I see the study of Drapeau et al. (2016), The algebra of conditional sets and the concepts of conditional topology and compacness, Journal of Mathematical Analysis and Applications. They bring the new concept of 'conditional set' (Definition 2.1):
A conditional set X of a non-empty set X and a complete Boolean algebra A is a collection of objects x|a for x in X and a in A such that
- if x|a = y|b, then a = b;
- if x,y in X and a,b in A with a
Dear Alex A.T. Rathke , based on ChatGPT's point of view can give this answer:
"No, it is not possible to reduce the new definition of 'conditional set' proposed by Drapeau et al. (2016) to the ordinary definition of set in existing set theory. The conditional set definition adds a new concept that is not present in the ordinary definition of set, where the elements of a set are defined by their properties rather than by their relationships with other sets. The conditional set definition introduces a dependency between two sets, where the elements of the second set depend on the elements of the first set satisfying a given condition. Therefore, the conditional set definition requires a new concept that goes beyond the existing set theory."
I hope that it was usefull for you.
Well, I see a circular response here, when the ChatGPT says "...The conditional set definition introduces a dependency between two sets, where the elements of the second set depend on the elements of the first set satisfying a given condition (my emphasis)". This is my point: any set is created by things satisfying some condition/property; this is already axiomatised by ZF.
The concept of a conditional set introduced by Drapeau et al. (2016) is based on the notion of a dependent type in type theory. It is a generalization of the traditional notion of a set in set theory, and it allows for the definition of sets in a more flexible and expressive way.
In traditional set theory, a set is defined as a collection of distinct objects, and it is often denoted using set-builder notation. For example, the set of all even natural numbers can be defined as {n | n is a natural number and n is even}.
In the conditional set theory introduced by Drapeau et al. (2016), a set is defined as a type, and it is often denoted using dependent type notation. The dependent type notation allows for the definition of a set in terms of a condition that the elements of the set must satisfy. For example, the set of all even natural numbers can be defined using dependent type notation as {n : N | n mod 2 = 0}, where N is the type of natural numbers and n mod 2 = 0 is the condition that the elements of the set must satisfy.
Therefore, the definition of a conditional set cannot be reduced to the traditional definition of a set in set theory. However, the two concepts are related, and the traditional notion of a set can be seen as a special case of the more general notion of a dependent type in conditional set theory.
- Badges
- Science topic
- Similar topics
- Mathematical Sciences
- Graphs