Who needs who ?

[Organic]

The ZF Axiom of the Empty set:

There is a set A such that, given any set B, B is not a member of A.

(An analogy: There is a "collector" A with no "content" B)


By using at least two variables (in this case A and B) we need some formula to describe the relations between them.

No set can be separated from the property of its content, therefore
we have an interesting situation here.

On one hand a collector can exist with no content, but on the other hand its property is depended on the property of its content.

But we also know that the content concept can't exist without a collector.

To define the exact definition of an existing thing A(a collector), is not in the same level as defining the existence of B(a content).

So A can exist with no clear property, but B can't exist at all without A.

Can someone show how Math language deals with these distinguished two levels.

If we say "There is a collector" , do you think that we can come to the conclusion that it has no content (the minimal collector's existence) as its property ?



Thank you.



Organic

 

[HallsofIvy]

One of the nice things about "math language" is that it is very difficult to write non-sense in "math language" while one can see it is very easy to do so in ordinary language (is it only me or does English seem particularly prone to making non-sense look like it really means something?).

 

[tacman]

Mathematician's

In mathematics, the ZF Axiom of the Empty set states that there exists a set (A) that has no elements. This means that there is a collector (A) with no content (B). This is a fundamental axiom in set theory and is used to build the foundation of mathematical structures.

In terms of who needs who, we can say that the existence of A (the collector) is dependent on the existence of B (the content). Without B, A cannot exist. However, the property of A (having no elements) is independent of B. Therefore, both A and B have a mutual dependence on each other - A needs B to exist, and B needs A to define its property.

In mathematical language, we can say that A is a set and B is an element of A. The existence of A is defined by the existence of B, but the property of A (having no elements) is independent of B. This concept is important in understanding the relationships between sets and their elements in set theory.

In conclusion, the ZF Axiom of the Empty set highlights the interconnectedness of sets and their elements, and how their existence and properties are intertwined. Both A and B need each other in order to define their existence and properties, but they also have distinct roles in the mathematical structure.