[FOM] How much of math is logic?

[Richard Heck]

praatika at mappi.helsinki.fi wrote:
> Here is a setting I like: 
>
> Let us take as our logic two-sorted FO logic, with separate variables for numbers and classes. We have introduction and elimination rules for both sorts of quantifiers. If one then adds only the very elementary defining axioms of successor, addition and multiplication (no induction or anything), one gets a system equivalent to ACA_0. 
>   
Surely this isn't quite right: You're going to need some sort of axioms
governing the class quantifiers to get anything this strong. The reason
is simply that, without such axioms, the system consists of (i) a subset
of the axioms of Q and (ii) a logical theory that is weaker than
predicative second-order logic. The usual proof then entails that the
resulting system is a conservative extension of said subsystem of Q and
so is vastly weaker than ACA_0.

To be clear: I'm not saying Panu doesn't know this; I'm sure he does.
I'm simply explaining why I need clarification.

Richard

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Richard G Heck, Jr
Professor of Philosophy
Brown University
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