**4**

votes

**1**answer

799 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 \...

**5**

votes

**3**answers

542 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 ...

**7**

votes

**1**answer

611 views

### How this set of functions is ordered?

Notation:
$k, m, n$ are non-negative integers
$f, g, h$ are functions $\mathbb{N} \to \mathbb{N}$
$f^k$ is $k$-th iterate of the function $f$: $f^0(n)=n, f^{k+1}(n)=f^k(f(n))$
$f \prec g$ means ...

**4**

votes

**2**answers

317 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 ...

**7**

votes

**2**answers

345 views

### Ordinals and complexity classes

What is the least recursive ordinal $\alpha$ such that there is no algorithm in complexity class $\mathsf{P}$ which implements a well-ordering of $\mathbb{N}$ with order type $\alpha$? (where the size ...

**4**

votes

**1**answer

416 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 $F$ ...

**7**

votes

**1**answer

781 views

### Ordinal category theory?

Just out of curiosity: Is there a notion of $\alpha$-category for an ordinal number $\alpha$, extending the given notions for $\alpha \leq \omega$? If there is none, which one would you propose? Feel ...

**4**

votes

**2**answers

2k 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 ...

**3**

votes

**1**answer

575 views

### Height of ordered set

In all "constructive" fixed-point theorems for functions on ordered sets that I am aware of, where the fixed point is obtained as the limit of a stationary increasing transfinite sequence, it is ...

**8**

votes

**2**answers

1k views

### Naturally occurring orderings

The are many orderings that naturally occur in interesting but seemingly unrelated circumstances. Here are some examples:
The volume spectrum of orientable hyperbolic 3-manifolds has order type $\...

**16**

votes

**3**answers

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 $\epsilon_{0}$-...

**3**

votes

**1**answer

963 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 ...

**14**

votes

**1**answer

465 views

### Ordinal-indexed homology theory?

Back when I was a grad student sitting in on Mike Freedman's topology seminar at UCSD, he posed the following question. Does there exist a good homology theory $H_{\alpha}(X)$ where $\alpha$ is an ...

**15**

votes

**1**answer

1k views

### Proof theoretic ordinal

In Ordinal Analysis, Proof-theoretic Ordinal of a theory is thought as measure of a consistency strength and computational power.
Is it always the case? I. e. are there some general results about ...

**1**

vote

**1**answer

279 views

### center depth of Birkhoff center

I saw a statement about the Birkhoff center. Namely let $X$ be a compact metric space and $f:X\to X$ be a homeomorphism on $X$. Then for each ordinal $\alpha$ we define
for $\alpha=0$, let $\Omega_0(...

**5**

votes

**1**answer

356 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 http://en.wikipedia.org/wiki/%CE%98_%...

**1**

vote

**0**answers

312 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

**5**answers

2k 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

**7**answers

2k 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 http://mathdl.maa.org/images/upload_library/22/Ford/Spencer669-...