Re: Goedel - interesting problem?

From: Acme Diagnostics (LFinezapthis_at_partpostmark.net)
Date: 06/11/04 

Date: 11 Jun 2004 07:17:08 -0500

Spike <ismy@friend.arf> wrote:
>"Acme Diagnostics" <LFinezapthis@partpostmark.net> wrote in message
>news:<40c741eb$0$55141$45beb828@newscene.com> the following;

BTW, I'm making good progress with the new and improved presentation
of the Dolan piece on Goedel and again thank all worthy critics. AFAIK
only one issue remains (decider of correctness). Then editing...

> I counted only two criticisms still standing:
>
> 1) "What does it mean for a statement to be true in a set of axioms?"
> This is also the second critic's outstanding question.
> 2) Explain "true" instead of the Gödel's "provable." See Chris's
> 200 line post on Sunday.
>
> Why not just answer these two questions?

Because so far only Goedel experts have asked them, apparently only
those unable to inference the entire first sentence in context (happens
to me too when I'm the expert). They only have the 600X lens whereas
the assumed reader only has the 50X lens. I repeat, when you look through
a telescope at 50X you see *completely different things* than you do at
600X. Saturn becomes a mere pixel in the larger picture. The first
sentence reads:

"...any set of axioms at least as rich as the axioms of arithmetic has
statements which are true in that set of axioms, but cannot be proved
by using that set of axioms."

For the verification of terms, see the second group of appended google
references. The entire sentence would be inferenced in the top level of
explanatory description by most educated laybpersons as (for one
example among myriad variations that say about the same thing:):

"Arithmetic or math, but not exactly, includes true statements that cannot
be proved by it's axioms."

To verify that the sentence is correct, and to make the distinction between
"true" and "provable," google: ["true statements" incompleteness] without
the brackets, and search "true statements" on each page you select.
Excerpts from sample pages are appended.

None of this is to imply that Dolan cannot supply a precise definition of
just "true in a set of axioms" at the 600X level of description. I have no
opinion on that.

>>If any poster can do better in 650 characters, his name will be Dolan.
>
> Post it in a few writer groups. See if that is true.

But they don't have the theoretical understanding. That's what is unique
about Dolan. He has the theoretical understanding (*at least* equal to the
math/ philosophy/ computer dept.) but additionally the editing talent.

But cross-posting to at least one such group might not be a bad
idea if only to increase focus on the editing component.

Larry
``

p.s. Some comments about the example inference, "Arithmetic or math,
but not exactly, includes true statements that cannot be proved by it's axioms."

There are several secondary inferences, but which do not apply to the two
questions you asked. Just to say that there is a lot of implication accomplished
in Dolan's few words. Also note the new level of description below "math
is not self-proving," now with the important concept of "true statements" and
"set of axioms," and "set of axioms" is somehow more precise than just
"math," and these "true statements" that can't be proved apply to "set of
axioms" more specifically than just "math." And this new level of description,
just one step down, is maintained exactly throughout the article. Obviously,
Dolan isn't attempting to teach set theory or define "axioms," but avoiding
that and other cans of worms. Logically, those who would know what an
axiom is would also know what statements are proved. For those who
don't, it doesn't matter. Can't teach that here, other than that "axiom" must
be the most significant technical word or starting point for more detail,
since it's the most technical word used, "set" being second.

"Math" comes from prior context. I think Dolan did a great job of implying
the "not exactly" by just the right amount with his phrasing and terminology
and judging the reader's context, and without using even 1 extra character
to do it over the other things he wanted to say. The second paragraph also
implies another arithmetic or math (not exactly) with "more powerful set
of axioms."

