Which axioms of ZF are needed for Cantor's theorem?

In summary, the conversation discusses Cantor's theorem, which states that the cardinality of the power set of any set is greater than that of the set. The Metamath proof of this theorem uses basic logic and several axioms, including the Axiom of Extensionality, Axiom of Separation, Null Set Axiom, Axiom of Power Sets, Axiom of Pairing, and Axiom of Union. However, the Axioms of Replacement, Regularity, and Infinity of ZF are not used in the proof. The proof also utilizes propositional and predicate calculus, as well as various rules and axioms related to equality and quantifiers. Some of the listed axioms, such as
  • #1
Dragonfall
1,030
4
Self explanatory. The Cantor's theorem which am referring to is that the cardinality of the power set of any set is greater than that of the set.
 
Physics news on Phys.org
  • #2
The Metamath proof of Cantor's theorem (canth) uses basic logic*, plus the following:
  • Axiom of Extensionality (ax-ext)
  • Axiom of Separation (ax-sep)
  • Null Set Axiom (ax-nul)
  • Axiom of Power Sets (ax-pow)
  • Axiom of Pairing (ax-pr)
  • Axiom of Union (ax-un)
The Axioms of Replacement, Regularity, and Infinity of ZF are not used in the proof.

* Propositional calculus (ax-1, ax-2, ax-3, ax-mp), basic predicate calculus (ax-4, ax-5, ax-6, ax-7, ax-gen), the equality and substitution rules (ax-8, ax-9, ax-10, ax-11, ax-12, ax-13, ax-14), and the second Axiom of Quantifier Introduction (ax-17).
** The page also lists the Axiom of Distinct Variables (ax-16) and the old Axiom of Variable Substitution (ax-11o), but these have been proven redundant with the others.
 
  • #3
Fascinating. Thanks.
 

1. What is ZF and how does it relate to Cantor's theorem?

ZF refers to the Zermelo-Fraenkel set theory, which is a foundational theory used in mathematics to define sets and their properties. Cantor's theorem is a result in set theory that states the cardinality (size) of the power set of a set is always strictly greater than the cardinality of the original set. This theorem relies on certain axioms from ZF in order to be proven.

2. What are axioms and why are they important in mathematics?

Axioms are fundamental statements or principles that are assumed to be true without proof in a particular mathematical system. They serve as the building blocks for the rest of the theory and provide a starting point for logical deductions. Axioms are important because they help to establish the consistency and coherence of a theory.

3. Which specific axioms of ZF are necessary for Cantor's theorem?

Cantor's theorem relies on the axioms of extensionality, pairing, union, power set, and infinity from ZF. These axioms ensure the existence and properties of sets that are necessary for the proof of Cantor's theorem.

4. Can Cantor's theorem be proven without using all of the axioms of ZF?

No, Cantor's theorem cannot be proven without using all of the axioms of ZF. Each of the axioms plays a crucial role in establishing the properties of sets and their cardinalities, which are essential for the proof of Cantor's theorem.

5. Are there any alternative set theories that can be used to prove Cantor's theorem?

Yes, there are alternative set theories such as ZFC (Zermelo-Fraenkel set theory with the axiom of choice) and NBG (von Neumann-Bernays-Gödel set theory) that can also be used to prove Cantor's theorem. However, these theories are extensions of ZF and still rely on the same axioms used in ZF.

Similar threads

  • Set Theory, Logic, Probability, Statistics
Replies
19
Views
3K
  • Set Theory, Logic, Probability, Statistics
Replies
3
Views
1K
Replies
72
Views
4K
  • Set Theory, Logic, Probability, Statistics
Replies
2
Views
1K
  • Set Theory, Logic, Probability, Statistics
Replies
1
Views
2K
  • Set Theory, Logic, Probability, Statistics
Replies
11
Views
2K
  • Set Theory, Logic, Probability, Statistics
Replies
1
Views
1K
  • Set Theory, Logic, Probability, Statistics
Replies
4
Views
1K
  • Set Theory, Logic, Probability, Statistics
Replies
10
Views
1K
  • Set Theory, Logic, Probability, Statistics
Replies
2
Views
2K
Back
Top