Kurt Gödel PROVED that ALL math must be kept within certain parameters
fitz
parameters.
The essential concept in Gödel's proof is that if you cannot see the
entire universe then what you believe are universal math & laws may
only be subset math & rules for your particular subset reference
frame.
This must be exactly what is happening if ALL astronomers in ALL our
universities KNOW Newton was right and gravity must be happening far,
far faster than the speed of light, or this universe could not
possibly be stable, but the entire other half of the scientific
community believes that gravity must be acting at the speed of light.
It IS exactly like this so something must be very, very wrong with
our rules and math if half the scientists believe one thing and the
other half, something entirely different.
Better READ:
http://www.rbduncan.com/schrod.htm
Cheers,
Fitz
Harry Mary Andruschak
> Kurt G�del PROVED that ALL math must be kept within certain
> parameters.
Are you sure? I was under the impression that his work applied to
certain well defined formal systems, but not all math, however that
may be interpreted.
Paul Ciszek
In article <93bd4021-73a7-4445...@c3g2000yqd.googlegroups.com>,
According to _Godel, Escher, Bach: An Eternal Golden Braid_, Godel
proved that *all* formal systems are complete. That for any formal
system, there will be statements for which neither a proof nor a
disproof can be constructed.
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Paul Ciszek
In article <hi6ri5$igf$3...@reader1.panix.com>,
Paul Ciszek <nos...@nospam.com> wrote:
>
>In article <93bd4021-73a7-4445...@c3g2000yqd.googlegroups.com>,
>Harry Mary Andruschak <adopts...@aol.com> wrote:
>>On Jan 7, 3:02�pm, fitz <zeus...@yahoo.com> wrote:
>>> Kurt G�del PROVED that ALL math must be kept within certain
>>> parameters.
>>
>>Are you sure? I was under the impression that his work applied to
>>certain well defined formal systems, but not all math, however that
>>may be interpreted.
>
>According to _Godel, Escher, Bach: An Eternal Golden Braid_, Godel
>proved that *all* formal systems are complete. That for any formal
>system, there will be statements for which neither a proof nor a
>disproof can be constructed.
Ack, that should be "*all* formal systems are INcomplete".
Seth
Paul Ciszek <nos...@nospam.com> wrote:
>In article <hi6ri5$igf$3...@reader1.panix.com>,
>Paul Ciszek <nos...@nospam.com> wrote:
>>According to _Godel, Escher, Bach: An Eternal Golden Braid_, Godel
>>proved that *all* formal systems are complete. That for any formal
>>system, there will be statements for which neither a proof nor a
>>disproof can be constructed.
>
>Ack, that should be "*all* formal systems are INcomplete".
And it isn't true.
Propositional calculus is complete.
All _sufficiently powerful but not too large_ formal systems are
incomplete.
Seth
Keith F. Lynch
> All _sufficiently powerful but not too large_ formal systems are
> incomplete.
Not too large? A large formal system can be both consistent and
complete? Please tell me more.
My understanding is that Goedel's work implied that math is completely
open-ended -- there's always more of it.
(Was I supposed to repair the MIME-garbled subject line?)
--
Keith F. Lynch - http://keithlynch.net/
Please see http://keithlynch.net/email.html before emailing me.
Dan Hoey
> Seth<se...@panix.com> wrote:
>> All _sufficiently powerful but not too large_ formal systems are
>> incomplete.
>
> Not too large? A large formal system can be both consistent and
> complete? Please tell me more.
Assuming a model M of some theory in the "sufficiently powerful"
realm, we could define a formal system whose axioms are exactly
the true statements of M.
Such a system would be "too large" in that it has no finite
axiomization.
Dan
Keith F. Lynch
> Keith F. Lynch wrote:
>> Not too large? A large formal system can be both consistent and
>> complete? Please tell me more.
> Assuming a model M of some theory in the "sufficiently powerful"
> realm, we could define a formal system whose axioms are exactly
> the true statements of M.
> Such a system would be "too large" in that it has no finite
> axiomization.
Thanks. I agree that having an infinite number of axioms is cheating.
Just as is having an infinitely long computer program. (Though at
least such a program wouldn't take up anywhere near as much space as
back in the days of punched cards. No, wait...)
Welcome back to rasff. How did you happen to find this thread?
Butch Malahide
> Dan Hoey <haoy...@aol.com> wrote:
> > Keith F. Lynch wrote:
> >> Not too large? A large formal system can be both consistent and
> >> complete? Please tell me more.
> > Assuming a model M of some theory in the "sufficiently powerful"
> > realm, we could define a formal system whose axioms are exactly
> > the true statements of M.
> > Such a system would be "too large" in that it has no finite
> > axiomization.
>
> Thanks. I agree that having an infinite number of axioms is cheating.
Actually, some very important theories (to which Goedel's
incompleteness theorem most certainly *does* apply) are *not* finitely
axiomatizable; namely, the standard first-order theory for arithmetic
(the principle of mathematical induction is an axiom *schema*
comprising infinitely many individual axioms), and standard set theory
(with axiom schemas for "separation" and "replacement"). The
requirement for Goedel's theorem is not a finitely axiomatizable
theory, but a *recursively* axiomatizable theory. You can take that to
mean that there is an effective procedure for deciding whether a given
sentence is an axiom or not. (Yes, a "recursively enumerable" set of
axioms would also work, but an r.e. set of axioms could be replaced by
a logically equivalent recursive set.)
An example of a theory which is "too large" for Goedel's Theorem is
"True Arithmetic", i.e., the set of all first-order sentences which
are true in the structure (N, +, *); take *all* of those sentences as
"axioms"; obviously any statement can be proved or refuted with a one-
line "proof"; but of course there is no way to recognize the axioms.
Butch Malahide
>
> Assuming a model M of some theory in the "sufficiently powerful"
> realm, we could define a formal system whose axioms are exactly
> the true statements of M.
I assume you mean the true *first-order* statements of M. Let's call
that Th(M), the (first-order) theory of M.
> Such a system would be "too large" in that it has no finite
> axiomization.
Not necessarily; it depends on what M is.
If M = (N, +, *), the natural numbers with addition and multiplication
(and of course the equality relation), then yes, Th(M) is not finitely
axiomatizable, and is undecidable.
If M = (N, +), the natural numbers with addition but without
multiplication (the so-called Presburger arithmetic), then Th(M) is
not finitely axiomatizable, but the theory is decidable. I.e., one can
define an effective proof procedure, so that every true statement (and
no false statement) can be proved.
If M = (Q, <), the rational numbers with the usual ordering, then Th
(M) is decidable and even *finitely* axiomatizable; namely, all true
statements are logical consequences of the axioms for a dense linear
order without endpoints.