ZFC and the Axiom of Power Sets ....

In summary, Peter is reading Micheal Searcoid's book "Elements of Abstract Analysis" and is currently focused on the treatment of ZFC in Chapter 1. He needs help understanding the Axiom of Power Sets and Definition 1.1.1. The relevant text from Searcoid explains that every set is a subset of itself and no set is a proper subset of itself, but it is not yet known if any set has a proper subset. Peter asks for an explanation of why this is the case, citing an example of a set with a proper subset guaranteed by the Axiom of Power Sets. Andre responds that the existence of the empty set as a proper subset is not yet introduced, so it is
  • #1
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I am reading Micheal Searcoid's book: Elements of Abstract Analysis ( Springer Undergraduate Mathematics Series) ...

I am currently focused on Searcoid's treatment of ZFC in Chapter 1: Sets ...

I need help in order to fully understand the Axiom of Power Sets and Definition 1.1.1 ...

The relevant text from Searcoid is as follows:
Searcoid - The Axioms ... Page 6 .png

At the end of the above text we read the following:

" ... ... Thus every set is a subset of itself and no set is a proper subset of itself. But we do not yet now that any set has a proper subset. ... ... "My question is as follows:

Can someone explain exactly why/how it is that we do not yet now that any set has a proper subset. ... ..?

My thinking is that surely we do know that any set has a proper subset ... for example if \(\displaystyle b = \{ s, t, r \}\) then \(\displaystyle a = \{ s, t \}\) is a proper subset of \(\displaystyle b\) ... and the existence of a is guaranteed by the Axiom of Power Sets ...

Help will be much appreciated ...

Peter
 
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  • #2
The set \[ \{s\} \] has power set \[\{\varnothing, \{s\}\}.\] Empty set is proper by the definition, but it is not yet introduced. So we don’t know if \[\{s\}\] has any proper subsets.
 
  • #3
Thanks for the help AndreI ...

... still reflecting on how what you have written answers my question ...

Thanks again ...

Peter
 

What is ZFC and the Axiom of Power Sets?

ZFC stands for Zermelo-Fraenkel set theory with the Axiom of Choice. It is a foundational theory in mathematics that provides a formal language and rules for reasoning about sets and their properties. The Axiom of Power Sets is one of the axioms of ZFC, which states that for any set, there exists a set containing all of its subsets.

Why is ZFC and the Axiom of Power Sets important in mathematics?

ZFC and the Axiom of Power Sets are important because they provide a rigorous and consistent framework for reasoning about sets and their properties. They are the basis for most of modern mathematics and are used in various fields such as logic, computer science, and theoretical physics.

Are there any controversies or criticisms surrounding ZFC and the Axiom of Power Sets?

Yes, there are some controversies and criticisms surrounding ZFC and the Axiom of Power Sets. Some mathematicians argue that the Axiom of Choice, which is part of ZFC, leads to counterintuitive results and should not be accepted as a fundamental axiom. There are also debates about the consistency and completeness of ZFC as a foundational theory.

What are some applications of ZFC and the Axiom of Power Sets?

ZFC and the Axiom of Power Sets have numerous applications in mathematics and other fields. They are used to prove theorems in set theory, topology, and analysis. In computer science, they are used for formal verification of programs and algorithms. In physics, they are used to model and study the foundations of quantum mechanics and other theories.

Is there ongoing research and developments in ZFC and the Axiom of Power Sets?

Yes, there is ongoing research and developments in ZFC and the Axiom of Power Sets. Mathematicians are constantly exploring new axioms and theories that can extend or modify ZFC, as well as investigating its limitations and implications. There are also efforts to find alternative foundations for mathematics that do not rely on ZFC and the Axiom of Power Sets.

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