**From:** Marc Geddes (*marc_geddes@yahoo.co.nz*)

**Date:** Wed Aug 18 2004 - 00:20:02 MDT

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--- Eliezer Yudkowsky <sentience@pobox.com> wrote:

*> Marc Geddes wrote:
*

*> >
*

*> > I don't see that an infinity of axioms (reality is
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*> > uncountable) is a problem. We are not limited to
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*> > axiomatic reasoning. Although higher level
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*> > mathematical axioms would not be derivable from
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*> lower
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*> > level mathetical axioms, we can still reason about
*

*> and
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*> > *prove* the higher level axioms using
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*> probabilistic
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*> > reasoning.
*

*>
*

*> IANAM but that sounded to me like nonsense.
*

*>
*

*> --
*

*> Eliezer S. Yudkowsky
*

*> http://intelligence.org/
*

*> Research Fellow, Singularity Institute for
*

*> Artificial Intelligence
*

*>
*

I don't think I was talking gibberish ;) I could

probably have worded it better but what I said still

seems O.K to me.

Let me try to rephrase:

Godel's theorem says that any system of finite axioms

(complex enough at least to incorporate both addition

and multipication) which is consistent, cannot be

complete. There will be some so-called 'undecidable'

truths, in the sense of perfectly sensible

propositions stated in the language of the system

which cannot be proved from within the system.

However are these truths really 'undecidable'? No,

not in any absolute sense. They would be perfectly

decidable from the perspective of a broader

mathematical system - one which contained the old

system of axioms plus some extra ones.

Now, if reality is a system of infinite axioms (which

cannot be finitely specified), then ALL mathematical

truths would in fact be decidable. Every mathematical

truth would be embedded in a broader axiomatic system

than the one needed to state it as a proposition.

All 'undecidable' actually means is that some

mathematical propositions cannot be proved with 100%

certainty. But there is nothing which says we can't

reason that the proposition is 70% likely to be true,

80% likely to be true, 90% likely to be true, or any

degree of certainty less than 100% (we simply deploy

non-axiomatic reasoning and treat mathematics like an

'experimental science' - using computers to perform

'experiments' on mathematical objects).

So you see, this whole notion of something being

'Godel undecidable' has been totally mis-interpreted

by the lay-man.

If a maths proposition is 'Godel undecidable' all that

means is that we can't deploy axiomatic reasoning and

achieve certainty regarding its truth status. But by

deploying non-axiomatic probabilistic reasoning, all

of those so-called 'undecidable' statements are in

fact decidable so long as we are prepared to accept

probabilities less than 100%

I repeat: the terms 'undecidable' and 'uncomputable'

have been totally misunderstood by mathematicians and

lay-men.

All of those so-called 'uncomputable' maths functions

are in fact computable to any degree of accuracy less

than 100% (so we can in fact compute the functions

with 95%, 99%, 99.9% or any degree of accuracy we

desire less than 100%)

Similairly, all of those so-called 'undecidable'

truths in maths are in fact decidable to any

confidence level less than 100% (so we could in fact

produce a non-axiomatic probabilistic argument to

achieve 95%, 99%, 99.9% or any degree of confidence we

desire less than 100%)

Make sense?

=====

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**Next message:**Christian Szegedy: "Re: All is information"**Previous message:**Keith Henson: "Re: Final draft of my philosophical platform now on line"**In reply to:**Eliezer Yudkowsky: "Re: All is information (was: All is number)"**Next in thread:**Christian Szegedy: "Re: All is information"**Reply:**Christian Szegedy: "Re: All is information"**Reply:**Eliezer Yudkowsky: "Re: All is information (was: All is number)"**Reply:**Thomas Buckner: "Re: All is information (was: All is number)"**Messages sorted by:**[ date ] [ thread ] [ subject ] [ author ] [ attachment ]

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