MHB Transfinite Induction: Theorem or Axiom?

Click For Summary
SUMMARY

Transfinite induction can be classified as either a theorem or an axiom depending on the logical system in which it is applied. In certain systems, it may be necessary to drop other axioms to maintain a minimal set if transfinite induction is treated as an axiom. Conversely, it may require the addition of other axioms to prove it as a theorem. This duality highlights the flexibility and context-dependence of foundational mathematical principles.

PREREQUISITES
  • Understanding of axiomatic systems
  • Familiarity with the concept of theorems
  • Knowledge of transfinite numbers
  • Basic principles of mathematical logic
NEXT STEPS
  • Research the role of axioms in set theory
  • Explore the implications of transfinite induction in different logical frameworks
  • Study the relationship between axioms and theorems in mathematical proofs
  • Investigate the historical development of transfinite induction
USEFUL FOR

Mathematicians, logicians, and students of advanced mathematics interested in the foundations of mathematical theory and the nature of induction in infinite sets.

facenian1
Messages
3
Reaction score
0
Is transfinite induction a theorem o an axiom?
 
Physics news on Phys.org
That depends. In any logical system you can always take any given statement as an "axiom" (you might need to drop at least one other axiom to keep a "minimal" set of axioms) or as a theorem (you might need to add at least one other axiom to be able to prove the theorem).
 

Thread 'My basic understanding of set theory'

If there are an infinite number of natural numbers, and an infinite number of fractions in between any two natural numbers, and an infinite number of fractions in between any two of those fractions, and an infinite number of fractions in between any two of those fractions, and an infinite number of fractions in between any two of those fractions, and... then that must mean that there are not only infinite infinities, but an infinite number of those infinities. and an infinite number of those...
View full post »

Similar threads

Goodstein theorem without transfinite numbers?
  • · Replies 2 ·
Replies
2
Views
2K
Vacuously True Starting Point in Transfinite Induction
  • · Replies 17 ·
Replies
17
Views
3K
Transfinite Induction: Understanding & Examples
  • · Replies 3 ·
Replies
3
Views
3K
Undergrad Question about transfinite numbers
  • · Replies 2 ·
Replies
2
Views
2K
Undergrad Venn diagram for the reals and transfinite numbers as sets
  • · Replies 10 ·
Replies
10
Views
3K
Undergrad What is before the first transfinite ordinal, omega?
  • · Replies 12 ·
Replies
12
Views
4K
MHB A question on Transfinite Recursion Theorem schema....
  • · Replies 11 ·
Replies
11
Views
4K
Undergrad Help with basic transfinite induction proof
  • · Replies 2 ·
Replies
2
Views
2K
Undergrad Formal derivation of statement from Peano Arithmetic system
  • · Replies 10 ·
Replies
10
Views
993
Transfinite Theory as an Extension of the Natural Numbers
  • · Replies 23 ·
Replies
23
Views
6K