### The Relevance of Predicativity

#### Posted by David Corfield

If I get around to writing a second book in philosophy of mathematics, one thing I’ll probably need to retract is the ill-advised claim made in the first book that the notion of predicativity is irrelevant to mainstream mathematics.

Here’s a passage which goes directly against such a thought, from Nik Weaver’s Is set theory indispensable?

…the predicative universe exhibits a strikingly exact fit with the universe of ordinary mathematics. This is particularly well illustrated by the dual Banach space construction in functional analysis. Classically, every Banach space $V$ has a dual Banach space $V'$, but predicatively this construction is only possible when $V$ is separable. Remarkably, it seems to be a general phenomenon that for any “standard” Banach space $V$, its dual is also “standard” if and only if $V$ is separable. In other words, start with any well-known Banach space $V$ that commonly appears in the functional analysis literature, and iteratively take duals to create a sequence $V$, $V'$, $V''$,… . It will generally be the case that if $V$ is in common use then $V'$, $V''$, etc., will also be in common use – up to the first nonseparable space in the sequence. All spaces after that point will be highly obscure.

For example, take $V = L^1(\mathbb{R})$ (separable). Then $V' = L^{\infty}(\mathbb{R})$ (nonseparable), and $V''$ is an obscure space that has no standard notation. Or take $V = C[0, 1]$ (separable). Then $V' = M[0, 1]$ (nonseparable), and $V''$ is an obscure space that has no standard notation. Take $V = K(\mathcal{H})$ (the compact operators on a separable Hilbert space $\mathcal{H}$; separable). Then $V' = T C(\mathcal{H})$ (the trace class operators on $\mathcal{H}$; also separable), $V'' = B(\mathcal{H})$ (the bounded operators on $\mathcal{H}$; nonseparable), and $V'''$ is an obscure space that has no standard notation. If $V = c_0$ (separable) then $V' = l^1$ (separable), $V'' = l^{\infty}$ (nonseparable), and $V'''$ arguably has a standard notation – it is the space of Borel measures on the Stone-Čech compactification of the natural numbers – but it is certainly an obscure space that appears in the literature with extreme rarity. Examples of this type could be multiplied endlessly.

The explanation of this phenomenon is simple: generally speaking, duals of nonseparable spaces are highly pathological objects about which little of value can be said. This is characteristic of impredicative mathematics generally.

Does this tally with people’s experience?

## Re: The Relevance of Predicativity

It matches my experience, back in the days when I did functional analysis.

I had a nice conversation about Nik Weaver’s work with Toby Bartels this Wednesday. He could summarize it a lot better than I could. But very roughly, it seems that Weaver has a version of logic that’s strong enough for most math that people actually do, but much weaker than ZFC. In this setup classical logic applies to countable sets, intuitionistic logic applies to larger sets, and the dual of a nonseparable Banach space doesn’t even exist.

I first heard of Weaver when learning about the Feferman-Schütte ordinal $\Gamma_0$.

According to Feferman, $\Gamma_0$ is the first ordinal that can’t be defined predicatively — and induction up to $\Gamma_0$ allows us to prove the consistency of predicative analysis. But Weaver has different views on this issue. Feferman has argued against these views, and Weaver has argued back. I’m too ignorant to have an opinion.