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From: marc.geddes <marc.geddes.domain.name.hidden>

Date: Sun, 30 Aug 2009 18:27:24 -0700 (PDT)

On Aug 31, 4:55 am, Bruno Marchal <marc....domain.name.hidden> wrote:

*> On 30 Aug 2009, at 10:34, marc.geddes wrote:
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*>
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*>
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*>
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*>
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*>
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*>
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*>
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*> > On Aug 30, 7:05 pm, Bruno Marchal <marc....domain.name.hidden> wrote:
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*>
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*> >> This does not make sense.
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*>
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*> > You said;
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*>
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*> >> The truth of Gödel sentences are formally trivial.
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*> >> The process of finding out its own Gödel sentence is
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*> > mechanical.
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*> >> The diagonilization is constructive. Gödel's
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*> > proof is constructive. That is what Penrose and Lucas are missing
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*> > (notably).
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*>
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*> > This contradicts Godel. The truth of any particular Godel sentence
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*> > cannot be formally determined from within the given particular formal
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*> > system - surely that's what Godel says?
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*>
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*> Not at all. Most theories can formally determined their Gödel
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*> sentences, and even bet on them.
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*> They can use them to transform themselves into more powerful, with
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*> respect to probability, machines, inheriting new Gödel sentences, and
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*> they can iterate this in the constructive transfinite. A very nice
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*> book is the "inexhaustibility" by Torkel Franzen.
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Yes, ok, fair enough, they can formally FIND the Godel sentences, but

can't formally PROVE them, that's what I meant.

*>
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*> Machine can determined their Gödel sentences. They cannot prove them,
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*> but proving is not the only way to know the truth of a proposition.
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*> The fact that G* is decidable shows that a very big set of unprovable
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*> but true sentences can be find by the self-infering machine. The
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*> machine can prove that if those sentences are true, she cannot prove
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*> them, and she can know, every day, that they don't have a proof of
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*> them. They can instinctively believe in some of them, and they can be
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*> aware of some necessity of believing in some other lately.
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Right, they can't formally prove them, and require an additional step

('instinctively believe')mathematical intuition (analogical reasoning)

to actualy believe in them.

*>
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*>
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*>
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*> > The points are addressed in ‘Shadows of The Mind’ (Section 2.6,
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*> > Q6).
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*>
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*> Hmm...
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*>
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*>
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*>
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*> > The point of Penrose/Lucs is that you can only formally determine the
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*> > Godel sentence of a given system from *outside* that system.
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*>
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*> The cute thing is that you can find them by inside. You just can prove
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*> them, unless you take them as new axiom, but then you are another
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*> machine and get some new Godel sentences. Machines can infer that some
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*> arithmetical sentences are "interesting question only". The machine
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*> can see the mystery, when she looks deep enough herself.
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Sorry, see above, I meant to say,

'the point of Penrose/Lucus is that you can only formally PROVE the

Godel senetence of a given systen from *outside* that system'

I accept that the 'machine can see the mystery', but not through any

ordinary reasoning methods. (you need analogical reasoning)

*>
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*> I would say it is very well known, by all logicians, that Penrose and
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*> Lucas reasoning are non valid. A good recent book is Torkel Franzen
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*> "Use and abuse of Gödel's theorem".
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*> Another "classic" is Judson Webb's book.
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*> Ten years before Gödel (and thus 16 years before Church, Turing, ...)
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*> Emil Post has dicovered Church thesis, its consequences in term of
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*> absolutely insoluble problem and relatively undecidable sentences, and
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*> the Gödelian argument against mechanism, and the main error in those
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*> type of argument. Judson Webb has seen the double razor edge feature
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*> of such argument. If you make them rigorous, they flash back and you
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*> help the machines to make their points.
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*>
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*> > We
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*> > cannot determine *our own* Godel sentences formally,
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*>
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*> We can, and this at each level of substitution we would choose. But
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*> higher third person level exists also (higher than the substitution
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*> level) and are close to philosophical paradoxes.
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OK, sorry, I should have said 'we cannot PROVE *our own* Godel

sentences fomally - getting to the 'higher order' levels needs

acccepting Godel sentences as an axiom, and this needs 'mathematical

intuition' at each step (analogies)

*>
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*> AUDA comes from the fact that ,not only machine can determined and
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*> study their Gödel sentences, but they can study how those sentences
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*> determined different geometries according to the points of view which
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*> is taken (cf the eight arithmetical hypostases in AUDA).
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*>
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OK.

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Received on Sun Aug 30 2009 - 18:27:24 PDT

Date: Sun, 30 Aug 2009 18:27:24 -0700 (PDT)

On Aug 31, 4:55 am, Bruno Marchal <marc....domain.name.hidden> wrote:

Yes, ok, fair enough, they can formally FIND the Godel sentences, but

can't formally PROVE them, that's what I meant.

Right, they can't formally prove them, and require an additional step

('instinctively believe')mathematical intuition (analogical reasoning)

to actualy believe in them.

Sorry, see above, I meant to say,

'the point of Penrose/Lucus is that you can only formally PROVE the

Godel senetence of a given systen from *outside* that system'

I accept that the 'machine can see the mystery', but not through any

ordinary reasoning methods. (you need analogical reasoning)

OK, sorry, I should have said 'we cannot PROVE *our own* Godel

sentences fomally - getting to the 'higher order' levels needs

acccepting Godel sentences as an axiom, and this needs 'mathematical

intuition' at each step (analogies)

OK.

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Received on Sun Aug 30 2009 - 18:27:24 PDT

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