Are There Infinite Permutations on an Infinite Set?

Click For Summary
SUMMARY

The discussion centers on the concept of permutations within a countably infinite set, specifically set A. It concludes that the set of all permutations of A is not countably infinite, but rather has the cardinality of {\aleph_0}^{\aleph_0}. This result is rooted in Cantor's diagonal argument, which demonstrates the existence of larger infinities beyond countable sets.

PREREQUISITES
  • Understanding of countably infinite sets
  • Familiarity with Cantor's diagonal argument
  • Knowledge of cardinality in set theory
  • Basic concepts of permutations and combinations
NEXT STEPS
  • Study the implications of Cantor's theorem on set theory
  • Explore the concept of cardinality in more depth
  • Learn about different types of infinities and their properties
  • Investigate advanced topics in combinatorial mathematics
USEFUL FOR

Mathematicians, computer scientists, and students of advanced mathematics interested in set theory and the nature of infinity.

guroten
Messages
32
Reaction score
0
Given a countably infinite set, A, is the set of all permutations of A also countably infinite?
 
Physics news on Phys.org
Have you learned about Cantor's diagonal slash?
 
No, there are [tex]{\aleph_0}^{\aleph_0}[/tex] permutations.
 

Similar threads

Undergrad The set of nonzero values of pmf is at most countable
  • · Replies 3 ·
Replies
3
Views
4K
Undergrad Countability of binary Strings
  • · Replies 18 ·
Replies
18
Views
4K
Undergrad Can you transform a countably infinite set to an uncountable one?
  • · Replies 15 ·
Replies
15
Views
2K
Undergrad I don't understand Dedekind cuts
  • · Replies 14 ·
Replies
14
Views
1K
Undergrad Proof of a fact about real numbers - transcendental and algebraic
  • · Replies 7 ·
Replies
7
Views
3K
High School Is this a finite argument of the countable infiniteness of the rationals?
  • · Replies 21 ·
Replies
21
Views
4K
Undergrad Add an exponential number of elements, what will be the final cardinality?
  • · Replies 16 ·
Replies
16
Views
2K
Undergrad Is an algorithm for a proof required to halt?
  • · Replies 3 ·
Replies
3
Views
2K
Undergrad Countably Infinite Unions and the Real Numbers: Can They Really Be Uncountable?
  • · Replies 4 ·
Replies
4
Views
3K
Undergrad My basic understanding of set theory
  • · Replies 20 ·
Replies
20
Views
3K