Sets vs Classes: Is Anything Beyond A Set?

  • Context: Graduate 
  • Thread starter Thread starter dodo
  • Start date Start date
  • Tags Tags
    Classes Sets
Click For Summary
SUMMARY

The discussion centers on the distinction between sets and classes in mathematical logic, specifically addressing the concept of proving the existence of proper classes by contradiction. It highlights that while sets can lead to contradictions when assumed to be classes, the reverse scenario—proving an object is not a class—rarely occurs in standard mathematics. The conversation emphasizes that sets serve as a simplified yet comprehensive framework for higher-order logic, which is a key application in mathematical theory.

PREREQUISITES
  • Understanding of set theory and its foundational principles.
  • Familiarity with higher-order logic concepts.
  • Knowledge of mathematical proof techniques, particularly proof by contradiction.
  • Basic comprehension of nonstandard analysis and its implications.
NEXT STEPS
  • Explore the principles of set theory and its axioms, focusing on Zermelo-Fraenkel set theory.
  • Study higher-order logic and its applications in mathematical reasoning.
  • Investigate proof techniques, particularly the method of proof by contradiction in mathematical contexts.
  • Learn about nonstandard analysis and its impact on the understanding of sets and classes.
USEFUL FOR

Mathematicians, logicians, and students of advanced mathematics who are interested in the foundational aspects of set theory and the philosophical implications of classes versus sets.

dodo
Messages
695
Reaction score
2
The stupid question of the day:

The existence of proper classes is often proven by contradiction: assume that some object is a set, you'll find a contradiction, therefore it is not a set. We baptized those as "classes".

Will (can) this even happen to classes? To find some object, assume it is a class, and get a contradiction, proving it is something else?
 
Physics news on Phys.org
Except for the question of internal vs external in nonstandard analysis, I can't think of any situation in ordinary mathematics where such a thing could come up. The only reason we see it with sets and classes is because one of the main applications of sets is to serve as a simplified yet extensive version of higher-order logic.
 

Similar threads

Graduate Defining the membership relation in set theory?
  • · Replies 14 ·
Replies
14
Views
6K
Graduate Equivalence classes of proper classes
  • · Replies 10 ·
Replies
10
Views
5K
Graduate Prove that points which are indistinguishable from 0 exist (using logic)
  • · Replies 17 ·
Replies
17
Views
2K
MHB Introduction to Set Theory Stream
  • · Replies 1 ·
Replies
1
Views
2K
Graduate Solve the paradox of set theory V7.4.1
  • · Replies 8 ·
Replies
8
Views
4K
Graduate Choices of Axiomatic and Number Systems / Sets and Alternatives
  • · Replies 2 ·
Replies
2
Views
2K
Is a Prof Failing 3/4 of the Senior Year Physics Class Normal? (Stat Phy)
  • · Replies 6 ·
Replies
6
Views
653
Graduate Example of transitive but not well ordered set needed
  • · Replies 2 ·
Replies
2
Views
5K
Proving Completeness property of ##\mathbb{R}## using Dedekind cuts
  • · Replies 20 ·
Replies
20
Views
4K
Graduate Is the set of nonregular languages larger than the set of regular languages?
  • · Replies 3 ·
Replies
3
Views
5K