Why statements in geometry are decidable (in Goedel sense) and in arithmetics not ? After all, are they not 2 languages for expressing same facts ? Does the decidability of geometry means that geometrs are obsolete now ?
> Why statements in geometry are decidable (in Goedel sense) > and in arithmetics not ? After all, are they not 2 languages > for expressing same facts ? Does the decidability of geometry > means that geometrs are obsolete now ?
I suppose that geometry is NOT decidable, since its assertions can be turned into aritmethical ones. Dunno too much, however...
> > Why statements in geometry are decidable (in Goedel sense) > > and in arithmetics not ? After all, are they not 2 languages > > for expressing same facts ? Does the decidability of geometry > > means that geometrs are obsolete now ?
> I suppose that geometry is NOT decidable, since its assertions can be turned > into aritmethical ones. > Dunno too much, however...
Elimination of quantifiers in theory makes real geometry decidable.
The difference with arithmetic is to do with induction ...
> Why statements in geometry are decidable (in Goedel sense) > and in arithmetics not ? After all, are they not 2 languages > for expressing same facts ? Does the decidability of geometry > means that geometrs are obsolete now ?
"Decidable" doesn't even make sense with respect to geometry. So it really only means that mathematarts are late again with their perpetual lamer, whiner, religous excuses:
"The dog ate my geometry homework".
So, it really only implicitly means that we need to go on to the next class, to discover how ignorant the mathematicians are in probabilistic terms.
Since wave functions are a priori obviously tales told by morons.
> > > Why statements in geometry are decidable (in Goedel sense) > > > and in arithmetics not ? After all, are they not 2 languages > > > for expressing same facts ? Does the decidability of geometry > > > means that geometrs are obsolete now ?
> > I suppose that geometry is NOT decidable, since its assertions can be > turned > > into aritmethical ones. > > Dunno too much, however...
> Elimination of quantifiers in theory makes real geometry decidable.
What quantifiers do you mean ?
> The difference with arithmetic is to do with induction ...
> Why statements in geometry are decidable (in Goedel sense) > and in arithmetics not ?
Geometry in general is undecidable. What is decidable are statements in Euclidean geometry that refer to a given finite number of points, lines, planes, angles and distances. That excludes statements about the length of a circle, or about regular n-gons (where n is a variable in the statement, but you can talk about regular figures with any specific number of sides such as triangle, pentagon, octagon, ...)
In <3D8D796F.F5997...@t-online.de>, on 09/22/2002 at 10:04 AM, Ariel Burbaickij <Ariel.Burbaic...@t-online.de> said:
>Why statements in geometry are decidable (in Goedel sense)
Who told you that it was? It isn't.
-- Shmuel (Seymour J.) Metz, SysProg and JOAT Atid/2, Team OS/2, Team PL/I
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> Why statements in geometry are decidable (in Goedel sense) > and in arithmetics not ?
Maybe so. I wouldn't know. But I have never seen a geometric proof that it is impossible to trisect an angle or square a circle. I would imagine that these are true but undecidable within geometry.
> > Why statements in geometry are decidable (in Goedel sense) > > and in arithmetics not ?
> Geometry in general is undecidable. What is decidable are > statements in Euclidean geometry that refer to a given > finite number of points, lines, planes, angles and distances. > That excludes statements about the length of a circle, or > about regular n-gons (where n is a variable in the statement, > but you can talk about regular figures with any specific > number of sides such as triangle, pentagon, octagon, ...)
I do not understand what are you talking about. 1) Why one geometry(i.e. Euclidean) should be prefered over another? What makes it so special?
2)On the one side you say : "What is decidable are statements in Euclidean geometry that refer to a given finite number of points, lines, planes, angles and distances." On the other side : " That excludes statements about the length of a circle, or about regular n-gons (where n is a variable in the statement, but you can talk about regular figures with any specific number of sides such as triangle, pentagon, octagon, ...) But are they not the collection of some finite number of lines, points, angles? So how can it it be that statements about constituting elements are decidable and statements about composed elements not ?
In <3d8e6b49$0$18874$afc38...@news.optusnet.com.au>, on 09/23/2002 at 11:17 AM, "Peter Webb" <pw...@REMOVESPAMopticon-aust.com.au> said:
>But I have never seen a geometric proof that it >is impossible to trisect an angle or square a circle.
And you never will. But you may see a proof that it is impossible to do it with compass and straightedge.
> would imagine that these are >true but undecidable within geometry.
No. Perfectly decidable.
-- Shmuel (Seymour J.) Metz, SysProg and JOAT Atid/2, Team OS/2, Team PL/I
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Ariel Burbaickij wrote: > > Geometry in general is undecidable. What is decidable are > > statements in Euclidean geometry that refer to a given > > finite number of points, lines, planes, angles and distances. > > That excludes statements about the length of a circle, or > > about regular n-gons (where n is a variable in the statement, > > but you can talk about regular figures with any specific > > number of sides such as triangle, pentagon, octagon, ...)
> [...]But are they not the collection of some finite number of lines, > points, > angles? So how can it it be that statements about constituting elements > are decidable and statements about composed elements not ?
Look up "first order theory" and "Tarski quantifier elimination". Slightly more precisely, what is decidable are geometric statements that can be translated into a finite system of polynomial equations and inequalities between real numbers (that is, coordinates of the points plus possible auxiliary variables), together with basic logical vocabulary "for all", "there exists", "and", "or", "not", "implies". In this language you cannot formulate the statement that a regular n-gon has a center, let alone discuss the circumference of a circle as the limit of lengths of inscribed polygons.
> > > Why statements in geometry are decidable (in Goedel sense) > > > and in arithmetics not ? After all, are they not 2 languages > > > for expressing same facts ? Does the decidability of geometry > > > means that geometrs are obsolete now ?
> > I suppose that geometry is NOT decidable, since its assertions can be > turned > > into aritmethical ones. > > Dunno too much, however...
> Elimination of quantifiers in theory makes real geometry decidable.
> The difference with arithmetic is to do with induction ...
> Charles
Elimination of quantifiers makes geometry 'dimensionless'. This is the real geometry. And if we want to make physics 'phenomenological' then we have to eliminate qunatifiers.
> > > Elimination of quantifiers in theory makes real geometry decidable.
> > What quantifiers do you mean ?
> 'Elimination of quantifiers' is a technical term in logic. In the relevant > case I believe the theorem is due to Tarski.
> Mr./Dr./Prof. Burbaickij, I'm sure curiosity is not in itself a bad thing. > But it is unrewarding in some cases to satisfy it.
> Charles
Elimination is not a technical term. I cannot understand whether 'quantifier' is the subject - of a predicate. OR, is 'quantifier' adjective - of quantity?
> "Charles Matthews" wrote > > "Ariel Burbaickij" wrote > > > Charles Matthews wrote:
> > > > Elimination of quantifiers in theory makes real geometry decidable.
> > > What quantifiers do you mean ?
> > 'Elimination of quantifiers' is a technical term in logic. In the relevant > > case I believe the theorem is due to Tarski. > Elimination is not a technical term. I cannot understand whether > 'quantifier' is the subject - of a predicate. OR, is 'quantifier' > adjective - of quantity?
Eliminating - getting rid of - universal and existential quantifiers in first order predicate calculus.
Geometrically the elimination of the existential quantifier says that the projection to R^m of R^n, where m < n and we can take the first m co-ordinates as defining the projection, of a set defined by algebraic equations and inequalities, is a set of the same type.
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