WWerner DePauli-Schimanovich Higher Cardinals are only a Convention, Ch64-SE11
Abstract: Zermelo’s Axiom of Separation is: Exist x: Forall y: (y in x <==> y in a & E(y)) with definite(E) for a parameter a. Thoralf Skolem suggested to characterize the terminus “definite” by “the property E should be representable by a FOL formula”. But that is trivial. “definite” must mean more. The author claims that “definite” means “in accordance with the theory of definitions of logic”. In this case the theorem of Cantor is no longer a theorem, but a undecidable sentence, and has to be established explicitly as axiom. This is not done by the community, but it is made a silent assumption that we can drop the appendix “definite(E)” from the axiom of separation at all. But this is a convention (even when it is silent) and it is nothing else than an axiom. Paper: At first, Georg Cantor, the inventor of set theory, still believed that “in the night of infinity all cats are grey!” (i.e.“at night, everything is the same color”). He knew that rational numbers are countable. But he could not demonstrate the cardinality of real numbers, and did not consider this question particularly important. Still, in his letter to Richard Dedekind on December 7 th in 1873 , Cantor provided proof for the uncountability of real numbers. The whole real numbers R are equal in cardinality to the interval [0,1] and can be seen as infinite binary fractions between 0 and 1 instead of as infinite decimal fractions. These binary fractions in turn constitute the number of functions of the natural numbers in {0,1}, which is nothing other than the power set (.) of . With this, the basis for generalization was already set, i.e., for Cantor’s theorem : [For all a:] |a| < | (a)|. The cardinality of a is smaller than the cardinality of its power set. For proving the uncountability of real numbers between 0 and 1, Cantor constructs a “diagonal element” from the list of all real numbers between 0 and 1, which is presupposed to exist, and from that he builds an “anti-diagonal element” ADE that also would have to be a real number, but nevertheless cannot be included in the list because of its construction. Here Cantor’s opponents brought a well-known counter-argument – that it is impossible to suddenly construct a new real number out of a closed list of all real numbers, but the new number is not a member of this closed list, in spite of the fact that it would have to be, since this is an implicit presupposition! Naturally, this objection comes from the constructivists. Platonists don’t give it a thought, since they see the list as an actual existing infinite set. This led to the wrong conclusion that the power set axiom should be rejected. But the problem is the axiom of separation (i.e., the partial-set axiom) which is formulated by Ernst Zermelo only for “definite” properties. But what should that mean: a definite property? The author is convinced that it can have only one meaning: “in accordance with the rules of the theory of definitions”. And this means mainly that the definiendum (= result of the definition) is not allowed to appear in the definiens except in recursive definitions. Thoralf Skolem gave another formulation to make the terminus “definite” precise: Definite properties are exactly those unary predicates which can be defined with parameters. Or: Definite(E) : = ∃ sets x , ...x n and a wff ϕ (z, x , ... x n ) such that “t(z) is valid ⇔ ϕ (z, x , ... x n )”. See Ebbinghaus [1994], page 30, or Skolem [1922]. But I would object this view. At least we have to add “… concerning the rules of definition”, because Skolem’s formulation alone constitutes only that the definite property E can be expressed as a predicate logical formula (of 1 st order). But since in logic and in set theory ZF (codified in FOL) all properties can be expressed as predicate logical formulas, Skolem’s formulation of “definite” is a tautology. Therefore it is dropped today in the axiom of separation and also replacement (i.e. image-set axiom). But if we drop it we say implicitely that “definite” has no meaning and that was definitely not Zermelo’s intention of the axiom of separation. Dept. of DB&AI, Technical University Vienna, Favoritenstraße 9, A-1040 Wien, Austria/Europe. [email protected] See Meschkowski [1973] page 16. Meschkowski [1973], page 17, considers this date to be the birthday of set theory. |a| is the cardinality (or power) of a, “<” means “…is of lower power than…”, and is the power set (i.e. set of all subsets) of a given basic set. Sometimes when we want to say it explicitely that “<” means really smaller, we write also “ ≠ <”. For an introduction to axiomatic set theory, see Suppes [1960] or Ebbinghaus [1994]. Juliet Floyd and Akihiro Kanamori write in [2006], page 42 “But what exactly is a “definite” property? This was a central vagary … of the Zermelo Set Theory.”