[FOM] Projective Determinacy and other topics: A reply to Harvey Friedman
Dmytro Taranovsky
dmytro at mit.edu
Tue Oct 28 23:21:13 EST 2003
I am grateful that you took the time to analyze my postings; below is
the reply to some of your comments.
In criticizing the consideration of projective determinacy as an axiom,
you wrote
>The notion of axiom you use here is completely nonstandard.
>Axiom normally refers to something that is self evident, and whose
>meaning is at least intuitively clear.
Unfortunately, solving the incompletenesses of set theory requires
statements that are much less self-evident than 2+2=4, so in set theory,
axiom is used in a broader sense as an assertion that (is supposed to
be) intuitively true but not provable in the base theory, most often
ZFC. It is also used to mean an assertion that is true in some
interesting structures and which is used as a starting point to study
the structures, e.g. axioms of abelian groups.
>Identifying second order arithmetic with the study of real numbers is
>rather unusual mathematically. Only a set theorist would make an
>identification like that.
Real numbers under addition, multiplication, and sine have exactly the
same expressive power as the standard second order arithmetic.
>I question that even professional set theorists think that PD is
>of fundamental importance in mathematics.
Just like the induction axiom scheme gives a true canonical theory of
first order arithmetic, projective determinacy gives a true canonical
theory of second order arithmetic. Whether this is of fundamental
importance is a question of debate.
>> Second, projective determinacy is an existence axiom:
>> Theorem: Projective Determinacy is true if and only if for every real
>> number z and natural number n, there is an iterable transitive model
>> that contains z and n Woodin cardinals.
>That is quite a definition of an existence axiom!! We have come a long
>way from such set existence axioms as the axiom of pairing, etc.!!!
We have, indeed. ZFC includes the complicated axiom scheme of
replacement, as well as the axiom of choice. One can get only so far
with simple existence axioms; complex axioms are needed to claim
existence of enough complicated infinite sets. Informally, projective
determinacy simply asserts that there are enough complicated sets to
code winning strategies for difficult games. Such sets are not
constructible, but winning (or drawing) strategies in the game of chess
do not appear to be constructible by computers either.
>>The proofs from determinacy are much more natural than
>> proofs from V=L.
>Of course you recognize that V = L not only answers all of these
>questions,
>many in the negative, but also answers continuum hypothesis, and all of
>those other related pesky set theoretic questions that have proved so
>difficult for the Platonists.
Not all of the questions; after all, V=L does not imply consistency of
ZFC. For sufficiently concrete questions, i.e. Sigma-1-2 questions,
projective determinacy is much more productive than V=L, which is
Sigma-1-2 conservative over ZFC. Projective determinacy can viewed as a
recursive set of statements about real numbers, and as such is not
supposed to solve the Continuum Hypothesis.
>So you say "much more natural" [proofs from PD than from V=L]. However
>you should also say that they are "much more difficult".
Some proofs from PD, like the proof that every uncountable projective
set has a perfect subset, are quite easy. Proofs from determinacy are
valued because they require mathematical insight rather than, as is the
case with V=L, complicated combinatorial machinery. Proofs from V=L are
easy when V=L trivializes the questions: Delta-1-2 well-ordering of the
reals trivializes most questions about definability. Similarly, it is
much easier to "answer" whether ZFC is consistent in PA + not con(ZF)
than in PA + Con(ZF) because the "axiom" not Con(ZF) trivializes such
questions.
>I close with a comment about V = L. V = L is not best construed as an
>axiom. Rather it is best construed as an announcement of intent.
Studies of the constructible universe are certainly a rich source of
ideas. However, mathematicians should refrain from doing ordinary work
in V=L.
If one cares about problems that are not concrete, then V=L provides
false and arbitrary solutions, and hence unacceptable. If one does not
care about such problems then it does not matter whether the theory
decides them; the fact that ZFC does not solve the continuum hypothesis
does not impede the study of differential equations. By contrast,
projective determinacy does help solving concrete questions; and it is
certainly more natural to work in ZFC+PD than in ZF + V=L + (ZF+PD is
Sigma-1-2 correct).
> I [an imaginary defender of V=L] have put a sign on my office door
>that reads: I use only constructible sets.
That sign is most likely false. For example, an ellipse is not a
constructible set since it is an uncountable set of pairs of real
numbers. ZF+V=L is conservative over ZFC for most important statements
about ellipses; however, absent a reason, one should refrain from
working in a false theory. Although a finitist may prefer to work in
WKL0 rather than PRA to obtain more intuitive proofs of Pi-0-2
statements, V=L does not appear illuminating about concrete statements.
Finally, the choice of ZF+V=L as a theory to work in is arbitrary; why
not ZF+V=L+(there is no transitive set model of ZFC), or perhaps
ZF+V=L[0#]?
-------------------------------------------
I will close this posting with some comments on Platonism and formalism.
>>It is difficult to even think about how to derive the
>>consequences of ZFC without invoking the notion of a set.
>That doesn't stop my correspondent from saying he is playing nice
>games.
While he may say that he is just playing games with feasible strings of
symbols, his thought process invokes real numbers and infinite series
and the explanations he gives of the games refer to infinite objects.
The existence of mathematical objects, even uncountable sets, appears to
be true at an almost logical level: Why cannot one iterate the
successor operation unboundedly? Why cannot one take all sets of
integers and collect them into a set? Why cannot one take all sets of
real numbers and collect them into a set?
If a philosopher argues that the only objects that exists are human
minds and a supreme being who manipulates their feelings to deceive them
into believing the existence of some physical world, then to prevail the
philosopher must give an extremely persuasive justification. Similarly,
(but to a lesser extent) in Platonism/formalism disputes, the heavy
burden of proof is on the formalist, at least if the formalist, unlike
Sazonov, denies any sense of existence to mathematical objects.
However, there is absolutely no evidence for the non-existence of the
least weakly compact cardinal. To be sure, important incompletenesses
exist in set theory, but the amount of incompletenesses is certainly
less than what one would expect from the fact that human beings neither
observe nor interact with mathematical objects. Epistemically, it is
possible that someday ZFC will be found inconsistent, but it appears
more likely that it will be found that dinosaurs have never existed.
In addition, there is strong evidence (arguments 5B and 5C in my posting
"Points of View in Philosophy of Mathematics", as well as similar
arguments by Martin Davis and others) for existence of real numbers.
-----------------------------------------------------
Best Wishes,
Dmytro Taranovsky
http://web.mit.edu/dmytro/www/main.htm
More information about the FOM
mailing list