crimboi[22] 你什么拓扑?

funny problem from model theory problem sheet

题面

Let L be a countable language, T a theory. Let c_1,c_2,... be new constants, L'=L\cup\lbrace c_1,...\rbrace, and let X be the set of all complete theories of L' with T\subset t. Enumerate the L' sentences as \lbrace\sigma_1,\sigma_2,...\rbrace Define a metric on X by d(t,t')=0 if t=t' or 2^{-n} where \sigma_n is the sentence with smallest index in t\triangle t'.

1) Show X is a complete metric space

2) Let \phi(x) be a formula in one variable. Show that the set of theories t\in X s.t. either \neg\exists x\phi(x) or t witnesses \phi, i.e. t\models\phi(c) for some constant c is open and dense in X. Using Baire Category Theorem to prove that there is a complete self-witnessing t containing T.

3) If P(x) is a nonprincipal partial type of T, show that there is a complete self-witnessing t whose named model omits P. Similar argument shows a countable set of nonprincipal partial types can be simultaneously omitted, which is known as Omitting Types Theorem.

先想再看提示哦

提示

瞎几把构造limit&sequence, 相信你可以！！

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标答

简单捏