Math 557 Sep 17

Completing Theories

Key Concepts

  • We previously defined the term model \(\mathcal{A}\). It holds that for any atomic sentence \(\sigma\), \[\mathcal{A} \models \sigma \quad \iff \quad T \vdash \sigma\]

  • Trying to extend this to arbitrary sentence via induction, the negation case looks like \[\mathcal{A} \models \neg\sigma \; \iff \; \mathcal{A} \nvDash \sigma \; \overset{\text{ind hyp}}{\iff} \; T \nvdash \sigma\] We would like to show that this is equivalent to \(T \vdash \neg\sigma\).

  • One direction follows from \(T\) being consistent, but for the other direction, \(T\) may not be strong enough to prove this.

  • We therefore need to extend \(T\) to a complete theory.

Problems

Exercise 1
Verify that indeed for all atomic sentences \(\sigma\), \[\mathcal{A} \models \sigma \quad \iff \quad T \vdash \sigma\]

Exercise 2
Recall that a theory \(T\) is maximally consistent if it is consistent but does not have any consistent proper extensions. \(T\) is called deductively closed if the deductive closure of \(T\), \[T^{\vdash} = \{ \sigma : T \vdash \sigma \}\] is equal to \(T\).

  • Show that a maximally consistent theory is complete and deductively closed.
  • Show that if \(T\) is complete, then \(T^{\vdash}\) is maximally consistent.

Exercise 3
Is every consistent, deductively closed theory complete?

Exercise 4
Show that the union of an increasing sequence of consistent theories is consistent.

Exercise 5
Extend Lindebaum’s theorem on the existence of maximally consistent extension from countable to arbitrary languages.

Exercise 6
Fix a language \(\mathcal{L}\). Let \(X\) be the set of all maximally consistent \(\mathcal{L}\)-theories. For an \(\mathcal{L}\)-sentence \(\sigma\), let \[\langle \sigma \rangle = \{ T \in X \colon \sigma \in T\}\]

Show that

  • \(\langle \sigma \land \tau \rangle = \langle \sigma \rangle \cap \langle \tau \rangle\)
  • \(\langle \neg\sigma \rangle = X \setminus \langle \sigma \rangle\)

Exercise 7
Continuing the previous exercise, let \(\mathcal{O}\) be the topology generated by the sets \(\langle \sigma \rangle\). Show that

  • each \(\langle \sigma \rangle\) is clopen,
  • the \(\langle \sigma \rangle\) form a basis for \(\mathcal{O}\),
  • \(\mathcal{O}\) is Hausdorff.