[FOM] set theory/CH
Harvey Friedman
hmflogic at gmail.com
Thu Aug 21 21:49:26 EDT 2014
On Aug 20, 2014, at 12:24 AM, Harvey Friedman <hmflogic at gmail.com> wrote:
I said that I would agree to the extent that CH research is not a
relatively promising area of research in the foundations of mathematics. Of
course, CH research is, and will continue to be, of interest to some or
many experts in the set theory community. And that, of course, will be
enough for the set theory community to sustain it - as long as there
remains a professional set theory community, which is doubtful in the long
run on the basis of what is going on now. Historically, math sheds areas
that do not substantially interact with other areas. That perhaps would not
be the case if math didn't continually produce striking interactions
between areas. But math is very strong in this respect, and there is no
prospect for math giving up on the deep interaction requirement.
I am wondering what current activities are leading you think this. Recent
activities such as the "Thematic Program on Forcing and its Applications"
at the Fields Institute in 2012 and the workshop on “Logic, Dynamics and
Their Interactions” at UNT indicate that there is a healthy amount of
interaction between set theory, analysis, topology, dynamical systems, and
group theory, and a strong interest among set theorists in building
connections with other areas.
*******************
I think I should have distinguished between descriptive set theory and
"higher" set theory. The former is going to be around for longer than the
latter. A sociological symptom that there is something wrong can be seen by
observing the almost total lack of interest in not only in "higher" set
theory but mathematical problems with any overtly "higher" set theoretic
content. (Mathematicians are willing to use limited doses of "higher" set
theoretic content in the service of proving theorems without such content,
but although this happens, it is quite unusual). I would be interested to
hear about any exceptions among the non logicians at the so called leading
math depts - Stanford, Berkeley, Chicago, MIT, Harvard, Yale, Columbia,
Princeton, IAS, etc. Of course, that's not the actual phenomenon, that's
just the sociological symptom.
As I said, there is significantly more interest in descriptive set theory
than higher set theory, but the interest even in descriptive set theory is
dangerously low.
If you think this is false, or even if you just think that it may be true,
but ought not to be true, why don't you report on a bunch of developments
that you find "striking" or "important" or whatever, and the FOM can have a
discussion. A possible outcome might be that people like me find that there
is more interest in such things than we thought, but I think it likely that
it is still very limited compared to the size of the math community.
Because of the rapidly increasing power of computing technology of various
kinds, mathematics is moving pretty strongly to the concrete. I.e., things
that have real computational meaning that can be put into real practical
action. Current work throughout (the main areas of) math logic is generally
quite weak in this regard. Of course, some math logic is used as a tool in
computer science areas, and that is another story.
Of course, if you follow what I am doing, I found ways to get very concrete
and natural consequences of higher set theory that I am starting to test on
mathematicians after 47 years of effort. Of course, I know of almost no one
in math logic, especially set theory, who pays the slightest bit of
attention to this development, as it doesn't impact their work. Also, as
you might know, I have designed desktop computer experiments whose outcome
can only be predicted using higher set theory. But these developments are
very very far removed from what anybody is doing in math logic, that I am
not even counting it.
E.g., I was in one of the largest math depts 1977-2012, and there was
never, or almost never, any active math faculty who had the slightest
interest in higher set theory, or probably even in descriptive set theory,
other than the math logicians - I was never even asked a single question
about such things (outside the logic group). The size of the Columbus dept
ranged from 60-80.
On Aug 20,2014, Martin Davis wrote:
The negation of CH may be stated as follows:\\
There is a set S of real numbers such that:
1. there is no bijection between S and the set of all real numbers, and
2. there is no bijection between S and any set of natural numbers.
It will be recognized that the concepts involved in making this statement
are all readily acceptable in the mathematical literature. So someone
claiming that CH is inherently vague should be prepared to tell us which if
these concepts are vague as well as how this purported vagueness avoids
contaminating ordinary mathematical discourse.
********
The issue is: arbitrary set of real numbers, and also 1.
Mathematicians have learned that this level of generality causes non
mathematical problems having nothing to do with the purpose of the
mathematics, stemming from the great generality of the notions.
Arbitrary sets of real numbers generally have no computational,
geometric, or analytic meaning. Of course, this level of generality is
quite pretty, attractive, and expositionally useful, IF it does not
cause non mathematical issues. When mathematicians see that it causes
non mathematical issues, then they back off that level of generality
and put conditions on the set and function. E.g., CH stated for Borel
sets and Borel bijections is provable, and that is more than enough
contact with CH for the overwhelming majority of mathematicians. For
this overwhelming majority, they don't lose much real mathematics by
cutting down the sets and functions. They don't lose the arithmetic,
geometry, analysis, computations, etc., that motivated the mathematics
in the first place.
It's Sol Feferman who consistently claims inherent vagueness for "sets of
integers" and "sets of reals". I don't like this way of talking about it
that much. I am perfectly open to the possibility that there could be some
new idea about set theory that allows for absolutely crystal clear
decisions about unresolved problems like CH. However, I regard this as very
unpromising - at least relative to other lines of research. There is Tony
Martin's old often quoted statement in print:
Throughout the latter part of my discussion, I have been assuming a naïve
and
uncritical attitude toward CH. While this is in fact my attitude, I by no
means
wish to dismiss the opposite viewpoint. Those who argue that the concept of
set is
not sufficiently clear to fix the truth-value of CH have a position which
is at
present difficult to assail. As long as no new axiom is found which decides
CH,
their case will continue to grow stronger, and our assertion that the
meaning of
CH is clear will sound more and more empty. (Martin 1976, 90-91)
****
It has been 38 years later, and I wonder what Tony Martin has to say now.
Perhaps the Editors should invite him to comment.
Harvey Friedman
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