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

I 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.

David C. Ullrich wrote:
|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.

This was taking "doubting the consistency of PA" to mean
"believing 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?

No, you replaced "PA" with "set theory" and "inconsistent"
with "consistent".

Rather than answer your question exactly as stated but
probably not as you meant to ask it ("no"), I thought I
could clear things up by explaining what I meant by
"doubting", which seemed to be the essential problem.

I wrote:
|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.

David C. Ullrich wrote:
|Sounds like you're justiifying the idea that [i] and [ii]
|are not the same.

No, I was telling you which (of [ii] and [iib] below)
I meant, since you'd taken me as meaning [ii] when I
actually meant [iib].

|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...

No. I think you mislabelled them.

[i] Believes that set theory is inconsistent.
[ii] Believes that PA is inconsistent.

[ib] Has doubts about the consistency of set theory.
[iib] Has doubts about PA's consistency.

I initially meant [i] is more remarkable than [iib].
I didn't mean [i] is more remarkable than [ib],
although this is also true. I didn't mean [i] is
more remarkable than [ii] which it obviously isn't.

Keith Ramsay

.

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