[FOM] What is second order ZFC?

Tue Sep 10 22:03:00 EDT 2013

On 08/31/2013 06:27 PM, Harvey Friedman wrote:
> I have always found that a good way to deal with issues of first order
> versus second order theories is to step back and ask
>
>           what is second order ZFC?
>
> We all agree on what ordinary ZFC is. We can call this first order
> ZFC. We know its syntax, what it means to be an axiom of ZFC, what it
> means to be a proof in ZFC, what it means to be a theorem of ZFC.
>
> I invite those who do not readily agree with what I have said about
> second order theories, to answer this question, and compare it with
> ordinary ZFC.
>
> To make this worthwhile, do not use, without complete explanation,
> phrases like "second order logic".

OK, I'll bite. References to various things I will mention are below.

For what it's worth, I don't care for the way Harvey draws the 
first-order/second-order distinction, but I'll acquiesce in his 
terminology for present purposes.

Second-order ZFC is the theory, formulated in a second-order language, 
whose axioms are the second-order versions of the axioms of first-order 
ZFC. Its theorems are the second-order consequences of those axioms. The 
problem is now to explain "second-order consequence". But, as always, 
consequence is truth-preservation, so what we need to know is how to 
define truth for a second-order language. Since quantification will be 
handled largely as Tarski taught, all we really need to know is what 
"instances" a quantified formula has.

I take it that the point of raising the question as one about ZFC 
specifically is that one typically talks about sets when trying to 
explain what the second-order quantifiers are supposed to mean. But if 
that is the problem, then it is a very old one, and one to which various 
responses have been available for some time now. Indeed, the most 
familiar of these responses, due to George Boolos, was motivated by 
exactly this sort of problem: there is something wrong with appealing to 
set-theory in trying to explain second-order quantification, because the 
intended interpretation of ZFC is one in which the first-order 
quantifers range over all sets; hence, there is no "power set of the 
first-order domain".

Boolos's solution was to appeal to plural quantification, the locus 
classicus for that discussion being his paper "To Be Is To Be a Value of 
a Variable". Boolos insists, plausibly enough, that there are some sets 
none of which is an element of itself, but denies, sensibly enough, that 
there is any set of all the sets that are not elements of themselves. 
The locution "there are some sets" should not, therefore, be regarded as 
quantifying over sets.

Boolos seems to have regarded plural quantification (i) as perfectly 
intelligible in its own right and (ii) as a reasonable way of 
interpreting second-order quantification. It is, in any event, obvious 
that there are expressive equivalences of various kinds here. The 
semantics of plural quantification has since been investigated 
extensively by linguists, and plural logic has since been developed by 
several people (among them, just off the top of my head, Agustín Rayo 
and Byeong-Uk Yi, with relevant work by many others).

Another option is Frege's: regard second-order quantifiers as 
quantifying over what he called "concepts", which are essentially a kind 
of function, not regarded as any kind of set. Frege's explanation of 
what these are appeals to an abstract notion of predication 
(unsaturatedness), which many have regarded as difficult to understand 
but which, in my own view, is actually pretty well understood nowadays. 
On this kind of view, the semantics of a second-order language is itself 
simply to be given in a higher-order language, as in work by Agustín 
Rayo and Gabriel Uzquiano.

There is, of course, an obvious question how it is to be guaranteed that 
the "second-order domain", if we want to speak that way (Boolos did 
not), is as "big" as it is supposed to be. There is literature on this 
question, as well. One option, pursued by Vann McGee (and, in a somewhat 
similar way, by me) is to appeal to a notion of "open-endedness" that is 
also found in some of Sol Feferman's work, though I'm sure Sol would not 
approve of the use we make of it. Here there are more questions than 
answers, no doubt, but the matter is hardly closed or hopeless.

As for Harvey's questions....

> 1. Is CH an axiom of ZFC?

Of course not.

> 2. Is CH an "axiom of second order ZFC"?

No. The only axioms of second-order ZFC are the second-order versions of 
the first-order axioms.

> 3. Is CH a theorem of ZFC?

Not unless ZFC is inconsistent.

> 4. Is CH a "theorem of second order ZFC"? [corrected]

I don't know. Second-order consequence is very hard to decide.

As Harvey has said, there are features of second-order logic, perhaps 
illustrated by (4), that make second-order theories useless for 
foundational purposes *of certain kinds*. There is very nice work by 
Peter Koellner exploring exactly why and how, but without simply 
assuming, as Harvey seems to do, that completeness is non-negotiable. I 
don't myself think completeness is non-negotiable: Partial 
axiomatization is perfectly possible, in many cases, and will do much of 
what we want.

Nonetheless, there are lots of ways in which logic can be used in the 
study of mathematics, mind, and world, and one should not confuse 
unsuitability for one's own purposes with unsuitability tout court. 
Solipsism, after all, is false.

Richard Heck

=====

References:

@ARTICLE{Boolos:ToBeJPhil,
   author = {George Boolos},
   title = {To Be Is To Be a Value of a Variable (Or Some Values of Some 
Variables)},
   journal = {Journal of Philosophy},
   year = {1984},
   volume = {81},
   pages = {430--49}
}

@ARTICLE{RayoUzq:SecondOrder,
   author = {Agust\'in Rayo and Gabriel Uzquiano},
   title = {Toward a Theory of Second-Order Consequence},
   journal = {Notre Dame Journal of Formal Logic},
   year = {1999},
   volume = {40},
   pages = {315--25}
}

@ARTICLE{McGee:Learn,
   author = {Vann McGee},
   title = {How We Learn Mathematical Language},
   journal = {Philosophical Review},
   year = {1997},
   volume = {106},
   pages = {35--68}
}

@ARTICLE{Feferman:Unfolding,
   author = {Solomon Feferman and Thomas Strahm},
   title = {The Unfolding of Non-finitist Arithmetic},
   journal = {Annals of Pure and Applied Logic},
   year = {2000},
   volume = {104},
   pages = {75--96},
   owner = {rgheck},
   timestamp = {2009.03.24}
}

@INCOLLECTION{Heck:LFTFT,
   author = {Richard G. Heck},
   title = {A Logic for {F}rege's {T}heorem},
   pages = {267--96},
   address = {Oxford},
   publisher = {Clarendon Press},
   year = {2011},
   booktitle = {Frege's Theorem}
}

@INCOLLECTION{HeckMay:FuncUnsat,
   author = {Richard G. Heck and Robert May},
   title = {The Function is Unsaturated},
   pages = {825--50},
   address = {Oxford},
   publisher = {Oxford University Press},
   year = {2013},
   editor = {Michael Beaney},
   booktitle = {The Oxford Handbook of the History of Analytic Philosophy}
}

@ARTICLE{Koellner:SecondOrder,
   author = {Peter Koellner},
   title = {Strong Logics of First and Second Order},
   journal = {Bulletin of Symbolic Logic},
   year = {2010},
   volume = {16},
   pages = {1--36}
}

-- 
-----------------------
Richard G Heck Jr
Romeo Elton Professor of Natural Theology
Brown University

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