Preparing for Logic Final: Union of T_n Satisfiable & Not Finitely Axiomatizable

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    Final Logic Review
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SUMMARY

The discussion focuses on the logic problem concerning the union of a strictly increasing sequence of satisfiable L-Theories, denoted as T_n. It is established that the union of T_n is satisfiable due to the compactness theorem, as every finite subset is satisfiable. However, the union is not finitely axiomatizable, as any finite set of axioms can only capture up to T_n, leaving T_n+1 unaccounted for, which demonstrates the lack of finite axiomatization.

PREREQUISITES
  • Understanding of L-Theories and their properties
  • Familiarity with the compactness theorem in logic
  • Knowledge of satisfiability and axiomatization concepts
  • Experience with sequences and their implications in logic
NEXT STEPS
  • Study the compactness theorem in detail
  • Explore the properties of satisfiable L-Theories
  • Research the concept of finitely axiomatizable theories
  • Examine examples of strictly increasing sequences of L-Theories
USEFUL FOR

Students of logic, particularly those preparing for exams in mathematical logic or theoretical computer science, as well as educators teaching these concepts.

moo5003
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I'm studying for my logic final and I can't seem to find an answer for this practice problem:

(Using < as proper subset since I don't have the right type)

T_1 < T_2 < T_3... be a strictly increasing sequence of satisfiable L-Theories.

a) Show that the union of T_n is satisfiable (over all n in the natural numbers).

b) Show that the union of T_n is not finitely axiomatizable.



A) Pretty simple, every finite subset is satisfiable since the largets T_n is satisfiable thus by compactness their entire union is satisfiable.

B) This is were I have some problems. I'm not sure how to go about showing this. I want to show that any finite amount of sentances can only axiomatize up to T_n and then we can simply show that T_n+1 is not axiomatized. Any ideas on this?

EDIT: I posted this in the wrong forum apparently, if anyone could move it to the logic section I would appreciate it.
 
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The midterm is over though I would still like to know how to solve this problem.
 

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