[FOM] An argument for V = L

[Timothy Y. Chow]

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