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 is a particular recursive method of constructing mathematical structures that live on a given ordinal ${\displaystyle \theta }$, using a single transformation ${\displaystyle \xi \mapsto C_{\xi ))$ which assigns to every ordinal ${\displaystyle \xi <\theta }$ a set ${\displaystyle C_{\xi ))$ of smaller ordinals that is closed and unbounded in the set of ordinals ${\displaystyle <\xi }$. The transfinite sequence

${\displaystyle C_{\xi }(\xi <\theta )}$

which we call a C sequence and on which we base our recursive constructions.

The method might be formally described this as follows:

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

Further

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

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

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

Minimal step from ${\displaystyle \beta }$ towards ${\displaystyle \alpha <\beta }$

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

where

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

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

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

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

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

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.

Method characteristics

Method use

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