For there are infinite vertices, we have infinite propositions. If they are all satisfiable. We do know there is a infinite path. Given a subset 2o, it is just a part of subtree with height k. Then we just add missed propositions into 2o and obtain the subtree represented by >o. As the tree has arbitrarily long finite paths. We know 2o is satisfiable and also 2o. Now we just apply compactness theorem to get final result In textbook, compactness theorem of proposition logic is proved based on Konig lemma. Here we prove inversely. It means that they are equivalent Exercises Let 2 be finite set of propositions and A2 the conjunction of its members. Prove that for any proposition a the following are equivalent (a)∑ (b)}∧∑→a. (d)A∑→a. 2. Suppose X is a finite set of propositions. Show that every CST from 2 is finite 3. Prove AV-B, BV-C, CV-DIF(D+A) 4. Suppose proposition could be infinite long. Define a new method to construct a CST and prove that every CST is finished 5. Design a circuit for multiply with two two bits input and four bits output. For example, we have1*1=1,10*11=110 6. Every set S can be(totally)ordered.(Hint: Use proposition to represent partial order and dichotomy, then try to apply compactness theorem. 7.Ex7/p46For there are infinite vertices, we have infinite propositions. If they are all satisfiable. We do know there is a infinite path. Given a subset Σ0, it is just a part of subtree with height k. Then we just add missed propositions into Σ0 and obtain the subtree represented by Σ′ 0 . As the tree has arbitrarily long finite paths. We know Σ′ 0 is satisfiable and also Σ0. Now we just apply compactness theorem to get final result. In textbook, compactness theorem of proposition logic is proved based on K¨onig lemma. Here we prove inversely. It means that they are equivalent. Exercises 1. Let Σ be finite set of propositions and ∧Σ the conjunction of its members. Prove that for any proposition α the following are equivalent: (a) Σ |= α. (b) |= ∧Σ → α. (c) Σ ⊢ α (d) ⊢ ∧Σ → α. 2. Suppose Σ is a finite set of propositions. Show that every CST from Σ is finite. 3. Prove {A ∨ ¬B, B ∨ ¬C, C ∨ ¬D} |= (D → A). 4. Suppose proposition could be infinite long. Define a new method to construct a CST and prove that every CST is finished. 5. Design a circuit for multiply with two two bits input and four bits output. For example, we have 1 ∗ 1 = 1, 10 ∗ 11 = 110. 6. Every set S can be (totally) ordered. (Hint: Use proposition to represent partial order and dichotomy, then try to apply compactness theorem.) 7. Ex 7/p46. 7