Let <math>\Omega</math> stand for the first [[Uncountablefirst set|uncountable]] [[Ordinalnumber|ordinal]] <math>\omega_1</math>, or, in fact, any ordinal which is (an <math>\varepsilon</math>-number and) guaranteed to be greater than all the countable ordinals which will be constructed (for example, the [[Church-Kleene ordinal]] is adequate for our purposes; but we will work with <math>\omega_1</math> because it allows the convenient use of the word ''countable'' in the definitions).
We define a function <math>\psi</math> (which will be [[Monotonic function|non-decreasing]] and [[Continuous function|continuous]]), taking an arbitrary ordinal <math>\alpha</math> to a countable ordinal <math>\psi(\alpha)</math>, recursively on <math>\alpha</math>, as follows:
