More on machine learning and CH

A few days ago I wrote about a paper establishing an independence result in the field of machine learning. Here I offer a few more comments.

In the paper the authors comment on the relation between their result and actual machine learning. That relation may seem tenuous because none of the functions involved in the arguments is related to any kind of algorithm.
Indeed the constructions of the compression schemes are very non-constructive in that they use repeated applications of the Axiom of Choice.

Now the inequality 20>ℵω implies there are no compression schemes whatsoever. But it may be of interest to know that consistent examples of compression schemes must be nonconstructive. It turns out that, with the aid of a few standard results from Descriptive Set Theory it is relatively easy to show outright that there are no Borel measurable monotone compression schemes and hence no Borel measurable learning functions for the class of problems studied in the paper mentioned above either.

The details can be found in this note.

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