MHB Transfinite Induction: Theorem or Axiom?

Click For Summary
Transfinite induction can be classified as either a theorem or an axiom, depending on the logical system in use. In any system, a statement can be treated as an axiom, though this may require dropping another axiom to maintain minimality. Conversely, it can also be considered a theorem, which might necessitate the addition of another axiom for proof. The classification ultimately hinges on the foundational framework being applied. Thus, the status of transfinite induction is context-dependent.
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 'Two interesting number theory tricks'

First trick I learned this one a long time ago and have used it to entertain and amuse young kids. Ask your friend to write down a three-digit number without showing it to you. Then ask him or her to rearrange the digits to form a new three-digit number. After that, write whichever is the larger number above the other number, and then subtract the smaller from the larger, making sure that you don't see any of the numbers. Then ask the young "victim" to tell you any two of the digits of the...
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
906
Transfinite Theory as an Extension of the Natural Numbers
  • · Replies 23 ·
Replies
23
Views
6K