Re: About Consistency in 1st Order Theories.



David C. Ullrich wrote:

On Mon, 19 Dec 2005 02:11:49 GMT, Nam Nguyen <namducnguyen@xxxxxxx>
wrote:



Chris Menzel wrote:


On Sun, 18 Dec 2005 23:06:50 GMT, Nam Nguyen <namducnguyen@xxxxxxx> said:


...


Suppose for discussion ZFC is genuienly inconsistent. Obvisouly
There have been only a finite number of theories that we have proven
so far, or that we could prove as long as humanity could last. Take
any of such theorem F, isn't it possible that the proof of ~F could
be too long for us to *know*? and hence as far as we understand the
nature of humanity, ZFC would exist as an inconsistent theory
without us ever knowing that?

Sure.

"Possible" in the sense of "conceivable", that is. It's consistent with
what we know that there are such contradictions in ZFC. This in itself
of course provides no reason for actually believing there are any.

I'm not quite sure I get what you intended to say here. What did you
really mean by "any" in "...provides no reason for actually believing
there are _any_"? I don't think you meant there isn't any pair F,~F
that are both provable from ZFC, since you stated that "we know there
are such contradictions in ZFC".


You seem to have a reading comprehension problem. 

Why is it that you could ask questions and I couldn't, in a
conversation?



The problem is that you don't seem to be reading what he
writes in reply to your questions - under those circumstances
continuing to reply to new questions seems pointless.

In particular, two things I noticed right away although I
haven't been following this:

(i) you say that he said "we know there are such contradictions in ZFC". He didn't say that. He said "It's consistent with
what we know that there are such contradictions in ZFC."
There's a big difference.

(ii) You claim not to understand what he meant by "any"
in ""...provides no reason for actually believing
there are any"? What he meant was _exactly_ what he said:


The fact that it's consistent with what we know that there
are contradictions in ZFC does not provide any reason for
believing that there _are_ any contradictions in ZFC.


First of all. Both he and you are wrong on this. What he and you
really meant is:

"The fact that it's consistent with what we know *about*
conceivable contradictions in ZFC does not provide any reason for
believing that there _actually are_ any contradictions in ZFC."

The key here is the preposition "about", or "of" of the verb "know",
which has 2 modes: transitive and intransitive.

(1) "I know Titan has life"
(2) "I know about [of] Titan having life"

There is a great difference in these 2 modes. If I were a renown
astro-biologist and stated (1), then the press would take that as a fact
and would have this headline "Beware, we are not alone!". If I only
stated (2), they should probably be careful and not publish that
headline: I probably only knew the rumor/suspicion/discussion *about*
Titan harbouring life - but that should not constitute a fact!

So before he or you accuse me of having "reading comprehension problem",
don't you think that both he and you ought to reflect on what you
exactly meant vs. what you exactly said, here? Don't you think that
in discussing logic, both the object language-expression and the
meta-language-expression have to be *precise*? [Or at least, when asked
one should be prepared to clarify it as a courtesy, instead of getting
mad!]

Secondly, remember that one of my two key questions is

> (2) Is it true then there is *a real danger* that there might exist an
>     inconsistent theory T in the sense that for all the theorems that
>     we - as human being - could conceivably prove, their negation
>     counterparts would have proof lengths that are beyond human capacity
>     to comprehend?

and remember that I clarified the question by *hypothetically
"supposing" ZFC is inconsistent?

So what happens to my key question about "a real danger"? I'm sorry:
I already supposed ZFC be genuinely inconsistent; so that meant his
"conclusion" about there is no reason to believe there are any contradiction in ZFC is a bit off of my expectation for an answer
about the "danger"! I made a tacit silence and not asked him whether
he actually understood the main questions and the main theme of my op.
Perhaps he should realize that more than accusing me when I simply
asked for a clarification.

I had the same problem when I was trying to discuss something
with you a little while ago - I'd say exactly what I meant,
in perfectly clear language, and you'd start speculating
about what I meant, not seeming to consider the possibility
that what I meant was exactly what I'd said. It happens a
lot that when _you_ write something here it's not at all
clear what you mean, because if we take what you say
literally it doesn't make any sense. But you shouldn't
necessarily assume that that applies to everyone else -
there _are_ people who do manage to say exactly what they
mean most of the time.


I don't agree: the precision in mathematics and logic requires
us to say exactly what we mean *all the time*. If we have to be
informal in order to make the conversation easy then that's fine.
But that doesn't mean we could abandon the courtesy of clarifying
things upon request. Humans (experts or otherwise) do have typo's,
misreading, overlook, mistakes... Naturally.

Or perhaps you write talking point memos for the Republican party.




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

David C. Ullrich 

--
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Time passes, there is no way we can hold it back.
Why then do thoughts linger, long after everything
else is gone?
                                              Ryokan
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