[FOM] Choice of new axioms 1

[Harvey Friedman]

On 2/2/06 2:42 PM, "Martin Davis" <martin at eipye.com> wrote:

> In a major address given some years ago in Champaign-Urbana, Harvey
> disparaged the elaborate picture of  the projective set hierarchy obtained
> by assuming projective determinacy as compared with the much simpler view
> under V=L. The only argument he gave for this was that mathematicians
> preferred the simpler view.
> 
> In his recent interchange with Arnon, Harvey explained carefully his
> reasons for soliciting impressions among "core" mathematicians of results
> he obtains as consequences of large cardinal axioms. He specifically
> discounted Arnon's suggestion that he was doing this to obtain a "kosher"
> stamp for his results.
> 
> In a private email to Harvey, I asked him whether this indicated a change
> of mind on his part, and he suggested that I raise the matter on FOM.
> 

I think I have maintained a consistent view on this. But the view is, and
always has been, dynamic.

In particular, what new axioms are to be preferred to others, or perhaps the
choice of adding no new axioms, may evolve over time depending on new
events. 

In this posting I will consider only the situation *AT PRESENT*,

***BEFORE ANY RECENT RESULTS OF MINE AND ITS EXPECTED IMPROVEMENTS AND
EXPANSIONS BECOME FULLY AVAILABLE TO THE MATHEMATICAL COMMUNITY AND ENOUGH
TIME HAS PASSED FOR THE PATTERN OF CONTINUING DEVELOPMENTS TO BE ABSORBED***

This could very substantially change the picture.

I will take this matter up under a posting in the future.

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

Let's start with general agreement that ZFC is currently the gold standard.

I personally discussed this gold standard with one of the famous principal
editors of the Annals of Mathematics (at the time: they rotate), who would
undoubtedly be actively involved if an issue concerning the acceptability of
a proof cropped up in the Annals of Mathematics related to the choice of
axioms for mathematics.

This is a multidimensional issue, where several important distinctions must
be drawn in order to have a profitable discussion.

1. Firstly, we may be talking about whether certain new axioms should be
adopted, or would be adopted. There is a difference.

2. Secondly, should be or would be adopted by who? We could be talking about
the set theory community, the f.o.m. community, the (wider) logic community,
or the mathematical community. Perhaps it is relevant to consider
subdivisions of the mathematical community. Right now, the mathematical
community is not divided on this - the de facto gold standard is ZFC.

When I addressed this issue in public, as Davis mentions, I was thinking
entirely of 

what the mathematical community would accept.

My views were as follows.

3. There was at present no compelling reasons yet put forth to the
mathematical community for expanding the current gold standard of ZFC.

4. I then expressed my opinion that if the mathematical community, or a
large subgroup thereof, would want to expand, then it would be because there
was a desire to be able to settle some of the more well known open questions
in set theory that they are readily familiar with.

5. Under this hypothetical, V = L has tremendous well known advantages to
any other proposal, which I will discuss below.

6. But of course V = L also has a perceived strong disadvantage by set
theorists, and some other logicians. Most specifically, a certain set of
questions are answered "the wrong way".

7. In addition, some interesting rationale is given that they are answered
by V = L in "the wrong way" that is rather involved, but argued to be
fundamental from a realist or Platonist viewpoint on set theory.

8. However, in terms of

what the mathematical community would accept

several factors make 6,7 rather weak.

9. One is that the realist or Platonist viewpoint is not particularly
appealing for the mathematical community when it comes to arbitrary sets of
reals, or even "complicated" sets of reals, and higher. Ditto for arbitrary
sets of countable ordinals and higher.

10. The reason is that after a brief flirtation with arbitrary and also
complicated sets of reals and higher, there was a very sharp move to
concreteness. In particular, the discrete and the continuous (on complete
separable metric spaces).

11. Actually, continuity understates this sharp move. After an interesting
discussion with one of the most famous analysts in the world, it became
clear that even analyticity is old fashioned, and one looks to specific more
special families associated with differential equations, and the like.

12. Borel measurability is still taught and certainly there are some
substantial analysts who work with Borel measurable sets and functions on
complete separable metric spaces. But the number of such is now way way down
over the years, and it is even smaller if you are looking for them to really
make use of the generality of Borel (even level 2) in the way that
descriptive set theorists do every day.

13. A justification for the sharp move away from Borel measurable functions
- even Baire classes 1 and 2 - is that it completely destroys
dimensionality, and in fact just about any topological or geometric or
analytic features. 

14. And whatever reasons have drawn mathematicians away from the Borel, have
drawn mathematicians even more strongly away from the non Borel - the higher
projective hierarchy - and certainly from the arbitrary.

15. So if you tell mathematicians who are really concerned about "inherently
open problems in mathematics" that there is one axiom that (just about)
settles all of the known problems involving sets of real numbers that have
been naturally proposed, and that has a certain overriding simple idea, and
that it is known not to lead to inconsistency, then they will jump at it.

16. Sure there are drawbacks. In order to grasp it, and its simplicity, one
has to talk about inductively generated sets and its extreme robustness,
etc. The fact that the real numbers have to be inductively generated, rather
than "just appear automatically" has been used as an argument against it by
set theorists. However, I don't see this as having that much force with the
relevant mathematicians (if they exist), since there are good defenses.

17. Clearly V = L looks like an axiom - and perhaps even more convincing for
some, it looks like a *convention* that we will only consider mathematical
objects that we can generate. After all, the idea of paying special
attention to just the objects that we can generate or get our hands on is
commonplace in mathematics. E.g., the algebraic elements of a field, the
elements of a finitely generated, or finitely presented, group, etc.

18. Of course, PD doesn't even pretend to look like an axiom or like a
convention. It is something that gets "justified" according to a rather
elaborate story that starts off with taking complicated sets of reals and
even arbitrary sets of reals as principal objects of mathematical study.
This goes against the move towards concreteness in mathematics.

19. In addition, there are dangers to it in the sense that its consistency
is enormously transcendental to the consistency of the current gold
standard, ZFC. The current story supporting its consistency is at present
very esoteric at best, for the mathematical community.

20. Furthermore, PD also does not touch the continuum hypothesis and all of
these related problems about sets of reals.

21. Item 20 is defended very strongly by set theorists asserting that an
additional axiom can be added to remedy 19 in a way that is analogous to V =
L in several important respects. But the analogy is too subtle and involved
compared to V = L, for the mathematical community.

22. In summary, V = L has immediate advantages. PD also has some advantages,
but they would not counteract the advantages of V = L for the mathematical
community under present circumstances. They do, however, more than fully
counteract the advantages of V = L for set theorists and many logicians, as
they are quite focused on, say, complicated and arbitrary sets of reals, and
sets of countable ordinals, etcetera. The fact that these mathematical
objects do not interact (much) with modern arithmetic, modern algebra,
modern geometry, modern topology, and modern analysis, is not of great
concern to set theorists and many logicians.

Harvey Friedman