About existence-as-consistency

Richard Heck richard_heck at brown.edu
Tue Jun 29 02:46:47 EDT 2021

On 6/27/21 2:57 PM, sambin at math.unipd.it wrote:
> I am deeply interested in the historical origin and explanation of the
> principle by which consistency of an axiomatic theory T (typically
> ZFC) is sufficient to  justify it and derive that what it speaks about
> exists (in the case of ZFC, sets satisfying the properties described
> by its axioms). I call this principle: existence-as-consistency,
> shortly EaC.
> I am thinking for instance of the appearance of EaC in Hilbert's
> program in the 1920s. I suspect that the standard model theoretic
> explanation of EaC (by which T is consistent iff it has a model) came
> later.
I think the usual view is that this originates with Hilbert's work on
geometry in the 1890s, and that the Frege-Hilbert correspondence is one
'locus classicus' of the disagreement it generates. Michael Hallett is
one of the best experts on this issue, but I don't know if he subscribes
to FOM. I'm cc'ing him here, but you might write to him directly.

>  A related question is: is there a way to avoid assuming EaC while
> keeping classical logic (and hence validity of LEM)?
Sure, Frege held just such a view, classical logic plus ~EaC. As he says
in the correspondence with Hilbert, either the axiom of parellels (AP)
is true or it is not true, tertium non datur, and that is the end of it.
Frege thinks Hilbert is saying that sometimes AP is true and sometimes
it is false, which he rightly regards as absurd: If AP expresses a
proposition (i.e., is interpreted), then either it's true or it's false,
and EaC makes no sense. And that seems clearly right, unless you're some
kind of relativist. (I enter that caveat because, at least formally,
relativism in the sense explored by John MacFarlane is, at least in
principle, an option here. I'm cc'ing him, too, as he's done wonderful
work on Frege, so may have something to add.)

It's Hilbert's move away from that idea---that there's a determinate
proposition that AP expresses---that is the great break (and
breakthrough). So, at least, says the usual story. There are
complications here, explored by many. I'll just mention Jamie
Tappenden's work as one good source. (I'm also cc'ing him.)

Just to ask the obvious question: Do we really think that Q + ~Con(Q) 
is *justified*? What does it even mean to talk of *uninterpreted*
formulas as 'justified'? If what's meant is that there is a /possible
meaning/ for '0', 'S', '+', and 'x' such that the syntactic forms that
are the axioms of Q + ~Con(Q) would be justified, so understood, then
sure: the completeness theorem establishes that fact. But who ever cared
about that claim?


Richard Kimberly (Riki) Heck
Professor of Philosophy
Brown University

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