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Las escuelas de pensamiento de Berlín y
Viena aportaron mucho en el desarrollo de las ciencias matemáticas, en el
periodo entre las dos grandes guerras del siglo XX. El neo-positivismo como
corriente filosófica, en su versión de la filosofía analítica, dejó su profunda
huella en todos los ámbitos del pensamiento científico hasta nuestros días.
El Círculo de Viena estaba formado por
filósofos, físicos y matemáticos, entre otros: Rudolf Carnap (1891-1970), Hans
Hahn (1879-1934) Moritz Schlick (1882-1936),Friedrich Waismann (1896-1959) Otto
Neurath (1882-1945) y el muy joven Goedel, discípulo de Hahn. Por cierto este
matemático es muy conocido por sus aportes al análisis funcional: el teorema de
Hahn- Banach en sus diversas formas, es parte integral de cualquier curso básico
de análisis funcional.
A este grupo de personas las influyó mucho Ludwig
Wittgenstein (1889-1951) y su precursor Ernest Mach (1838-1910).
Con el asesinato del Prof, Moritz
Schlick a manos de su antiguo alumno, ahora nazi: Dr. Hans Nelboeeck, se inicia
el fin del Círculo de Viena. Este asesinato brutal de un académico en el campus
por razones raciales era un presagio de lo que vendría pocos años después.
Muchas de las “eminencias académicas” de
Viena ignoraron las componentes políticas del asesinato y trataron de muchas
maneras de presentarlo como un acto demencial.
Desafortunadamente los documentos se
encuentran en alemán, la referencia en la red es:
http://zeit1.uibk.ac.at/quellen/stadler3.htm para quienes deseen leer la
ignorancia de los sabios.
Este asesinato conmocionó profundamente
a Goedel.
En 1929 el Círculo publica su manifiesto
titulado “ Concepción Científica del Mundo” en el cual definen los puntos
fundamentales de sus creencias. Este idealismo en concordancia con el
pensamiento del Obispo Berkeley pero negando toda forma de metafísica no fue
totalmente del agrado del joven Goedel. El Prof. Manuel Valdivia el 2 de mayo
del 2000 con motivo de su investidura como Doctor Honoris Causa de la
Universidad de Alicante dijo:
... "
Cuenta Gödel que cuando ingresó en la Universidad de Viena, en 1924, su
intención era graduarse en Física, pero al recibir clases del matemático
Phillips Furtwängler, especialista en teoría de números, quedó tan impresionado
que cambió la física por las matemáticas. Un maestro de Gödel, Hans Hahn, al que
conocemos muy bien los que trabajamos en análisis funcional, pues su nombre va
unido al de Stephan Banach, en el llamado teorema de Hahn-Banach, puso en
contacto a Gödel con el famoso Círculo de Viena, entre cuyos componentes
figuraban científicos relevantes, uno de ellos el mismo Hahn. En el año 1926,
cuando Gödel se incorporó a este grupo, se reunían en un seminario de la
Facultad de Matemáticas de la Universidad de Viena. Allí se dedicaban a
construir una filosofía de la ciencia que se conoce ahora con el nombre de
"positivismo lógico" y que sostiene que una afirmación para que tenga sentido ha
de ser verificable por la experiencia física. Esto chocaba, obviamente, con el
platonismo de Gödel. Por otra parte, los componentes del círculo de Viena
eliminaron el concepto de Dios. Entonces Gödel a pesar de que les respetaba
mucho científicamente, como era profundamente religioso, les abandonó.
Gödel afirma que su
platonismo le ayudó mucho en sus investigaciones. Sobre todo en su trabajo sobre
la incompletitud de la aritmética. El hallazgo de Gödel sobre la incompletitud
de todo sistema axiomático que contenga la aritmética elemental es uno de los
más profundos e importantes de la lógica matemática...."
El volumen
en páginas de la obra completa de Goedel cabe en un pequeño
libro, su significado ha llenado y llenará revistas y libros populares.
El teorema de la incompletitud como dice
el prof. Valdivia es de los más importantes resultados de la historia de la
matemática. En un sistema formal axiomático que contenga la aritmética
elemental existe una expresión bien construida –de acuerdo a las reglas
establecidas- que no se puede demostrar si es verdadera o falsa. Goedel publicó
este teorema en 1931 con el título:
“Ueber formal unentscheidbare Saetze der Principia
Mathematica und verwandter Systeme” que en castellano se podría traducir por :”
Sobre los Teoremas formales no decidibles de la Principia Matemática y
Sistemas relacionados“ En este contexto Principa Matemática se refiere a la obra
de Bertrand Russell publicada con ese título en 1910,1912,1913 cuyo coautor fue
Alfred North Whitehead
Veamos una versión de dominio público del célebre artículo
de Goedel en idioma inglés. Usa las notaciones de Russell-Whitehead en la
Principa Matemática.
