How can I prove the compactness theorem for sets of sentences?

In summary, the Compactness Theorem is a fundamental theorem in mathematical logic that states that if an infinite set of sentences has a model, then it must have a finite subset that also has a model. The proof of the Compactness Theorem involves constructing a larger model from a collection of smaller models using the compactness property of ultrafilters. It is closely related to completeness in mathematical logic and has numerous applications in fields such as model theory and set theory. However, it can only be applied to first-order logic systems due to its reliance on the properties of this logical system.
  • #1
rainwyz0706
36
0
An expression of the compactness theorem for sets of sentences is that: let T be a set of sentences in L. Then T has a model iff every finite subset of T has a model.
Could anyone give me some hints how to prove this?
The first direction is straightforward: every model of T is a model of every subset of T. But what about the opposite direction? Any help is appreciated!
 
Physics news on Phys.org
  • #2
My advice would be to not try to reinvent the wheel, and study the proofs given in any logic textbook.
 

1. What is the Compactness Theorem?

The Compactness Theorem is a fundamental theorem in mathematical logic that states that if an infinite set of sentences has a model, then it must have a finite subset that also has a model. In other words, if a set of sentences is consistent, then it must have a consistent finite subset.

2. What is the proof of the Compactness Theorem?

The proof of the Compactness Theorem involves constructing a larger model from a collection of smaller models using the compactness property of ultrafilters. This allows us to show that if an infinite set of sentences has a model, then it must have a finite subset that also has a model.

3. How does the Compactness Theorem relate to completeness?

The Compactness Theorem and completeness are closely related concepts in mathematical logic. While completeness deals with the existence of proofs for all logically valid statements, the Compactness Theorem deals with the existence of models for infinite sets of sentences. Both are important properties of first-order logic.

4. Can the Compactness Theorem be applied to non-first-order logic systems?

No, the Compactness Theorem only holds for first-order logic systems. It relies on the properties of first-order logic, such as the compactness of ultrafilters, and cannot be applied to other logical systems that do not have these properties.

5. How is the Compactness Theorem used in mathematics?

The Compactness Theorem has numerous applications in mathematics, particularly in the fields of model theory and set theory. It is often used to prove the existence of mathematical structures, such as infinite graphs or algebraic structures, by showing the existence of a model for a set of axioms. It is also used in proofs of the completeness and decidability of logical systems.

Similar threads

I Is an algorithm for a proof required to halt?
    • Set Theory, Logic, Probability, Statistics
Replies
3
Views
1K
MHB ZFC and the Axiom of Power Sets ....
    • Set Theory, Logic, Probability, Statistics
Replies
2
Views
1K
I Direct limit of multiverse models of ZFC
    • Set Theory, Logic, Probability, Statistics
Replies
2
Views
1K
MHB Proving Relatively Compactness in C([a, b]) using Arzelá-Ascoli Theorem
    • Topology and Analysis
Replies
2
Views
1K
I An easy proof of Gödel's first incompleteness theorem?
    • Set Theory, Logic, Probability, Statistics
Replies
16
Views
2K
A Somewhat difficult set theory proof
    • Set Theory, Logic, Probability, Statistics
Replies
6
Views
1K
Compactness in Topology and in Logic
    • Set Theory, Logic, Probability, Statistics
Replies
4
Views
3K
MHB Proving that a subset of a countable set is countable
    • Set Theory, Logic, Probability, Statistics
Replies
1
Views
3K
I Bloch Analysis proof of Theorem 2.5.5 (Definition by recursion)
    • Topology and Analysis
Replies
2
Views
314
MHB A Subset of a Graph with No Perfect Matching
    • Set Theory, Logic, Probability, Statistics
Replies
1
Views
1K

Hot Threads

Recent Insights

Back
Top