ZFC and the Axiom of Power Sets ....

  • Context: MHB 
  • Thread starter Thread starter Math Amateur
  • Start date Start date
  • Tags Tags
    Axiom Power Sets Zfc
Click For Summary
SUMMARY

The discussion centers on the Axiom of Power Sets as presented in Micheal Searcoid's "Elements of Abstract Analysis." The Axiom states that for any set, there exists a power set containing all possible subsets, including the empty set. However, participants clarify that while a set can have subsets, the existence of proper subsets is not guaranteed until the Axiom of Power Sets is fully established. This distinction is crucial for understanding foundational concepts in Zermelo-Fraenkel set theory (ZFC).

PREREQUISITES
  • Understanding of Zermelo-Fraenkel set theory (ZFC)
  • Familiarity with the concept of power sets
  • Knowledge of set notation and terminology
  • Basic comprehension of mathematical logic
NEXT STEPS
  • Study the Axiom of Power Sets in detail
  • Explore the implications of the empty set in set theory
  • Review Chapter 1 of "Elements of Abstract Analysis" by Micheal Searcoid
  • Investigate the concept of proper subsets and their definitions
USEFUL FOR

This discussion is beneficial for students of mathematics, particularly those studying set theory, as well as educators and anyone seeking to deepen their understanding of foundational mathematical concepts in ZFC.

Math Amateur
Gold Member
MHB
Messages
3,920
Reaction score
48
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 $$b = \{ s, t, r \}$$ then $$a = \{ s, t \}$$ is a proper subset of $$b$$ ... and the existence of a is guaranteed by the Axiom of Power Sets ...

Help will be much appreciated ...

Peter
 
Physics news on Phys.org
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.
 
Thanks for the help AndreI ...

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

Thanks again ...

Peter
 

Similar threads

MHB ZFC and the Pairing Principle .... Searcoid Theorem 1.1.5 ....
  • · Replies 1 ·
Replies
1
Views
2K
MHB Set Theory and ZFC - The Subset Principal - Searcoid, Page 7
  • · Replies 5 ·
Replies
5
Views
2K
MHB Set Theory and ZFC - The Axiom of Replacement - Searcoid, Pages 6-7
  • · Replies 14 ·
Replies
14
Views
4K
MHB ZFC and the Axiom of Foundation
  • · Replies 3 ·
Replies
3
Views
3K
MHB Understanding ZFC and the Axiom of Infinity: Simple Explanation and Examples
  • · Replies 2 ·
Replies
2
Views
2K
Undergrad Direct limit of multiverse models of ZFC
  • · Replies 2 ·
Replies
2
Views
2K
Graduate On soundness and completeness of ZFC set theory
  • · Replies 6 ·
Replies
6
Views
2K
MHB Properties of the Ordinals ....
  • · Replies 1 ·
Replies
1
Views
1K
MHB Well Orders and Total Orders .... Searcoid Definition 1.3.10 ....
  • · Replies 5 ·
Replies
5
Views
2K
MHB Axioms of Set Theory .... and the Union of Two Sets ....
  • · Replies 2 ·
Replies
2
Views
2K