proof-theoretic consistency in the introduction

[mikeshulman]

James Hanson has pointed out that the comment

One of the original virtues of type theory, relative to set theory, was that it can be seen to be consistent by proof-theoretic means.

in the introduction is misleading: what enables a proof-theoretic consistency proof is not type theory versus set theory but the proof-theoretic strength, and in particular predicativity. In particular, some predicative set theories like KP and CZF have proof-theoretic consistency proofs. So perhaps we could change it to something like

One of the virtues of weak constructive theories, relative to strong ones that include axioms like AC and LEM, is that they can often be shown to be consistent by proof-theoretic means.

(The notion of predicativity has not been introduced at that point, but maybe "weak constructive" is a suitable handwavy stand-in.)

[James-Hanson]

KP set theory is classical. While there are examples of predicative theories that become non-predicative after the addition of LEM (like CZF), this isn't always the case.

[mikeshulman]

Do you mean to say that KP isn't "predicative" because it is classical? I think a theory can be both predicative and classical, since without function-sets you can't construct impredicative things like powersets from LEM.

Or did you mean to say that since KP is both predicative and classical, it isn't included in my suggested phrase "weak constructive"? I think that's a valid objection, but I don't think we can get into the meaning of predicativity in the introduction. Maybe say "weak constructive" with a footnote that says "more precisely, predicative"?

[James-Hanson]

I meant the second point about KP not being included in the phrase 'weak constructive.' It is true that classical mathematical logicians do tend to think of theories like KP as being 'more constructive' than theories like ZF, but I also was under the impression that constructive mathematicians probably wouldn't really see the distinction the same way.

This is also probably too much nuance for the introduction, but LEM and AC have something of a subtle relationship with consistency strength. Obviously LEM/AC adds consistency strength to something like CZF, but in a fair number of situations they don't (i.e., LEM doesn't add strength to HA or IZF and AC doesn't add strength to second-order arithmetic, Z, ZF, and somewhat surprisingly even HA^\omega).

[mikeshulman]

What do you think of my proposed solution with the footnote?

[James-Hanson]

I'm not sure. This was something Andrej was agonizing over right up to the last minute with our topos theory paper (specifically the use of words like 'constructive' and 'intuitionistic'), so I get the impression that people think of them as being fairly loaded words. I just think that the current wording implies that 'weak constructive theory' precludes having LEM or AC, which strictly speaking 'predicative' does not.

[mikeshulman]

What if the footnote says "more precisely 'predicative', which includes some theories with classical logic"?