Una construcción arquitectónica que se eleva vertiginosamente...‏

La ciencia, 

en su práctica cotidiana,

está mucho más cerca del arte 

que de la filosofía.

Cuando analizo 

la demostración que hizo Gödel

para su teorema de indecibilidad, 

no veo un argumento filosófico.

Esta demostración

es una construcción arquitectónica

que se eleva vertiginosamente,

una estructura única y maravillosa

como la catedral de Chartres.

Gödel tomó los axiomas matemáticos

formalizados de Hilbert como piezas

con las que montar su edificio

y construyó con ellos

una elevada estructura de ideas

en la que pudo finalmente insertar

su proposición aritmética indecidible

como clave del arco.

La demostración del teorema

es una grandiosa obra de arte.

Se trata de una construcción,

no de una reducción.

Arruinó el sueño de Hilbert

de reducir la totalidad de las matemáticas

a unas pocas ecuaciones

y lo sustituyó por un sueño mayor:

las matemáticas como un dominio 

de ideas que crece sin fin.

Gödel demostró que en matemáticas

el todo es siempre mayor

que la suma de las partes.

Toda formalización de las matemáticas

plantea cuestiones que  van más allá

de los limites del formalismo

y se internan en un territorio no explorado...

[The Scientist as Rebel, Freeman Dyson

traducido al castellano como 

El Científico Rebelde

Debate /Random House Mondadori (Barcelona)

Editorial Sudamericana (Buenos] Aires, 2008)

---------------------------------------

Synopsis

Kurt Gödel

[Reprinted in Kurt Gödel  - Das Album/The Album

Karl Sigmund, John Dawson & Kurt Mühlberger

Vieweg  & Sohn Verlag (Wiesbaden, 2006)]

The editors of  Erkenntnis have invited me

to give a synopsis of the results of my paper

«On formally undecidable propositions 

of Principia Mathematica and related systems» (1931),

which has recently appeared in 

Monathshefte für Mathematik und Physik 38,

but was not yet available at the Königsberg conference.

The paper deals with problems of two kinds, namely:

(1) the question of the completeness (decidability)

of formal systems of mathematics;

(2) the question of consistency proofs for such systems.

A formal system 

is said to be complete

if every proposition 

expressible by means of its symbols

is formally decidable from the axioms, 

that is, if for each such proposition A 

there exists a finite chain of inferences, 

proceeding according

to the rules of the logical calculus,

that begins with some of the axioms

and ends with the proposition A

or the proposition not-A.

A system G is said to be complete 

with respect to a certain kind of proposition R

if at least every statement of  R 

is decidable from the axioms of G.

What is shown 

in the paper cited above

is that there is no system

with finitely many axioms 

that is complete

even with respect only

to arithmetically propositions

[under the assumption that no false

(that is, contentually refutable)

arithmetical propositions

are derivable from the axioms

of the system in question.].

Here by "arithmetical propositions"

one has to understand those propositions

in which no notions occur other than

+, •, = (addition, multiplication, identity;

with respect to the natural numbers),

as well as the logical connectives

of the propositional calculus and, finally, 

the universal and existential quantifiers,

restricted to variables whose domains

are the natural numbers.

(In arithmetic propositions, 

therefore, no variables 

other than those

for natural numbers occur.)

Even in systems 

having infinitely many axioms,

there are always 

undecidable arithmetical propositions,

provided the "axiom scheme" satisfies 

certain (very general) requirements.

In particular, it follows from what has been said

that  there are undecidable arithmetical propositions

in all known formal systems of mathematics

-for example, Principia mathematica (including 

the axioms of reducibility, choice and infinity),

the Zermelo-Fraenkel and von Neumann 

axiom systems for set theory, and the

formal systems of Hilbert's school.

Concerning the results on consistency proofs,

it has first to be noted that they have to do

with consistency of the formal (Hilbertian) sense,

i.e. consistency is conceived as a purely

combinatorial property of certain systems of signs

and the corresponding "rules of the game".

Combinatorial facts can, however, 

be expressed in the symbols of mathematical systems

(for example Principia mathematica).

Hence the statement that a certain 

formal system G is consistent 

will often be expressible 

in the symbols of that system itself

(in particular, this holds for

all of the systems mentioned above).

What is shown is the following:

For all formal systems for which the existence 

of undedidable mathematical propositions

was claimed above, the assertion 

of the consistency of the system itself 

belong to the propositions undecidable in that system.

That is, consistency proof 

for one of these systems G

can be carried out 

only by means of inference 

that are not formalized in G itself.

Thus for a system in which all finitary 

(i.e. intuitionistically correct)

forms of proof are formalized,

a finitely consistency proof,

such as the formalists are seeking,

would be altogether impossible.

However, it seems doubtful 

whether one of the systems hitherto set up,

for instance Principia mathematica,

is so all-embracing (or whether such

an all-embracing system exists at all).