[FOM] Finite to Infinite

[Richard Kimberly Heck]

On 05/21/2018 01:36 AM, joeshipman at aol.com wrote:
> In his numbered posting 808 from May 1, 2018, Harvey wrote:
>
> "One thing I have worked on, announced in various talks, discussed on
> the FOM, but did not write up proofs, is that, in a certain hard nosed
> and perhaps surprising sense,
>
> ZFC RESULTS FROM TAKING ALL TRUE STATEMENTS ABOUT THE HEREDITARILY
> FINITE SETS OF A CERTAIN NATURAL FORM AND ADDING THE AXIOM OF
> INFINITY.
>
> I.e., ZFC naturally arises from going from the finite to the infinite.
>
> This result(s) go back decades, but I want to revisit this with a
> fresh (older) mind."

I'd like to take the opportunity to express a bit of skepticism about
this project. Not technically, but philosophically. The technical result
may well be very useful and significant. But I'm not at all sure that,
as Joe puts it, such a result might allow us to "put ZFC on a firmer
foundation". The relevant comparison in this case seems to me to be
Zermelo set theory, Z, since it contains an axiom of infinity but is, of
course, far weaker than ZFC. The immediate observation is that Harvey's
'truths about HF of a certain natural form' need to include (enough of)
the axioms of replacement, and also choice.

My difficulty is that both choice and replacement seem to be trivial
when restricted to V_ω. To put it differently: The worries that people
have had about these axioms do not seem to be ones that would arise if
we were restricting attention to V_ω. I hope that seems obvious, but to
give some examples: Whatever I think of full choice, I happen to think
that countable choice is a *lot* more plausible, and I am quite
convinced of finite choice. And the sorts of worries that Boolos
expresses about replacement in "Must We Believe in Set Theory" are
explicitly ones about what happens when we leave the familiar ground of
the finite. Indeed, I think it's fair to say that worries about
replacement and choice have typically been of the form: It's dangerous
(and so unwarranted) to generalize from the finite to the infinite.

To put the worry more directly: It's very unclear to me why the fact
that choice and replacement are true in V_ω should give us *any reason
whatsoever* to think they are true in V. Indeed, in general, that seems
like a very strange inference.

To ask a concrete question, here is a truth about V_ω: If A⊊B, then
there is no 1-1 function from A onto B. I assume this must not be of the
'natural form', but (i) what is that form and (ii) why should we think
that this truth's not being of that form has any epistemological
relevancd? Let me emphasize that I ask this
question...inquisitively...and curiously.

Riki


-- 
-----------------------
Richard Kimberly Heck
Professor of Philosophy
Brown University

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