
315 views

How strong are large cardinal properties of Ord?

Ordinal numbers are generalizations of natural numbers. In this sense the "proper class" of all ordinals ($Ord$) is very similar to "infinite" set of all natural numbers ($\omega$). In the other ...
12
votes
2answers
315 views

What is the least ordinal than cannot be embedded in $\mathbb{R}^\mathbb{R}$?

Let $\mathbb{R}^\mathbb{R}$ be the set of functions $\mathbb{R}\to\mathbb{R}$ patially ordered by eventual domination. Obviously, every ordinal below $\omega_1$ can be embedded in ...
23
votes
2answers
407 views

Order type of the smallest set containing the identity function and closed under exponentiation

Let $E$ be the smallest set of functions $\mathbb{N}^+\to\mathbb{N}^+$ containing the identity function $n \mapsto n$ and closed under exponentiation $(f,g) \mapsto \left(n \mapsto ...
0
votes
1answer
228 views

Is there a name for the smallest ordinal $\alpha$ such that $X \subseteq \alpha$

Let $X$ be a set of ordinals. If $X$ has no largest element, then \[ \sup X \notin X \subseteq \sup X, \] and $\sup X$ is the smallest ordinal $\alpha$ such that ...
3
votes
1answer
322 views

Cardinality of $\omega\uparrow^\omega\omega$

I was wondering what the cardinality of $\omega\uparrow^\omega\omega$ is, with $\uparrow$ being Knuth's up-arrow notation. I ask this purely out of curiosity; after finding out about set theory I feel ...
0
votes
1answer
187 views

Good set theory in which to study ordinal-indexed sequences?

I'd like to "model" the absolute complement of a set $X$ as the ordinal-indexed sequence $\alpha \mapsto V_\alpha \setminus X$ where $V_\alpha$ is the $\alpha$ stage of the cumulative hierarchy. My ...
17
votes
1answer
935 views

Number of distinct values taken by $\alpha$ ^ $\alpha$ ^ $\dots$ ^ $\alpha$ with parentheses inserted in all possible ways, $\alpha\in\mathbf{Ord}$

Let $\alpha\in\mathbf{Ord}$ and $n\in\mathbb{N}^+$. Let $F_\alpha(n)$ be the number of distinct values taken by ordinal exponentiation $\underbrace{\alpha \hat{\phantom{\hat{}}} \alpha ...
13
votes
1answer
774 views

Are Conway's omnific integers the Grothendieck group of the ordinals under commutative addition?

This is a question in two parts. Say that $\mathbf{On}$ is the proper class of all ordinal numbers in ZFC. We can define a binary operator over $\mathbf{On}$ which corresponds to the commutative ...
7
votes
1answer
416 views

Does this “jumping-ahead” ordinal function exist?

While working on a project in operator algebras with a collaborator (and fellow MO user), we are able to successfully complete a transfinite induction assuming that the following has an affirmative ...
4
votes
1answer
582 views

Cantor's Normal Form and Aleph_1

The Cantor Normal Form Theorem states that every ordinal $\alpha > 0$ can be uniquely expressed in the form $$\omega^{\beta_1}k_1 + \omega^{\beta_2}k_2 + \dots + \omega^{\beta_n}k_n$$ for some $n ...
4
votes
3answers
489 views

Is it consistent with ZFC that for all ordinals $\alpha, \beta < \omega$ it holds that $2^{\aleph_\alpha} = 2^{\aleph_\beta}$?

Let $\gamma=\omega$ (the first transfinite ordinal). Is it consistent with ZFC that for all ordinals $\alpha, \beta < \gamma$ it holds that $2^{\aleph_\alpha} = 2^{\aleph_\beta}$? If yes, can the ...
4
votes
2answers
293 views

Heights of several interesting posets

Let the height of a poset $P$ be the supremum of ordinals that are order types of all well-ordered subsets of $P$ (with order inherited from $P$). Define several sets of total functions, in each ...
4
votes
1answer
314 views

Cardinality of cofinal set of normal functions $f \colon \omega_1 \to \omega_1$

What is the cardinality of the set $F$ of all normal functions $f \colon \omega_1 \to \omega_1$, where $\omega_1$ is the first uncountable ordinal? What is the least cardinality of a subset of $F$ ...
3
votes
2answers
1k views

Foundations: Existence of uncountable ordinals.

This isn't really a research question, but at least it's research-level mathematics. I'm talking with some other people about the first uncountable ordinal, and I want some facts to inform this ...
13
votes
3answers
2k views

Understanding the countable ordinals up to $\epsilon_{0}$

Hello, in a recent MO question, link, discussing the current foundations of mathematics, the author linked a video lecture by Prof. Voevodsky, which argues against the principle of ...
3
votes
1answer
714 views

Well ordering of countably branching well founded trees

Hello everybody, I would like to say, before stating the question, that I'm not a mathematician, and therefore i apologize in advance if the question is trivial, or well-known. I'll try to state it ...
5
votes
1answer
282 views

value of Theta in ZF+AD

Since I found out about it, I've always been interested in the Axiom of Determinacy rather than the Axiom of Choice. Along these lines, I've kept flipping back to ...
1
vote
0answers
296 views

Transfinite Sums Related to a Sequence

Hello, Given a sequence $S$ indexed by the finite ordinals, a limit ordinal $\alpha$, and $k \in \mathbb{N}$, define $S_{\alpha+k}$(the extension of $S$ to $\alpha+k$) to be the sum over the products ...
7
votes
5answers
1k views

What is the idea behind stationary sets?

Let $\kappa$ be a cardinal (of uncountable cofinality). A subset $S \subseteq \kappa$ is called stationary if it intersects every club, i.e. closed unbounded subset of $\kappa$. Now my question is ...
13
votes
5answers
1k views

Finding the largest integer describable with a string of symbols of predefined length

(This question is motivated by the reading of the article Large numbers and unprovable theorems by Joel Spencer, which can be found at ...