The uncountable transfinite subway

We explore a more sophisticated version of the infinite subway paradox, with stations all the way to the uncountable ordinals and beyond

In recent posts we have been exploring the infinite subway paradox, starting with an introduction, followed by a fuller range of paradox, and then the transfinite extension of the paradox into the countable ordinals.

In this essay, we extend the paradox to the uncountably infinite. This essay engages therefore with a few more serious set-theoretic ideas, finding in the infinite subway paradox a call for the notions of stationarity, the club filter, and Fodor’s lemma on regressive functions. Please enjoy!

So let us consider the fully uncountable extension of the transfinite subway line. The train starts at station 0 and proceeds to station 1, station 2, through all the finite stations, to station ω, station ω + 1, station ω + 2, and so forth, eventually reaching station ω², station ω³, station ω^ω, station ε₀, and so on far beyond through all the countable ordinals. The train finally concludes its trip at the terminal station of the first uncountable ordinal, station ω₁. Which patterns of embarkment and disembarkment up to ω₁ are possible?

In the previous essays we saw how to reach any given countable ordinal with a finitely extensible train car, capable of holding any finite number of passengers at one time (but never infinitely many), while still having a disembarkment at every station stop along the way. The citizens of Infinitopolis can therefore fulfill the infinite subway challenge to reach any given countable ordinal.

Can we reach the first uncountable ordinal ω₁ itself this way? That is, can we unify all the various particular solutions reaching the various countable ordinals with a single uncountable schedule of passenger itineraries that proceeds all the way to ω₁? We know how to reach any particular countable ordinal α, yes, each with a separately defined schedule aimed at reaching that particular ordinal, but the question I am asking is whether we can do so in a fully uniform manner, whether we can describe a single uncountable trip, a single schedule of passenger itineraries in the finite extensible train that proceeds through every countable ordinal station in one run, while still having a disembarkment at every station stop along the way.

Interlude

We shall answer this question about the ω₁ subway, indeed, with a surprising follow-up second answer, and then proceed to explore the infinite subway at higher cardinals, where a new surprising phenomenon awaits. In fact, the infinite subway paradox reveals a peculiar situation in set theory, a situation where features of the first infinite cardinal ω generalize exactly to the uncountable singular cardinals and not at all to the uncountable regular cardinals. Very unusual. Read on to find out what I mean.