[FOM] An argument for V = L
Timothy Y. Chow
tchow at alum.mit.edu
Tue Aug 26 12:17:38 EDT 2014
Here's an argument for V = L, from the point of view of the "ordinary
mathematician," that I haven't seen articulated in quite this way before.
A fundamental and widespread activity in mathematics is to identify the
precise hypotheses needed to arrive at certain desirable conclusions. For
example, a mathematician studying doohickeys might want to know if every
doohickey with Property A necessarily also has Property B. If the answer
turns out to be yes, then a natural followup question will be whether
Property A can be weakened to Property A_0; if the answer is no, then a
natural followup question will be whether Property A can be strengthened
to Property A_1. The motivating idea is that we are seeking to understand
exactly what structural properties of doohickeys cause Property B.
For example, in a slightly fictionalized history of general topology, we
could imagine that continuous functions are initially defined in terms of
epsilons and deltas in real analysis. Later it is recognized that you can
generalize the concept of continuity to metric spaces, and even later it
is recognized that all you need to carry out continuity arguments are the
axioms for a topological space. The mental picture is of a topology as
being a "structure" on an otherwise structureless underlying set. By
stripping away everything but this structure, one has clarified that
certain kinds of properties of real functions depend only on a small
number of abstract properties of the reals; other idiosyncratic properties
of the reals are revealed to be irrelevant for pure continuity arguments.
Now let's look at the trickier example of the axiom of choice. The
Hahn-Banach theorem is a useful tool in functional analysis and therefore
one would like to understand what properties a function space needs to
have in order for the Hahn-Banach theorem to hold. Inspecting the
argument, one finds that in addition to properties that make specific
mention of relevant mathematical structure (linear maps, subadditivity),
one also needs an additional principle---Zorn's lemma, or at least
ultrafilters. On the face of it, this doesn't look like a *property of
linear spaces* per se. The analyst is therefore tempted *not* to phrase
these findings in the form, "the Hahn-Banach theorem holds only for
`choosable' real or complex vector spaces," but instead wants to say that
the Zorn's lemma argument is a (logical?) principle that holds in general,
leaving it out of the list of *structural assumptions* that are needed for
the Hahn-Banach theorem to go through.
Trouble in paradise begins with the following
Unpleasant Surprise: Infinite sets have structure.
This is news that the ordinary mathematician doesn't want to hear. Sets
are supposed to be the structureless substratum upon which mathematical
structure is built. The Unpleasant Surprise means that the mathematician
now has to worry about the distinction between set theory and logic, and
has to decide what kinds of set-theoretical structure to accept. These
are questions that the mathematician doesn't want to think about.
The argument for V = L, then, is that it does a tolerably good job of
making the Unpleasant Surprise go away. For example, confronted with the
axiom of measurable cardinals for the first time, the average
mathematician's instinct is that it is asserting the existence of some
kind of structure on an infinite set, and therefore it makes no sense to
accept the existence of such structure unless it is forced on us by
logical necessity. If anything, the fact that V = L rules out measurable
cardinals is a point in favor of V = L.
Set theorists, of course, have a different attitude. For them, the
Unpleasant Surprise is a Pleasant Surprise. They make their living
studying the structure of infinite sets. They quite reasonably object to
the ostrich-head-in-the-sand attitude of the ordinary mathematician who
wants, irrationally, to deny the existence of structure on infinite sets,
and who sometimes seems to make contradictory demands. For them, V = L is
just one option among many, and they don't see any particular reason to
choose it. Certainly its role as a magic spell to wave away undesirable
structure on infinite sets carries no weight with them. First of all,
they understand that V = L doesn't completely eliminate questions about
set-theoretic structure, and secondly, they don't see anything undesirable
about set-theoretic structure.
FOM readers, of course, will mostly deplore the attitude of the ordinary
mathematician who regards foundational issues as unpleasant things to be
ignored or minimized. However, in my opinion, the honest thing to do is
help the mathematical community understand the extent to which V = L will
allow them to bypass foundational issues. I think that we have not done a
particularly good job of this, primarily because we have vested interests.
Tim
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