ON FORMALLY UNDECIDABLE
PROPOSITIONS
OF PRINCIPIA MATHEMATICA AND RELATED
SYSTEMS 11
by Kurt
Gödel, Vienna
1
The
development of mathematics in the direction of greater exactness has–as is well
known–led to large tracts of it becoming formalized, so that proofs can be
carried out according to a few mechanical rules. The most comprehensive formal
systems yet set up are, on the one hand, the system of Principia Mathematica (PM)2
and, on the other, the axiom system for set theory of Zermelo-Fraenkel (later
extended by J. v. Neumann).3
These two systems are so extensive that all methods of proof used in mathematics
today have been formalized in them, i.e. reduced to a few axioms and rules of
inference. It may therefore be surmised that these axioms and rules of inference
are also sufficient to decide all mathematical questions which can in any
way at all be expressed formally in the systems concerned. It is shown below
that this is not the case, and that in both the systems mentioned there are in
fact relatively simple problems in the theory of ordinary whole numbers4
which
[174]
cannot be
decided from the axioms. This situation is not due in some way to the special
nature of the systems set up, but holds for a very extensive class of formal
systems, including, in particular, all those arising from the addition of a
finite number of axioms to the two systems mentioned,5
provided that thereby no false propositions of the kind described in footnote 4
become provable.
Before going
into details, we shall first indicate the main lines of the proof, naturally
without laying claim to exactness. The formulae of a formal system–we restrict
ourselves here to the system PM–are, looked at from outside, finite
series of basic signs (variables, logical constants and brackets or separation
points), and it is easy to state precisely just which series of basic
signs are meaningful formulae and which are not.6
Proofs, from the formal standpoint, are likewise nothing but finite series of
formulae (with certain specifiable characteristics). For metamathematical
purposes it is naturally immaterial what objects are taken as basic signs, and
we propose to use natural numbers7
for them. Accordingly, then, a formula is a finite series of natural numbers,8
and a particular proof-schema is a finite series of finite series of natural
numbers. Metamathematical concepts and propositions thereby become concepts and
propositions concerning natural numbers, or series of them,9
and therefore at least partially expressible in the symbols of the system PM
itself. In particular, it can be shown that the concepts, "formula",
"proof-schema", "provable formula" are definable in the system PM, i.e.
one can give10
a formula F(v) of PM–for example–with one free variable v
(of the type of a series of numbers), such that F(v)–interpreted as to
content–states: v is a provable formula. We now obtain an undecidable
proposition of the system PM, i.e. a proposition A, for which neither A
nor not-A are provable, in the following manner:
[175]
A formula of
PM with just one free variable, and that of the type of the natural
numbers (class of classes), we shall designate a class-sign. We think of
the class-signs as being somehow arranged in a series,11
and denote the n-th one by R(n); and we note that the concept
"class-sign" as well as the ordering relation R are definable in the
system PM. Let
a
be any class-sign; by [a;
n]
we designate that formula which is derived on replacing the free variable in the
class-sign
a
by the sign for the natural number n. The three-term relation x = [y;
z] also proves to be definable in PM. We now define a class K
of natural numbers, as follows:
n
Î
K ş
~(Bew
[R(n); n])11a (1)
(where Bew
x means: x is a provable formula). Since the concepts which appear in
the definiens are all definable in PM, so too is the concept K
which is constituted from them, i.e. there is a class-sign S,12
such that the formula [S; n]–interpreted as to its content–states that
the natural number n belongs to K. S, being a class-sign,
is identical with some determinate R(q), i.e.
S = R(q)
holds for
some determinate natural number q. We now show that the proposition [R(q);
q]13
is undecidable in PM. For supposing the proposition [R(q); q] were
provable, it would also be correct; but that means, as has been said, that q
would belong to K, i.e. according to (1),
~(Bew
[R(q); q])
would hold good, in contradiction to our initial assumption. If, on the
contrary, the negation of [R(q); q] were provable, then
~(n
Î
K),
i.e. Bew [R(q); q] would hold good. [R(q); q] would thus be
provable at the same time as its negation, which again is impossible.
