[FOM] bug fix/Re: Projective Determinacy and other topics: A reply to Harvey Friedman

[Harvey Friedman]

NOTICE: In my last posting, Less or more news?!, 10/28/03 8:39PM, there
still is, as far as I know, exactly one bug. In Proposition 1 I need to say
that A is nonempty.

***************************************

Reply to Taranovsky. This is the rather conventional and rather well known
and rather intellectually arrogant, I must say, party line that seems only
to be convincing to some, perhaps many, specialists in set theory. I have
already criticized this approach, and many of my points seem to have been
ignored. I will try again.

On 10/28/03 11:21 PM, "Dmytro Taranovsky" <dmytro at MIT.EDU> 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,

Perhaps you should consider the possibility that you have not "solved" the
incompleteness of set theory. From the point of view of the mathematical
community, the last thing they are going to even consider is any kind of
additional axiomatic principle beyond what they intuitively grasp as part as
their natural abilities, UNLESS the reasons for doing so are overwhelming.

They are certainly not even going to consider new axioms - tantamount to
changing the "rules of the game" - solely for the purpose of getting truth
values to what they consider as esoteric nonconcrete questions that seem to
have nothing to do with what they really care about.

The only way the mathematical community would even consider the possibility
of additional axioms is either

i) these axioms are intuitively clear and evident; or
ii) they need to do so, unavoidably.

In particular, they are not going to even consider changing the rules of the
game for any of the purposes you have discussed - unless i) holds.

I spent decades on ii), and make progress more or less continuously, and am
certain that this will eventually force the issue. It will likely take at
least one, perhaps two successors to me, who are as good at this as I am.

If I can resign myself to the hard fact that ii) requires decades of
research - perhaps decades more from here - to get the matter of new axioms
fully on the table as a major issue in the mathematical community, then you
should at least avoid discarding i).

Alternatively, where is your proof that there are no new intuitively clear
and evident axioms to be found?

>so in set theory,
> axiom is used in a broader sense as an assertion that (is supposed to
> be) intuitively true

Intuitively true? The claim that measurable cardinals exist is intuitively
true, as well as far weaker axioms, seems difficult to justify.

Since "intuitively true" has proved too difficult - perhaps impossible - set
theorists have switched to "have beautiful consequences, etc."

>> 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.

Identifying the study of real numbers with the first order theory of
addition, multiplication, and sine is something only a set theorist would
do.
 
>> 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,

This is totally new to me. What on earth does this mean?

>projective determinacy gives a true canonical
> theory of second order arithmetic.

For this, as opposed to the above, I have a better idea of what you mean. Of
course, I haven't really seen any clear account of what it means. V = L also
does this.

Please don't issue arrogant dicta that this or that is false. You have only
standard party line arguments for that, which are not convincing.

>Whether this is of fundamental
> importance is a question of debate.

At the moment, the number of mathematicians who know what the projective
hierarchy is is rather small. So if it is of fundamental importance for
mathematics, then this case has not been made effectively.
> 
>>> 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,

No, replacement is NOT complicated. I have some projects involving simple
schemes, and note that ZF is axiomatized by simple schemes - including
replacement. Instances of replacement may be complicated, but not the
scheme.

> as well as the axiom of choice.  One can get only so far
> with simple existence axioms;

Do you have a proof of this?

>complex axioms are needed to claim
> existence of enough complicated infinite sets.

Do you have a proof of this?

>Informally, projective
> determinacy simply asserts that there are enough complicated sets to
> code winning strategies for difficult games.

Put in this way, this is rather complicated and technical and specialized.
Far too much to be any kind of axiom in the normal sense of the word.

>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 two uses of the word "constructible" seem to be unrelated to each other.

>>> 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.  

That is not a question in the sense that you mean when you say that PD
"settles all questions about reals". Don't change the standards here.

Neither PD or V = L answer all consistency statements, although PD answers
more of them than ZFC + V = L. They do seem to answer the same questions of
the kind you are concerned with "about reals". Also V = L settles many more
statements than PD and large cardinals do of a set theoretic nature.

In fact, does V = L settle every single directly set theoretic problem ever
asked by set theorists acting as set theorists?

>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 what? One can obviously just add to ZFC + V = L, the existence of (even
well founded) structures satisfying various large cardinal axioms.

This process will derive all of the Sigma-1-2 sentences that large cardinals
do, and also solve the continuum hypothesis and perhaps all directly set
theoretic problems ever posed.

NOTE: As I said in my last reply, V = L is not an axiom, but a statement of
intent. 
> 
>> 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.

The proofs that you refer to require set theoretic insight. A mathematician
not working in set theory would not recognize the manipulations involved as
mathematical in any sense that he/she could relate to. That doesn't mean
that it is isn't deep and interesting to logicians. It just means that there
is yet another obstacle to getting mathematicians interested in PD or large
cardinals as axioms, via the projective hierarchy.

>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.

The analogy between V = L and not Con(ZF) makes no sense. V = L is a clear
announcement of intent to simply restrict the construct of mathematical
objects to methods that are more explicit. It is a sensible reaction to the
totally unconvincing state of affairs in pure set theory, where one seems to
run out quickly of any convincing ideas to deal with so many of the
independence results.

