抄録
We tried to analyse on the alternatives in the Field of Public Choice from the view point of the B-G set theory, and got some interesting conclusions as follows.
1. Two types of the alternatives
There are two types of alternatives in the choice problems.
‹two types of alternatives in case of choosing one from two›
type A1
A1= [x|x=a∨x≠a]
(“a”or“not a”)
type A2
A2= [x|x=a∨x=b]
(“a”or“b”, another choice is not available)
‹two types of alternatives of in the choice problems in general›
type A1*
A1*= [x|x∈A∨x∉A]
(A case where it is not clear what does it mean when a concrete type of alternatives called A is given, but no choiceis made out of it.)
type A2*
A2*= [x|x∈A∨x∈B]
whereB=φ∨B= [b] ∨B= [b1, b2, ...] (A case where it is implicitly or explicitly clear what does it mean when A is given, but no choice is made out of it.)
where, B=φ : We must choose at least one from “A”, and to choose no one from “A”is impossible.
B= [b] : To choose no one from “A” means to choose “b”which is not a element of “A”.
B= [b1, ... ] : If we don't choose from “A”, it is an obvious fact that we may choose from some concrete sphere which is independent from “A”.
2. Conclusion in the set theory
A1 and A1* raise Bertrand Russel's paradox. And they can not satisfy the axiom of regularity (or axiom of foundation) . Hence A1 and A1* neither class nor set.
But A2 and A2* can be considerd as class as well as set.
Then we get following two conclusions of alternatives.
[conclusion 1]
Generally alternatives is neither class nor set.
[conclusion 2]
If there exist some restrictions which limits sphare of alternatives to the concrete limited sphere, then the alternatives is a set
3. Alternatives in the field of public choice As I explained in section 4, about the example in the representative assemblies, we actually come across the alternatives type A1 or A1*. And now and then we play the kind of game to make alternatives type A1 (or A1*) or A2 (or A2*) .
This type of alternatives A1 (or A1*) is the concept that even Paul Bernays and Kurt Godel excluded from context of set theory to avoid the dilemma. But unfortunately we must treat the same concept in the context of the theory of public choice.
From another view point, to make alternatives set or not obviously depends upon the restriction called the rule of the choice. Hence, problems around the rule of choice or decision making are quite interesting in the theoretical analysis as well as in the applied analysis on politics, (to be continued)
