How to understand compactness? - Mathematics Stack Exchange
This is precisely what compactness ensures. Other examples: Peano's theorem about the existence of solutions to ordinary differential equations relies on compactness to construct a good candidate. A commonly used technique for showing the existence of solution for partial differential equations is the calculus variations.
general topology - Why is compactness in logic called …
2012年6月12日 · 95. In logic, a semantics is said to be compact iff if every finite subset of a set of sentences has a model, then so to does the entire set. Most logic texts either don't explain the terminology, or allude to the topological property of compactness. I see an analogy as, given a topological space X and a subset of it S, S is compact iff for ...
Most astonishing applications of compactness theorem outside logic
$\begingroup$ @user: in fact the compactness theorem is equivalent to Tychonoff's theorem for compact Hausdorff spaces. (In that sense it is known to be strictly weaker than Tychonoff's theorem, which is equivalent to the axiom of choice, so I …
Why is compactness so important? - Mathematics Stack Exchange
2013年9月7日 · 11. Compactness is important because: 1)It behaves greatly when using topological operations. a)It's a condition that is carried on by continuous functions on any topological space, that is, if C is compact and f: C → Y where Y …
model theory - Is the compactness theorem (from mathematical …
The compactness theorem for first-order logic is equivalent (in $\sf ZF$) to the completeness theorem, as well to the ultrafilter lemma, and several other interesting principles. The proof is due to Henkin [1], but I could not find the paper online, the proof appears in [2, Theorem 2.2].
Proof of the Compactness Theorem for propositional logic by …
2017年1月13日 · This theorem provides the link between the definition of compactness in terms of finite subcovers and the view of compactness that we use when we claim that propositional logic enjoys compactness. Share
set theory - Clarifications on proof of compactness theorem ...
4. You've identified the one place in the Henkin proof of compactness where choice is used, namely in completing an arbitrary theory T T. If you don't want to use the full Axiom of Choice, you can simply assume "every first-order theory has a completion" (call this TC for theory completion) and move on with your day.
general topology - Compactness theorem and Tychonoff theorem ...
Then, by Tychonoff's Theorem, {T, F}I is compact. If every finite set of wffs is satisfiable, the Jf s have the finite intersection property (every finite subset of the Jf s has a nonempty intersection.) Each Jf is closed, so by compactness, ⋂f∈ΣJf ≠ ∅. Therefore, Σ is satisfiable. This is the Compactness Theorem.
Compactness Theorem explanation - Mathematics Stack Exchange
You might rephrase the compactness theorem as follows: If $\Sigma$ is a theory of propositional logic, then $\Sigma$ is satisfiable if every finite subset of $\Sigma$ is satisfiable. In first-order logic, saying that a theory is satisfiable is the same as saying it has a model. (This fact is called the completeness theorem.)
Application of compactness theorem. - Mathematics Stack Exchange
Compactness theorem: A set of wffs is satisfiable iff every finite subset is satisfiable. Attempt: Pick any finite subset X ⊆ A. Let < be restricted to X and has dimension less than n + 1. Fix an x, y. Let < be any linear order. Define truth assignment v as follow, v(<):= {T x <y F otherwise. So now I want to mould the assumption I made so ...