Re: Who believes/believed that set theory is/was inconsistent?

David C. Ullrich wrote:
|On 16 Jul 2005 11:07:26 -0700, "Keith Ramsay" <kramsay@xxxxxxx> wrote:
[...]
|>The most respected person I've heard of doubting the
|>consistency of PA is Ed Nelson. He expressed his doubt
|>in his book _Predicative Arithmetic_. He spent some time
|>trying to find an inconsistency.
|>
|>Now, saying that someone believes that set theory is
|>inconsistent is a different (and more remarkable) story.
|
|My first reading of this was that saying someone believes
|that set theory is inconsistent is more remarkable than
|saying that someone believes that PA is inconsistent.
|That seemed like a remarkable thing for you to say...
|
|You actually meant that saying that someone believes that
|set theory is consistent is more remarkable than saying
|that someone has doubts about the consistency of set theory, right?

Under the usage of the verb "to doubt" that I'm accustomed
to (and this is compatible with dictionary definitions),
to doubt something doesn't require belief that it's false,
merely lack of conviction that it's true. So someone who
thinks there's a 1/10 chance of PA's being inconsistent can
be said to doubt that it's consistent, but not to believe
that it's inconsistent.

I've heard people say, "I doubt [X]. In fact, I think it's
not true." This suggests that they're not treating belief
in ~X as necessarily following from doubting X. Native
English speakers have a habit of understatement. Stan
Ulam wrote once that when he spoke in English, he always
felt like he was understating himself. (In French he felt
like he was overstating himself, and Polish of course felt
just right.) Instead of saying we believe something is false,
we'll say we don't believe it's true, and things like that.
The literal meaning is very different however. The fact that
something stronger is intended is merely implied.

On the other hand, saying someone "has doubts" is probably
less liable to cause a misunderstanding if one is not
indicating that they disbelieve.

There are contexts like some religious ones, where
expressions of "doubt" are interpreted very differently
depending on whether the presumption is that one is a
believer doubting the existence of God, say, or a
nonbeliever doubting it. I must say I'm not sure that
this is fair.

In any case, while I remember his expressing doubts about
the consistency of PA, I don't remember his writing
anything that indicates he considers ZF more likely to be
inconsistent than consistent. It could easily be that I
just don't remember it. I'd say that it would be much
more of a "far out" view than merely not being convinced
of the consistency of PA (let alone not being convinced
of the consistency of ZFC or higher cardinals etc.).

Note that one of the things Nelson is most famous for is
his work in "internal set theory", a system usable for
nonstandard analysis, consisting of ZFC augmented with
axioms for "standardness". It's consistent if and only if
ZFC is consistent. It would be odder yet to be doing this
if he had an expectation that it was inconsistent. Not
impossible, of course, merely pretty odd. I mean, if ZFC
turned out to be inconsistent, it wouldn't necessarily
make all that work useless, but it would be a problem.

>>From his home page,

http://www.math.princeton.edu/~nelson/index.html

one can get to a paper of his on "faith in mathematics"
where on page 7 he writes:

I must relate how I lost my faith in Pythagorean numbers.
One morning at the 1976 Summer Meeting of the American
Mathematical Society in Toronto, I woke early. As I lay
meditating about numbers, I felt the momentary overwhelming
presence of one who convicted me of arrogance for my belief
in the real existence of an infinite world of numbers,
leaving me like an infant in a crib reduced to counting on
my fingers. Now I live in a world in which there are no
numbers save those that human beings on occasion construct.

That was a few years before Jim Spriggs heard Mayberry's
remark.

I remember a similar but less impressive experience of my
own. As you know, in constructive mathematics we don't
assume that propositions in general are always either true
or false. One morning, out of the blue, I felt this
thunderous affirmation that the Riemann hypothesis was
definitely either true or false, independent of our knowledge
of which it is. It had no lasting effects, however.

Keith Ramsay

.

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