[FOM] V = L/crises

[Harvey Friedman]

This is an edited version of an email that has been sent by me to another
email list.

Someone wrote that V = L has been rightly rejected by the set theory
community for identifiable mathematical reasons. I responded that V = L has
tremendous merits for the general mathematical community:

I wonder if you would agree that for mathematicians not interested in set
theory or logic, but only arithmetic, geometry, differential equations, and
so forth, and this is the overwhelming majority of mathematicians,

1. They should care about having universal foundations of mathematics. But
the overwhelming majority don't, and the exceptional people don't
understand, and don't bother to become familiar, with the basic material
known about it.
2. For the ones who care, and not interested in set theory or logic, their
very best clear choice for universal foundations has been ZFC + V = L, at
least perhaps up until about now. V = L takes care of all of these annoying
cases for them where there is the appearance of a real mathematical problem
which can be after the fact rejected on the grounds that it has a disguised
set theory component, but rather than get into after the fact
pronouncements about how it was not the right problem after all, and that
one should reformulate it to cut down the generality, etc. the problem can
invariably be completely nuked by ZFC + V = L, period.
3. Only until about now, with perfect Pi01 incompleteness, is this "putting
an end to natural (prima facie) mathematical incompleteness" starting to be
seriously challenged, for the overwhelming majority of mathematicians who
consider set theory and logic not part of interesting mathematics like
arithmetic, geometry, differential equations, etcetera.

Someone asked whether there is a difference between what the community
thinks is important and what's really important. I responded as follows.

In general, the overwhelming majority of the math community has no concept
of "foundational exposition", which is key to making math intelligible to
outsiders, and particular math areas intelligible to other areas. Not a
clue. Turning to the set theory community, and even the wider math logic
community, the search for perfect Pi01 mathematical incompleteness was,
fairly soon after Cohen, regarded as the premiere blockbuster issue by most
of the people in and out of math logic who were struck by Goedel/Cohen on
AxC and CH. Admittedly, most of them were probably not engaged in the issue
of whether this premiere blockbuster issue should be couched in terms of a
pre existing very concrete statement like FLT, RH, and of course lesser but
known problems like that, or whether it should admit "perfect mathematical
Pi01", either as an end in itself or just as a  crucial initial step
towards something fully integrated already in mathematics.

The initial excitement about this prospect waned as there did not appear to
be any way of approaching this. Cohen was even dismissive of his own work
in comparison to this prospect - which he put this way to me: "I figured
out how to add sets to models of set theory. But how do you add integers to
a model of set theory? That's much deeper." I.e., he envisioned the method
for obtaining mathematically natural arithmetic incompleteness (or
obtaining pre existing mathematically natural arithmetic incompleteness) to
be a matter of enlarging a model of set theory, sort of like the enlarging
that he accomplished through forcing. This now still appears hopeless to
this day, and I use another method, tying perfect Pi01 mathematical
incompleteness to large cardinals (equivalence with the consistency of LCs).

Then there was excitement in 1977 again about this in the math logic
community, when Harrington greatly improved on Paris to get the Paris
Harrington incompleteness from FINITE set theory. I remember Jack Silver
saying to me "now we should go for incompleteness from COUNTABLE set
theory", not quite in those words - I think he used the usual equivalent
Z_2 formulation. Over the next few years, there was utter failure with this
for arithmetic sentences. (However, progress much higher up in the Borel
world had already started and continued. Let me not digress).

Probably already by the 1980s, there was a sharp drop in interest in the
math logic and set theory communities generally in "concrete mathematical
incompleteness", and I doubt if "concrete mathematical incompleteness" was
even seriously mentioned as an overwhelmingly crucial issue for f.o.m. to
students. For example, I don't recall being invited to a single meeting in
set theory since the early 1980s - even to report on plans, prospects, and
progress.

Looking at other communities, beyond set theory, beyond math logic, beyond
math, beyond science, I get the distinct impression that they all overlook
"something important or think something is important when it's not". This
of course includes completely wrong headed socially reinforced judgments
about relative importance of issues as well.

I asked: I would like to get your take on whether we are witnessing an
emerging "foundational crisis" or whether this is best viewed as ordinary
business as usual.

Someone responded indicating that it may not be a crisis. That we have
already been forced to face that right and wrong in math is more subtle
than we might prefer, given a choice, and that physicists coped with such
things, and mathematicians will also cope. They said that it might mean
that mathematicians will void areas like higher set theory, but they hoped
that the attractions will keep it alive. I responded as follows:

I see good news from yours (and mine) point of view.

The physicists recovered in spectacular fashion from the qm crisis.
Conventional wisdom there is that - after the fact - certain notions don't
make any sense, and are replaced by other notions that can be handled with
great facility and accuracy using statistics. Randomness is fully
predictable statistically. Rejection of prior notions replacing them with
new notions is very much something that physicists have been greatly
successful with in order to climb out of crises. E.g., this happened all
through special and general relativity. with rejection of absolute space
and time.

But I think the situation with math, assuming perfect Pi01 mathematical
incompleteness proceeds as planned, seems to be different. It is hard to
imagine how mathematicians can replace old notions with new ones to deal
with this. This can be done for set theoretic statements - specifically
replacing "set" by "constructible set", which is attractive if you are
trying to get to the core mathematical issues and don't want them mucked up
by set theoretic and logical issues. However, there does not appear to be
any prospect for doing something like that - changing the rules or the
ontology - to overcome the coming onslaught of perfect Pi01 mathematical
incompleteness.

Of course, what will be available to the math community IS to get engaged
with large cardinals. I think that they will find it much more attractive
to think about the existence of models of large cardinals, since that is
enough to prove Pi01 consequences. Of course there is also perfect Pi02
mathematical incompleteness surely coming, with "large cardinal" growth
rates. So probably they will be compelled to think about the existence of
omega models of large cardinals. There will be a move to extract the finite
combinatorial content of large cardinals into some new combinatorial
principle that has some plausibility argument attached to it which, in
large finite contexts, can be confirmed by computer.

So in contrast to your "lots of mathematicians prefer not to work in areas
like higher set theory", they will be forced to deal with higher set theory
- at least with models of LC, probably omega models of LC, probably with an
effort to extract the essential finite combinatorial
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Harvey Friedman
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