ZF Set Theory and Law of the Excluded Middle

Click For Summary
SUMMARY

The discussion centers on the relationship between ZF set theory and the law of the excluded middle (LEM), particularly in the context of the axiom of choice. It is established that LEM is implied in ZFC set theory due to the axiom of choice. The inquiry specifically seeks to determine whether LEM holds in ZF set theory without the axiom of choice, and requests proofs or references regarding this implication. A related discussion is also referenced on Math Stack Exchange for further insights.

PREREQUISITES
  • Understanding of ZF set theory and its axioms
  • Familiarity with the axiom of choice in set theory
  • Knowledge of the law of the excluded middle in classical logic
  • Basic comprehension of mathematical proofs and their structure
NEXT STEPS
  • Research the implications of the axiom of choice on set theory
  • Study the differences between ZF and ZFC set theories
  • Explore proofs related to the law of the excluded middle in infinite sets
  • Visit Math Stack Exchange for community discussions on set theory topics
USEFUL FOR

Mathematicians, logicians, and students of set theory who are exploring foundational concepts in mathematical logic and the implications of axioms in set theory.

HJ Farnsworth
Messages
126
Reaction score
1
Hello,

I know that the law of the excluded middle is implied in ZFC set theory, since it is implied by the axiom of choice. Taking away the axiom of choice, does ZF set theory (with axioms as stated in the Wikipedia article http://en.wikipedia.org/wiki/Zermelo–Fraenkel_set_theory), imply the law of the excluded middle [for infinite sets]?

If LEM does follow from ZF, could you please provide the proof if you know it, or point me to the proof if you know where it is, or tell me what ZF axioms it follows from if you don't know of theproof or its location?

Thanks very much.

-HJ Farnsworth
 
Physics news on Phys.org

Similar threads

  • · Replies 5 ·
Replies
5
Views
1K
  • · Replies 40 ·
2
Replies
40
Views
4K
  • · Replies 10 ·
Replies
10
Views
2K
  • · Replies 14 ·
Replies
14
Views
5K
  • · Replies 7 ·
Replies
7
Views
4K
  • · Replies 11 ·
Replies
11
Views
3K
Replies
11
Views
10K
  • · Replies 132 ·
5
Replies
132
Views
20K
  • · Replies 17 ·
Replies
17
Views
2K
  • · Replies 13 ·
Replies
13
Views
6K