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.
(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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