|Paradoxes: are contradictions within formally correct statements or sets of statements that lead to an existence assumption, which initially seemed plausible, to be withdrawn. Paradoxes are not errors, but challenges that may lead to a re-formulation of the prerequisites and assumptions, or to a change in the language, the subject domain, and the logical system. See also Russellian paradox, contradictions, range, consistency.|
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|Thiel I 322
Russell's Antinomy/solution: an attempt to avoid the Russellian paradox would be to say instead of "all" always "all, which". So now the suspicion is centered on "all".
Poincaré saw this suspicion confirmed and claimed:
Conditions such as "~ (x ε x)" are unsuitable to determine a set, for they require a circulus vitiosus. He did not get to this diagnosis with the help of the Russellian antinomy, but with the antinomy constructed by Jules Richard.
Richard's Antinomy: The totality E of all the decimal fractions which can be defined by a finite number of words (from the letters of a finite alphabet). Also, the totality E of the decimal fractions is countable. But then we can define a new decimal fraction d by the rule:
Is the n-th number of the n-th decimal fraction of E
so the the corresponding number of d is
Since, by definition, d differs from the n-th decimal fraction from E at the n-th place, and this applies to an arbitrary n, d is different from every decimal fraction from E, and therefore does not belong to E.
On the other hand, d must lie in E because we have defined it with finitely many words, and E was the totality of all such decimal fractions.
Solution/Poincaré: he generalized the solution provided by Richard himself that E can be correctly explained only as the totality of not all but the decimal fractions which can be defined with a finite number of words without already introducing the concept of the totality E itself.
Burali-Forti/Poincaré: Poincaré transferred this explanation also to other antinomies e.g. the antinomy of Burali-Forti: of the "set Ω of all ordinals". They can only be applied correctly to the set of all ordinals which can be defined without the introduction of the set Ω. (Otherwise, Ω + 1 always results).
Thiel I 324
Poincaré: believed that he had found the decisive criterion: illegitimate, "non-predicative" conditions are those that contain such a circle. (> impredicative, Russell).
At first, it seemed sufficient to demand of expressions the relation between element and set that in "x ∈ y" the second relation term y should belong to exactly one step higher than x (simple > type theory), thus the requirement that every permissible expression should be formed not only "predicatively" (i.e. not impredicatively), but also all arguments occurring in it must satisfy this condition, to lead to a "branched type theory."
VsType Theory: Its complications included not only the fact that such a theory must also consider orders in addition to types, but also the more than annoying fact that now, for example, the upper limit of a non-empty set of real numbers (whose existence is presupposed in all continuity considerations in classical analysis) is of higher order than the real numbers whose upper limit it is.
The consequence is that one can no longer quantify simply via "all real numbers", but only via all real numbers of a certain order. This is unacceptable for the field mathematics, and a huge obstacle to the "arithmetic program" of classical basic research.
Especially for the logicism which follows.
Poincaré's analysis carries even further than he himself presumed.
(1) (1) is wrong
With the variant "the only sentence numbered on this page is wrong". Or in the form of
"I lie (now)".
one accepts the necessary empirical regressions on book pages and "now", this leads to formal contradictions.
The "liar" is weaker, originally in the letter of the apostle Paul to Titus, verse 12 of the first chapter. Luther: Z "One of them always said, their own prophet: the Cretans are always liars, evil beasts, and idle bellies."
A <> "All Cretans lie (always)"
Synonymous with the statement: "for this statement applies: if it is made by a Cretan, its opposite is true."
K(A) > ~A
(>separation rule: A, A > B >> B I 92)
According to the separation rule, the statement ~ A becomes a true statement. This implies, however, that A is false, while we have derived this demand from the assumption that A is true. Since this is only assumed hypothetically, the reasoning (also I 315 Zermelo-Russell's antinomy) shows, with reference to the reductio ad absurdum: (A> A)> A, that A is indeed false.
This does not lead to any formal contradiction, if there is a Cretan who makes at least one single true statement, A is then simply wrong. Nevertheless, Poincaré would dispute the admissibility: the definition of the abbreviation sign A is a universal statement, in which the variability range of the quantifier consists of all propositions, and therefore also contains the statement A itself, A is therefore impredicatively defined and therefore inadmissible.
The applicability of the Poincaré criterion comes unexpectedly because the liar antinomy, due to the occurrence of metalogical terms such as "true" and "false" belongs to another, actually non-mathematical, type of conclusions that Peano classified as "linguistic".
Philosophie und Mathematik Darmstadt 1995