|Set Theory: set theory is the system of rules and axioms, which regulates the formation of sets. The elements are exclusively numbers. Sets contain individual objects, that is, numbers as elements. Furthermore, sets contain sub-sets, that is, again sets of elements. The set of all sub-sets of a set is called the power set. Each set contains the empty set as a subset, but not as an element. The size of sets is called the cardinality. Sets containing the same elements are identical. See also comprehension, comprehension axiom, selection axiom, infinity axiom, couple set axiom, extensionality principle.|
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|P. Lorenzen Ein dialogisches Konstruktivitätskriterium (1959) in Karel Berka/L. Kreiser Logik Texte Berlin, 1983
Berka I 269
Inductive Definition/Set Theory/LorenzenVsSet Theory: For example, an inductive definition of a set M by
a(y) > y ε M, x ε M u b(x,y) > y ε M
whereby a (x) and b(x,y) are already defined formulas in which M does not occur, is "explained" set theoretically that M should be the average of all sets N satisfying these implications with N instead of M.
Lorenzen: whoever wants to defend a claim n ε M (sic) will hardly attempt all these sets N. As P, he will rather defend O against either directly a(n), or he will first give an m which he will defend b (m, n) and m ∈ M.
Step number/Lorenzen: in order to determine this procedure as the the dialogical sense of the inductive definition of M, we must also require of P to indicate the number of steps required for complete proof for each assertion of the form x ∈ M.
E.g. suppose, for example, he traces n ε M back to the assertion m ∈ M and has stated the step number v for n ε M...
...so he must specify a step number μ
0 < y for positive numbers y
x < y _> x +/ 1 < y +/ 1
e.g. 1 < 0, and begin a "proof" with the aid of 0 < 1, 1 < 2, 2 < 3 .... Of course, the proof could not be finished, but O could not prove this.
Dialogical logic/Lorenzen: in these dialogues, it is never permitted to intervene suddenly in the "free speech" of the opponent. If, on the other hand, P has to specify a step number v, he will have lost his assertion at the latest after v steps.
Step number: the steps are, of course, natural numbers. If one wants to give infinite inductive definitions, i.e. such with an infinite number of premisses, a dialogical meaning, one must allow transfinite ordinal numbers as the step numbers.
Inductive Definition/LorenzenVsHerbrand: For example, a function sequence f1, f2 ... is already defined and the induction scheme
a(y) > y ε M (x)fx(y) ε M > y ε M
is adressed. This definition is by no means "impredicative". But it is also not really constructive either. We have infinitely many premises here
f1 (y) ∈ M, f2 (y) ε M ... which are necessary to prove y ∈ M.
Infinite: in dialogue one cannot defend every premise, one will therefore allow O to select an fm(y) e M. This must then be claimed and defended by P. In addition, P must specify a generally transfinite ordinal number as the step number.
Step number: the step number of a premise must always be specified as less than the step number of the conclusion.
Winning strategy: of P: must provide the step numbers for all opponent's elections.
II. Number-class/second/Lorenzen: set-theoretically one can prove easily the existence of suitable ordinal numbers of the II. number class. One can define transfinite recursion through this:
y ε M0 <> a(y) y ε Mλ <> (x)fx(y) ε Ux x < λ Mx. .
Then M = Ul l> μ Ml for a suitable μ and if M is to be a set of natural numbers, μ can be taken from the II. number class.
Constructively, if the inductive definition is to be constructive, the ordinal numbers used must also be "constructive". Here it is obvious to limit oneself to the recursive ordinal numbers of Church and Kleene.
Constructive Philosophy Cambridge 1987
K. Berka/L. Kreiser
Logik Texte Berlin 1983