[FOM] V = L/crises

Sat Aug 23 22:31:52 EDT 2014

In a debate I had with Maclane some 30 years ago, I pointed to the use
of diamond in Shelah's work on the Whitehead problem.

Kaplansky remarked,  "Now diamond is a mathematical property".

John T. Baldwin
Professor Emeritus
Department of Mathematics, Statistics,
and Computer Science M/C 249
jbaldwin at uic.edu
851 S. Morgan
Chicago IL
60607

On Fri, Aug 22, 2014 at 9:38 PM, Joe Shipman <JoeShipman at aol.com> wrote:

> I disagree that ordinary mathematicians would regard V=L as a good axiom,
> because it cannot even be STATED without bringing in much more set theory
> than most mathematicians are comfortable with.
>
> Can you provide an alternative to V=L, consistent with it, that has
> similar consequences at the level of statements of descriptive set theory,
> but which is recognizably "mathematical"?
>
> -- JS
>
> Sent from my iPhone
>
> On Aug 22, 2014, at 5:45 PM, Harvey Friedman <hmflogic at gmail.com> wrote:
>
> 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
>  content.
>
>
> Harvey Friedman
>
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