Past exam of the mathematics course of the University of Cambridge / 2015 / iii / Paper 24 / 1 / a

A strict partial order is a well-founded relation when every nonempty subset $Y\subseteq P$ has a minimal member:In ZFC this is equivalent to the absence of an infinite descending sequence $p_0{>}_P{p}_1{>}_P\cdots$. A descending sequence violates the minimal-member property; conversely, repeatedly choosing a predecessor from a subset with no minimal member constructs such a sequence using the axiom of choice.

For a Sierpiński decomposition of the plane, $A,B$ partition ${\mathbb{R}}^2$, and one has countable vertical sections while the other has countable horizontal sections. With the orientation fixed as follows,for every real $x,y$. Interchanging the coordinate directions and the names $A,B$ gives the equivalent convention. Here “countable” includes finite sets.

The relation of eventual domination isA dominating family $C\subseteq {\omega }^{\omega }$ satisfiesIt must dominate every function, and the exceptional finite initial interval may depend on both functions.