Draft:Walks on ordinals

In mathematics, the method of walks on ordinals, often called minimal walks on ordinals, was invented and introduced by Stevo Todorčević as a part of his new proof of the existence of the Countryman line in May 1984.

The method is a tool for constructing uncountable objects (from set-theoretic trees to separable Banach spaces) by utilizing an analysis of certain descending sequences of ordinals known as minimal walks.

The method might be formally described this as follows:

Let $\alpha$ be an ordinal and let $\beta$ be another one defined as $\beta =\{\alpha :\alpha <\beta \}$

Further

${\begin{aligned}&0=\emptyset ;\\&1=\{0\};\\&2=\{0,1\};\\&\,\,\,\vdots \\&\omega =\{0,1,2,\ldots \};\\&\omega +1=\omega \cup \{\omega \};\\&\omega +2=\omega \cup \{\omega ,\omega +1\};\\&\,\,\,\vdots \end{aligned}}$

and the Cantor's normal form of an ordinal is $\alpha =n_{1}\omega ^{\alpha _{1}}+n_{2}\omega ^{\alpha _{2}}+\cdots +n_{k}\omega ^{\alpha _{k}}$ where $\alpha _{1}\geq \alpha _{2}\geq \cdots \geq \alpha _{k}\geq 0$ are ordinals and $n_{1},n_{2},\ldots ,n_{k}$ are natural numbers. For more details see Ordinal arithmetic.

Ordinals from the class $\varepsilon _{0}=\{\alpha :\alpha =\omega ^{\alpha }\}$ – in this case the ordinals $\alpha _{1}>\alpha _{2}>\cdots >\alpha _{k}$ are all in the Cantor normal form of $\alpha$ and smaller than $\alpha$. For each limit countable ordinal $\alpha <\varepsilon _{0}$ we'll create a sequence $C_{\alpha }=\{c_{\alpha }(n):\alpha <\omega \}\subseteq \alpha$ such that $c_{\alpha +1}(n)=\alpha$ for all $n$ and such that $c_{\alpha }(n)<c_{\alpha }(n+1)$ for all $n$ and $\alpha =\lim _{n\to \infty }c_{\alpha }(n)$ when $\alpha$ is limit.

Minimal step from $\beta$ towards $\alpha <\beta$

$\beta \curvearrowright c_{\beta }(n(\alpha ,\beta ))$

where

$n(\alpha ,\beta )=\min\{n:c_{\beta }(n)\geq \alpha \}$

Minimal walk from $\beta$ towards $\alpha$ is a finite decreasing sequence

$\beta =\beta _{0}\curvearrowright \beta _{1}\curvearrowright \cdots \curvearrowright \beta _{k}=\alpha$

such that for all $i<k$ the step $\beta _{i}\curvearrowright \beta _{i-1}$ is the minimal step from $\beta _{i}$ towards $\alpha$ i.e.

$\beta _{i+1}=c_{\beta }(n(\alpha ,\beta _{i}))$

The method definition given above belongs to Todorcevic. Different, but equivalent method definitions, can be found in papers.

Many applications of the method have been found in combinatorial set theory, in general topology and in Banach space theory.

Sources

• Todorčević, Stevo (1987), "Partitioning pairs of countable ordinals", Acta Mathematica, 159 (3–4): 261–294, doi:10.1007/BF02392561, MR 0908147
• Todorčević, Stevo (2007), Walks on ordinals and their characteristics, Progress in Mathematics, vol. 263, Basel: Birkhäuser Verlag, ISBN 978-3-7643-8528-6, MR 2355670
• Hudson, William Russell (May 2007), Minimal walks and Countryman lines, MSc thesis, Boise State University
• Rinot, Assaf (2012), "Transforming rectangles into squares, with applications to strong colorings", Advances in Mathematics, 231 (2): 1085–1099, doi:10.1016/j.aim.2012.06.013, MR 2955203
• Lücke, Philipp (2017), "Ascending paths and forcings that specialize higher Aronszajn trees", Fundamenta Mathematicae, 239 (1): 51–84, doi:10.4064/fm224-11-2016, MR 3667758

There are the three topics related to this method definition: equivalent method definition (Hudson's MSc thesis), method characteristics (Todorcevic) and the method uses (the last two references)

However whoever might be tempted to work on this stub may select any references handling the three topics listed above