# Das Kalenderblatt 110429

Gödels first incompleteness theorem proves that formal systems T
satisfying "certain conditions" are incomplete, i.e. that there is a
sentence A in the language of the T which can neither be proved, nor
disproved in T. Among the "certain conditions" must be some condition
implying that T is consistent.

Gödel's second incompleteness theorem proves that formal systems T
satisfying certain other conditions "cannot prove their own
consistency", in the sense that a suitable formalization in the
language of T of the statement "T is consistent" cannot be proved in
T. Again one necessary condition is that T is in fact consistent,
since otherwise everything is provable in T.

The second incompleteness theorem applies in particular to those
formal systems that can be used to develop all of the ordinary
mathematics that one finds in textbooks. One such system is the
axiomatic set theory called ZFC. Since all the theorems ordinarily
proved in mathematics can be proved in ZFC {{das ist ein weit
verbreiteter Irrtum; die Mathematik der Lehrbücher hat nur bei
oberflächlicher Betrachtung etwas mit dem zu tun, was ZFC und andere
formale Systeme abzuleiten gestatten}}, and since the consistency of
ZFC cannot be proved in ZFC (unless ZFC is inconsistent), it is often
concluded that we cannot expect to prove, and therefore can't know,
that ZFC is consistent. "We can't know that mathematics is
consistent."

Clearly, for any theory T, there is another theory T' in which "T is
consistent" can be proved. For example, we can trivially define such a
theory T' obtained by adding "T is consistent" as a new axiom to T.
{{Wenn aber die Theorie, in der dieser Konsistenzbeweis erfolgt,
selbst inkonsistent ist? Was ist der "Beweis" dann wert?

Ist es wirklich sinnvoll, von einem "Beweis" zu sprechen, wenn eine
nach logischen Regeln erlaubte Kette von Schlüssen in einem
inkonsistenten System geführt werden kann? Sollte man den Adelsbrief
"Beweis" nicht besser für richtige Beweise reservieren, also für
solche, die unabhängig vom System zu einer wirklich wahren Aussage
führen. Dazu muss die Wahrheit einer Aussage absolut feststellbar
sein. Das ist möglich - durch Vergleich mit der Realität. Dann wäre
allerdings ein "Beweis" wie der berühmte von Zermelo, auch offiziell
nichts mehr wert, denn dass nicht jede überabzählbare Menge
wohlgeordnet werden kann (sofern solche Mengen existieren), ist eine
Tatsache, also eine absolut wahre Aussage - trotz des gegenteiligen
"Beweises".}}

http://www.geometry.net/theorems_and_conjectures/incompleteness_theorem_page_no_2.html

Gruß, WM
.

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