Whereas not Con(ZF) is not any announcement of intent - in particular to
restrict the ontology. If anything, it increases the ontology in a
mysterious way. 
> 
>> 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.

They only need to know the background information that if they run into set
theoretic difficulties, then these difficulties are immediately taken care
of. This is rather comforting to people who are not set theorists.

Mathematicians generally have also forgotten what the axioms of ZFC are.
Again the axioms of ZFC are only background information for them. They know
that if there is a logical dispute about the legalities in a proof, then
they can rely on ZFC as a gold standard.

NOTE: Of course, for decades I have been trying to make sure that
mathematicians are as uncomfortable as possible!! This probably can only be
done through explicitly Pi-0-1 sentences.

> If one cares about problems that are not concrete, then V=L provides
> false and arbitrary solutions, and hence unacceptable.

You say they are false. That is not only intellectually arrogant, it is also
irrelevant - since V = L is not an axiom, but a statement of intent.

LIKE THIS: "such and such is true in the universe of constructible sets, and
I don't care about any other kinds of sets".

AND LIKE THIS: 'i only care about people in Europe, and i don't care about
any people elsewhere'.

>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.

But it is you who is trying to get the mathematical community to adopt new
axioms. 

Most common reaction is: I don't need any new axioms, since I don't care
about these problems.

However you might be lucky enough to find people who say

well, for my own research, I don't care about these problems. but they are
there, and it would be nice to know how to eliminate these problems.

And there you have some audience, and with such people, you have to consider
the merits of V = L as a statement of intent.

>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).

It may be more natural to a set theorist, but it doesn't solve all kinds of
questions that ZFC + V = L does. More natural might be certain Borel forms
of PD that I gave here on the FOM some time ago, that are compatible with V
= L and prove a lot of correctness of ZFC + PD.
> 
>> 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.  

Speak for yourself. I may have the usual definition of ellipse, which
ultimately involves "set of natural numbers", but I only mean "constructible
set of natural numbers", and there is no point in telling anybody that since
it never comes up in my proofs.

>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.

Since V = L is taken here to be a statement of intent, ZFC + V = L is NOT a
false theory. I have given you its interpretation.

> 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.

ZFC is very illuminating about concrete statements, and doesn't need any
help from V = L, which takes care of all of the nonconcrete statements.

NOTE: Of course, I am trying to show that concrete statements, even very
attractive Pi-0-1 sentences, are tied up with large cardinals. By the way, I
think soon tied up with the entire large cardinal hierarchy, including
Reinhardt's cardinal over ZF.
> 
> 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#]?

We don't know that 0# exists. Creates more trouble than it's worth?? Also,
ZFC + V = L does so much already.

NOTE: There is a real question, after my program is taken thru all the way
by successors, as to whether any large cardinals are enough to take care of
gorgeous Pi-0-1 sentences. E.g., one should be able to go thru higher and
higher forms of elementary embedding axioms inconsistent with choice.
> 
> -------------------------------------------
> 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.

I think it is quite possible to avoid referring to infinite objects in order
to define these games.
> 
> The existence of mathematical objects, even uncountable sets, appears to
> be true at an almost logical level:

Appears to whom?

>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?

Instead of making declarations, the real issue is to say something new and
more interesting and/or more convincing than people have up to now.
> 
> 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.

There are a lot of mathematicians and scientists who don't believe that, but
regard the notion of arbitrary set of natural numbers as meaningless.

I am under the impression that most scientists are formalists with regard to
mathematics - that it is all a game, that turns out to be useful. I talked
to a Nobel Prize winner in physics who strongly held this view.

As I have said many times on the FOM, I am agnostic. I only am interested in
criticism and defense of various positions.

Your party line needs some criticism.

>Similarly,
> (but to a lesser extent) in Platonism/formalism disputes, the heavy
> burden of proof is on the formalist,

and the heavy burden of proof is on the Platonist.

>at least if the formalist, unlike
> Sazonov, denies any sense of existence to mathematical objects.

There are all sorts of views about which mathematical objects exist and
which are fictions...

> However, there is absolutely no evidence for the non-existence of the
> least weakly compact cardinal.

However, there is absolutely no evidence for the existence of the least
weakly compact cardinal. Show me one. Where is it? Is it in my living room?
Or in your head? Perhaps it's in my basement!

>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.

A little bit goes a long way. At the concrete level, this is true to an
extreme. As one goes further away, into serious set theory, this stops being
true. A formalist interprets this differently than a Platonist.

>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.

There are two ways to look at this. A formalist might say that
inconsistencies are just very hard to find. We discard the inconsistent ones
and we are left with the hard ones. We discarded the full comprehension
axiom. Then we discarded Reinhardt's axiom over ZFC. Now we are simply stuck
with harder inconsistencies.

The Platonist looks at it differently. The cases where we found
inconsistencies are special cases, where we were either in a delusional
state, or we just didn't have a well developed enough theory to even know
what we were dealing with. The Platonist probably feels that the days of
inconsistencies are long over, because what is left is the TRUTH. And as we
get more and more knowledge and lots of structure theory, then we will be
confident that we have the TRUTH. And by soundness, we therefore have
consistency.

Note how the same phenomena can lead to totally different interpretations by
formalists and Platonists.
> 
> 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.

Which real numbers? 0,1?

Harvey Friedman