Le 13-sept.-07, à 19:52, Brent Meeker a écrit :

> A theory also can be contradicted by a fact.  The theory need not be
> contradictory, i.e. capable of proving false, in order to be 
> contradicted.


Yes sure! Actually the second incompleteness theorem (GODEL II) makes 
this remark genuine even just for a simple theory like Peano Arithmetic 
(the escherichia coli of the Lobian Machines).

For example, anyone who knows a bit PA is confident that PA's theorems 
fit the facts (here: the true arithmetical statements). In particular 
everyone believes that PA is consistent. So the proposition "PA is 
consistent" fits the facts (once tranlated into an arithmetical 
proposition like Godel did).. So the proposition "PA is not consistent" 
does not fit the arithmetical facts. But by GODEL II, PA + (as new 
axiom) "PA is not consistent" is consistent (if not PA would be able to 
prove its consistency). It is just that, like in the situation you 
describe, PA cannot prove false the proposition PA is inconsistent, 
which is indeed false.

But then that is why I prefer to say that a machine is sound when it is 
correct relatively to its intended (standard) model, and to keep the 
notion of consistency (and inconsistency, contradiction) in a pure 
syntactical sense: a machine is inconsistent if proves a contradiction. 
Of course Godel "completeness" relates the two notions for the first 
order theories (such a theory has a model (logician's sense, see below) 
iff it is consistent).

For the non-mathematically minded, perhaps this could help: imagine a 
museum in which there is the VENUS DE MILO sculpture, and machine 
describing, "by heart" but accurately,  the VENUS DE MILO sculpture. 
Now, unless you believe there is something contradictory in the reality 
itself of the VENUS DE MILO sculpture, you can understand that the 
machine is consistent and sound (correct). Now take that machine and 
put in front of some other sculpture in the museum. There is no reason 
the machine becomes inconsistent by that move, but the machine become 
unsound, it loses correctness ... relatively to its intended model.

(Note that in the average, when a physician talks on "model", he means 
what a logician calls a "theory". Logicians use the word "model" for 
mathematical structures "satisfying" (= "making true" in some precise 
mathematical sense) formula belonging to a theory (=, in my context, 
provable by a lobian machine). This significant departure in vocabulary 
does not help the dialogs between logicians and physicians 'course.

Those mathematical structures plays the role of (and actually *are*, 
for platonists) the realities *realities*. For example, the natural 
number structure (N, +, x) plays the role of the intended reality for 
the PA theory (mechanical or syntactical generator of theorems).

Bruno

http://iridia.ulb.ac.be/~marchal/


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