30/07/2015 - 23:52 von Rudolf Sponsel | Report spam
Hallo Wolfgang (WM),
wie beurteilst Du das Fundierungsaxiom?
Gruß: Rudolf

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#1 WM
31/07/2015 - 20:31 | Warnen spam
Am Donnerstag, 30. Juli 2015 23:52:27 UTC+2 schrieb Rudolf Sponsel:

Hallo Rudolf,

Ich bin gerade dabei, eine Quellensammlung aller kritischen Aspekte des Transfiniten anzufertigen. Dort steht:
S [S    ( X  S) S  X = ]
No set can be an element of itself. As an example consider the set {X} which has only the element X. X e {X} but X ~e X. This implies the difference between "contained as an element" and "contained as a subset". Always X subset X and { } subset X but never X e X and rarely { } e X.

Da ich den Unterschied "contained as an element" and "contained as a subset" für unnatürlich halte, benötige ich das Axiom nicht.

In 1901 Bertrand Russell devised the first antinomy that became widely known: He constructed the set of all sets that do not contain themselves as an element S = {X | X ~e X}. With unrestricted comprehension and without the axiom of foundation it is possible to define such a set. For instance, the set of all abstract notions is also an abstract notion, hence contains itself as an element. Russell concluded an antinomy. He mentioned this in 1902 in a letter to Frege who in 1879 had devised the axiom of (unrestricted) comprehension. The same antinomy had been found by Zermelo independently.

Stimmen dazu:
This axiom guarantees that every nonempty set S contains an element X which has no element in common with S. This excludes the formation of sets which contain themselves, in order to avoid Russell's antinomy (cp. chapt. III). There is no set that contains a set that contains a set that ... is the set. Such a set would not be contradicted by the other axioms. Since this axiom is not essential for mathematics, it cannot be regarded as fundamental by the traditional axiomatic attitude [A.A. Fraenkel, Y. Bar-HillelL, A. Levy: "Foundations of Set Theory", 2nd ed., North Holland, Amsterdam (1984) p. 89].

Gruß, WM

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