I realised yesterday that, if I want an accurate picture of how bad Man Utd currently are, I can track 27 live match stats on the BBC. People CAN in fact comprehend complex numerical information - it takes real effort to create and maintain economic ignorance at the level our media does.

We cannot out-Reform Reform. But we can be more Labour.

We must stand up for working people with clear policies that tackle inequality, protect communities and put peace, justice and welfare at the heart of all we do.

We must take a much harder line with Israel.

I should mention that the paper is open access, and preprints of both submitted versions are available on the arXiv. arxiv.org/abs/2306.06471

This was something I enjoyed digging into when I was writing the paper, but I kept it very short in the submitted version because I was worried the paper was already long. We often complain that referees make papers worse, but in this case they definitely made it better, at least in my view.

I very much appreciate the job the referees did, not just because they found a few small mistakes in the submitted version, but because they asked me to expand on history of how Arrow’s theorem was extended to infinite, computable societies by Alain Lewis in the 1980s and Reiju Mihara in the 1990s.

One corollary of the results in the paper is that Arrow’s theorem is provable in PRA, the formal system of primitive recursive arithmetic. I suspect it’s actually provable in a weaker system still.

In the paper I show that arithmetical comprehension is exactly the axiom needed to construct infinitary counterexamples to Arrow’s theorem. In doing so I set up a framework of countable societies that can be used to analyse the strength of other theorems in social choice theory.

David Lodge did this joke already in 1984 in ‘Small World’ with the ELIZA chatbot.

I realise editors need a shorthand for encouraging authors to polish their language but this can be done without using nativist language. Some of us who learned English later in life are bigger snobs about it than many native speakers. Let’s just encourage attention to clarity of expression.

A precise result here is: The statement that there exists an ultrafilter on every countable algebra of sets of natural numbers is Weihrauch equivalent to the jump of weak König's lemma.

I suppose I worry that readers might see the word 'theory' associated with ultrafilters and come away with the mistaken impression that these are things we can reason with, rather than objects which are in general non-computable.

Also unfortunate not to have any mention of computability-theoretic issues, although the axiom of choice is discussed. The existence of ultrafilters for Lindenbaum–Tarski algebras of countable languages doesn't require choice, and in fact is provable in a system conservative over PA.