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

On 16 Jul 2005 18:44:51 -0700, "Keith Ramsay" <kramsay@xxxxxxx> wrote:

>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 [i] someone believes
>|that set theory is inconsistent is more remarkable than
>|saying that [ii] 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.

Sounds like you're justiifying the idea that [i] and [ii]
are not the same. I doubt that anyone doubts that - I was
just asking whether you meant to be asserting that
[i] is more remarkable than [ii]. I gather the answer is yes...

>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


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

David C. Ullrich
.

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