The analogy
between this result and Richard's antinomy leaps to the eye; there is also a
close relationship with the "liar" antinomy,14
since the undecidable proposition [R(q); q] states precisely that q
belongs to K, i.e. according to (1), that [R(q); q] is not
provable. We are therefore confronted with a proposition which asserts its own
unprovability.15
The method of proof just exhibited can clearly
[176]
be applied to
every formal system having the following features: firstly, interpreted as to
content, it disposes of sufficient means of expression to define the concepts
occurring in the above argument (in particular the concept "provable formula");
secondly, every provable formula in it is also correct as regards content. The
exact statement of the above proof, which now follows, will have among others
the task of substituting for the second of these assumptions a purely formal and
much weaker one.
From the
remark that [R(q); q] asserts its own unprovability, it follows at once
that [R(q); q] is correct, since [R(q); q] is certainly unprovable
(because undecidable). So the proposition which is undecidable in the system
PM yet turns out to be decided by metamathematical considerations. The
close analysis of this remarkable circumstance leads to surprising results
concerning proofs of consistency of formal systems, which are dealt with in more
detail in Section 4 (Proposition
XI).
1
Cf. the summary of the results of this work, published in Anzeiger der Akad.
d. Wiss. in Wien (math.-naturw. Kl.) 1930, No. 19.
2
A. Whitehead and B. Russell, Principia Mathematica, 2nd edition,
Cambridge 1925. In particular, we also reckon among the axioms of PM the axiom
of infinity (in the form: there exist denumerably many individuals), and the
axioms of reducibility and of choice (for all types).
3
Cf. A. Fraenkel, 'Zehn Vorlesungen über die Grundlegung der Mengenlehre',
Wissensch. u. Hyp., Vol. XXXI; J. v. Neumann, 'Die Axiomatisierung der
Mengenlehre', Math. Zeitschr. 27, 1928, Journ. f. reine u. angew.
Math. 154 (1925), 160 (1929). We may note that in order to complete the
formalization, the axioms and rules of inference of the logical calculus must be
added to the axioms of set-theory given in the above-mentioned papers. The
remarks that follow also apply to the formal systems presented in recent years
by D. Hilbert and his colleagues (so far as these have yet been published). Cf.
D. Hilbert, Math. Ann. 88, Abh. aus d. math.
Sem. der Univ.
Hamburg
I (1922), VI (1928); P. Bernays, Math. Ann. 90; J. v. Neumann, Math.
Zeitsehr. 26 (1927); W. Ackermann, Math. Ann. 93.
4
I.e., more precisely, there are undecidable propositions in which, besides the
logical constants
~
(not), Ú
(or), (x) (for all) and = (identical with), there are no other
concepts beyond + (addition) and . (multiplication), both referred
to natural numbers, and where the prefixes (x) can also refer only to
natural numbers.
5
In this connection, only such axioms in PM are counted as distinct as do
not arise from each other purely by change of type.
6
Here and in what follows, we shall always understand the term "formula of PM"
to mean a formula written without abbreviations (i.e. without use of
definitions). Definitions serve only to abridge the written text and are
therefore in principle superfluous.
7
I.e. we map the basic signs in one-to-one fashion on the natural numbers (as is
actually done on [179]).
8
I.e. a covering of a section of the number series by natural numbers. (Numbers
cannot in fact be put into a spatial order.)
9
In other words, the above-described procedure provides an isomorphic image of
the system PM in the domain of arithmetic, and all metamathematical
arguments can equally well be conducted in this isomorphic image. This occurs in
the following outline proof, i.e. "formula", "proposition", "variable", etc.
are always to be understood as the corresponding objects of the isomorphic
image.
10
It would be very simple (though rather laborious) actually to write out this
formula.
11
Perhaps according to the increasing sums of their terms and, for equal sums, in
alphabetical order.
11a
The bar-sign indicates negation. [Replaced with
~.]
so12
Again there is not the slightest difficulty in actually writing out the formula
S.
13
Note that "[R(q); q]" (or–what comes to the same thing–"[S; q]")
is merely a metamathematical description of the undecidable proposition. But as
soon as one has ascertained the formula S, one can naturally also
determine the number q, and thereby effectively write out the undecidable
proposition itself.
14
Every epistemological antinomy can likewise be used for a similar undecidability
proof.
15
In spite of appearances, there is nothing circular about such a proposition,
since it begins by asserting the unprovability of a wholly determinate formula
(namely the q-th in the alphabetical arrangement with a definite
substitution), and only subsequently (and in some way by accident) does it
emerge that this formula is precisely that by which the proposition was itself
expressed.
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