[FOM] What is second order ZFC?

Colin McLarty colin.mclarty at case.edu
Wed Sep 11 11:17:01 EDT 2013

```On Tue, Sep 10, 2013 at 10:03 PM, Richard Heck
<richard_heck at brown.edu> responded
to Harvey's challenge to say what is second order ZFC, but did not so much
give an answer as indicate that there are several.

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".....
>
> 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.

Yes, precisely.  There have been various responses around for a very long
time, because none of them can inspire a consensus.  And this is not a
quibble.  It is the whole point of Harvey saying

HF> For any of the usual classical foundational purposes, you need to be
able
HF> to get down to finite representations that are completely non
problematic....
HF>
HF> Furthermore, first order logic is apparently the unique vehicle for such
HF> foundational purposes. (I'm not talking about arbitrary interesting
HF> foundational purposes).

These various versions have been prominently problematic throughout their
existence.  And one common problem has been to tell whether a given version
can produce finite representations of proofs or not.  That problem
is highlighted by Richard's answer to Harvey's specific question

> HF> 4. Is CH a "theorem of second order ZFC"? [corrected]
>
> I don't know. Second-order consequence is very hard to decide.
>

I think this answer is imprecise.  For example, it is not hard to decide CH
by the ZF axioms -- it is provably impossible.  Some definitions of second
order consequence explicitly rest on ZF, so that deciding second order
consequence is not hard, but provably impossible on these definitions.

I do not know the issues of decidability for plural quantification.  Is
there a standard theory of that?  Frege's option, invoked by Richard,
of regarding
second-order quantifiers as quantifying over what Frege called "concepts"
offers no serious prospect of decidability.

And towards the end of his post Richard offers the following as if it
disagreed with Harvey:

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.
>

But Harvey had already said what I quote from him above: "I'm not talking
about arbitrary interesting foundational purposes."  And he is obviously
not talking about other non-foundational purposes.  This is not a point of
disagreement.  Taking it to be a disagreement may be one source
of confusion in this discussion.

Colin

> 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},
>   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},
>   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
>
> Website:   http://rgheck.frege.org/
> Blog:      http://rgheck.blogspot.com/
> Amazon:    http://amazon.com/author/**richardgheckjr<http://amazon.com/author/richardgheckjr>
>
> Check out my books "Reading Frege's Grundgesetze"
> and "Frege's Theorem":
>   http://tinyurl.com/**FregesTheorem <http://tinyurl.com/FregesTheorem>
> or my Amazon author page:
>   amazon.com/author/**richardgheckjr<http://amazon.com/author/richardgheckjr>
>
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