=======================================
Example google sites:

Example sites for meaning of "Arithmetic or math, but not exactly,
includes true statements that cannot be proved by it's axioms." (and similar):

"each of them contains, at any given time, more true statements than it
can possibly prove" "I know a truth that UTM can never utter," Gödel
says. "I know that G is true. UTM is not truly universal."
http://www.miskatonic.org/godel.html

of mathematics from which all and
only true statements can be deduced non-achievable.
http://pespmc1.vub.ac.be/ASC/INCOMP_THEOR.html

"Therefore, the assumption that every true statement is provable must
be rejected." http:
//mulhauser.net/research/tutorials/computability/completeness.html

Gödel showed that there are propositions which are true but which
cannot be proved. The finite methods of proof are unable to reach all
true statements.
http://cs.wwc.edu/~aabyan/CII/BOOK/book/node159.html

t is then perhaps not surprising that the set of axioms obtained by
replacing 7 by 7 cannot be used to prove all the true statements about
N. The surprising thing is that we cannot even prove all the true
statements that can be expressed using the more restricted language.
http://216.239.39.104/search?q=cache:MInKUciPrvcJ:www.mis.coventry.ac.uk/~mtx014/goedel.pdf+%227+by+7

--------------------------------------------------------
Googling for terms in the Dolan article:

"set of axioms" incompleteness 3,180
"axioms of arithmetic"" incompleteness 242
"cannot be proved" incompleteness 2,000
"proved in mathematics" incompleteness 14
"undecidable statements" incompleteness 261
"true statements" incompleteness 1,870
"true statements of arithmetic" incompleteness 60

Example selections, not cherry-picked, but from interesting-looking
links on the beginning google listings from above search words:

"...we can never be completely sure that any reasonable set of axioms
is actually consistent unless we take a *more powerful set of axioms*
on faith." - Department of Mathematics Trent University Peterborough,
Ontario Canada

"Will it mean that every other theorem can be proven or disproven
with this *stronger set of axioms*? Martin Plenio, Imperial College
London

"proves all true statements of arithmetic. But now is a proof of from
..defines the set of Goedel numbers of) true sentences of arithmetic"
http://planetmath.org/encyclopedia/GodelsIncompletenessTheorems.html

"not all true statements of arithmetic are provable from any
effectively given set of axioms" Department of Mathematics, Bristol
Univ., UK

"set of theorems of T will either contain some false arithmetical
statements or not contain some true arithmetical statements." Com Sci
Inst for Adv Comp Studies, U of Maryland

"showed, roughly, that a formal logic must be incapable of proving all
the true statements of arithmetic"
http://216.239.39.104/search?
q=cache:8lzAZlH9P9wJ:www.cs.ucl.ac.uk/staff/R.Hirsch/papers/dialectics/dialectics.ps+%

"set of axioms that proves all such *statements true* in N and refutes
all false ones."
http://www.math.niu.edu/~rusin/known-math/97/goedel

Relevant Pages

  • Re: Goedel - interesting problem?
    ... For obvious reasons, ... >no set of axioms such that we can tell what's an axiom and what isn't. ... If that person rejects "axioms" as the "true statements which are ... the basic logic sense of a logical truth condition. ...
    (sci.logic)
  • Re: Goedel - interesting problem?
    ... >>any given axioms, there are true sentences NOT provable from that set. ... There are two reasons for this. ... for "truth of mathematics". ... there are true statements that are unprovable. ...
    (sci.logic)
  • Re: Godels theorem is uninteresting?
    ... about an S1 in which only interesting true statements are provable ... it is only provable by a proof dependent on the axioms and interesting ... This means interestingness and provability are ... > Do you feel that Godel's Theorems contain self-reference, ...
    (sci.logic)
  • Re: Goedel - interesting problem?
    ... >> will contain true statements that cannot be proved by it's axioms? ... >the set of consequences of some set of axioms, ... The usual language for arithmetic contains the numeral ... other explanation about the math theory. ...
    (sci.logic)
  • Re: Goedel - interesting problem?
    ... >> statements which are true in that set of axioms, ... I just intuitively assumed theorems associated ... I can't think of any intended reader that would help, ... supersede Dolan on all issues except the editing talent. ...
    (sci.logic)
Loading