FOM: "Axiom" of constructibility

[Harvey Friedman]

Martin Davis Sat, 12 Feb 2000 14:31:43 wrote:

>Maddy's interest in V=L has been along the lines of: how can we use
>mathematical practice to refute it? She's never proposed it as an
>appropriate new axiom.

in response to Joe Shipman 1:00 AM 2/12/00.

Davis is completely right, and Maddy's views can be seen, e.g., from the
following two books:

Former Maddy - Realism in Mathematics,  Oxford : Clarendon Press ; New York
: Oxford University Press, 1990.

Present Maddy - Naturalism in Mathematics, Oxford : Clarendon Press ; New
York : Oxford University Press, 1997.

as well as the many references to her work and other work contained therin.

Feferman's views can be seen, e.g., from

Does Mathematics Need New Axioms?, American Mathematcal Monthly, volume
106, no. 2, pp. 99-111.

PLEASE NOTE!! I have slightly(?) misstaed the title of the panel
discussion. I wrote:

DO WE NEED NEW AXIOMS?

It is actually:

#######################
DOES MATHEMATICS NEED NEW AXIOMS?#
#######################

as in the title of Feferman's paper.

$$$$$$$$$$$

I would like to make a comment on the axiom of constructibility (V = L).

The most sensible way to view the "axiom" of constructibility is not as an
axiom in the usual sense of the word, but rather as a

***clarification.***

This is an important distinction. For many people (not me), implicit in the
use of the phrase:

axiom of constructibility

is the suggestion that there is something compellingly true or even obvious
upon reflection about V = L (i.e., every set is constructible). But that
appears to me to be completely indefensible, and I have never defended such
a viewpoint.

Rather, the axiom of constructibility is a CLARIFICATION of what notion of
set one is talking about.

Of course, the very idea of focusing attention in this way on another
notion of set (the constructible sets) suggests the view that the most
general notion of set - for which contructibility appears merely as a
special restriction with no special foundational status - is somehow
suspect. E.g., that the most general notion of set is deficient in one or
more of the following ways:

1. It is incoherent.
2. It is insufficiently clear to support the much desired appropriate
settling of many of the most basic questions couched in its terms.
3. It makes the underlying foundations of mathematics needlessley and
unproductively general, fraught with weak points that would not be present
under more refined presentations of the foundations of mathematics.

>From this point of view, some crucial issues about the axiom of
constructibility remain:

a. Do the constructible sets correspond to a fundamentally significant
notion of set? After all, the standard presentations of the constructible
sets are technical, and do not prima facie look fundamentally significant.
However, scholars have long been struck by the extreme robustness of the
notion. As a research project, I have previously tried to present the
constructible sets, and axioms about them, in fundamentally significant
ways, with some - but by no means complete - success. The upcoming panel is
a good excuse for me to go back to this effort.

b. To what extent does the move to constructible sets really address 1, 2,
and 3 above? Of course, the most superficial answer that puts V = L into
play from this point of view is that the axiom of choice and the continuum
hypothesis are both appropriately demonstrable when interpreted in the
constructible sets (this is technically closely related, but not identical
to the fact that ZFC + V = L proves the axiom of choice and the continuum
hypothesis).