Re: all the incompleteness proofs are worthless untill...



Marshall wrote:
Aatu Koskensilta wrote:
Marshall wrote:

But thinking about it some more, it really highlights what
"provable" means: derivable from axioms via inference rules.
Everything is relative to the context of a particular formal
system.

Results such as "there exists an uncountable set", "Peano arithmetic
is incomplete", "If T + A |- P then T |- A --> P", the Heine-Borel
theorem and so on are "relative to the context of a particular formal
system"? What system is that?

The device of asking rhetorical questions certainly has its place,
whether in debate or pedagogy or what have you. And clearly
we each have to use whatever techniques suit us, and clearly
rhetorical questions are a good match for your detached
personality. And yet I cannot help but feel that you overuse
the technique. Often the effect is that, having read a post
of yours, I feel somehow like I know *less* than I did before,
because my knowledge hasn't increased any, but
my uncertainty has.

Yes, in matters of mathematical logic I'm a beginner, for
which I do not apologize. Perhaps seeing a connection
between provability and derivability from axioms via inference
rules is altogether risible from your position.

Around here provability is indeed the same thing as
derivability from axioms via inference rules. The most
charitable assumption here is that Aatu wishes you to
reflect on the meaning of the phrase "mathematically
proven". He accepts some things as mathematically true,
and implicitly asks, "Don't you?".

Forgive me if
that was the only conclusion I was able to draw from your
earlier response; it was too terse to admit much else.
Perhaps you would consider the option of informing me
of a more nuanced interpretation?

As to your specific questions, I respond:

"There exists an uncountable set" is a statement relative
to a particular set theory. If a system has, say, the set of
natural numbers, and a power set operation, then it has
an uncountable set. (Probably this statement represents
an incomplete understanding; please feel free to point
that out, if you are also going to supply some clarifying
further information.) If a system has sets that can only
contain truth values, then there are only four possible
sets, so in that system the statement is false.

Quite so.

My understanding is that "Peano arithmetic is incomplete"
can be shown in Peano arithmetic.

The sentence "Peano arithmetic is incomplete" can be
_expressed_ in Peano arithmetic, but it cannot be
proven in Peano arithmetic, if Peano arithmetic
is consistent. What can be proven in Peano arithmetic
is "If PA is consistent, then PA is incomplete".

"Peano arithmetic is incomplete" can be both expressed
and proven in ZFC. So you are correct that the proof of
that sentence is system dependent.

I don't see how it
could be shown in, say, propositional logic, since that
system does not seem expressive enough to represent
Peano arithmetic. But perhaps I am mistaken.

"If T + A |- P then T |- A --> P". Is there some sense in
which that statement is provable? Is it a theorem, and
if so, how was it derived without reference to axioms
or rules of inference?

It is a meta-theorem of classical logic. I am quite
unclear as to what formal system is required to give it
a formal proof.

If it is not *provable,* then I don't
see how it is relevant to my earlier statement. To my
untrained eyes it looks something like a rule of inference
itself, or perhaps a metarule is a better description.
Does it hold in nonmonotonic logic?

As to Heine-Borel I have nothing to say, being unqualified.

It's a theorem of ZFC. I'm sure we could think up some
stupid system in which it's expressible and refutable.
So the proof of Heine-Borel is also system dependent.

Which actually is kind of freeing; we can just study
the systems instead of having to worry about any kind of
absolute truth.

Everyone is of course free to study any random formal system and
mechanically churn out meaningless formulas, but such an activity
would not have much to do with mathematics.

That, sir, is altogether unfair and uncalled for. Nowhere did I
suggest the pursuit of randomness or meaninglessness,

Well, I made a parallel post thet could be construed in this
fashion. Aatu may have been responding in part to that.

and an *honest* assessment of my statement would at least
admit the possibility that when someone at my level speaks
of "the systems" he means the established ones.

What you said in your previous post (quoted above) could,
with all charity, stand some clarification, as could what
you said immediately thereafter:

"Everything is relative to the context of a particular formal
system."

Personally, it was nice to see a sort of "aha" experience
in somebody; it didn't strike me as silly.

For your rebuke to stand, you shall have to demonstrate your
own connection to absolute truth, since it was only the worry
of that that I was disclaiming. Or do you rather intend that we
should *worry* about it merely, despite not being able to do
anything about it?

Marshall

Personally, I'm willing to take logical validity as being as
close to a knowledege of absolute truth as the intellect can
aspire to. It is worthwhile to reflect on what one might
mean by, or accept as, "absolute truth".

--
hz
.

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