Thread: Fol 9

Enderton 2.6.2

Let $\displaystyle T_1$ and $\displaystyle T_2$ be theories (in the same language) such that

(i)$\displaystyle T_1 \subseteq T_2$,
(ii)$\displaystyle T_1$ is complete, and
(iii)$\displaystyle T_2$ is satisfiable. Show that $\displaystyle T_1 = T_2$.

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Before I get started, I need to understand some definitions related to this problem.

(*) A theory T is said to be complete iff for every sentence $\displaystyle \sigma$, either $\displaystyle \sigma \in T$ or $\displaystyle \neg \sigma \in T$.

I am confused with an unsatisfiable theory, where a theory is defined to be a set of sentences closed under logical implication.

The textbook gives an example of an unsatisfiable theory, which is the theory consisting of all sentences of the language. Why is this unsatisfiable theory? Any description or example?

Thank you.

A theory is unsatisfiable if no structure makes all sentences from the theory true. Any theory that contains a formula and its negation is unsatisfiable.

A theory is defined to be a set of sentences closed under logical implications.

As you said, if I am not mistaken, for an unsatisfiable theory T, there exists a sentence $\displaystyle \sigma$ such that

$\displaystyle T \models \sigma$ and $\displaystyle T \models \neg \sigma$, where $\displaystyle \sigma \in T$ and $\displaystyle \neg \sigma \in T$.

I was having hard time finding an example of T and $\displaystyle \sigma$. Is there any concrete example of an unsatisfiable theory T and $\displaystyle \sigma$ for the above?

Additionally, is there any example of an inconsistent theory T and a sentence $\displaystyle \sigma$ such that

$\displaystyle T \vdash \sigma$ and $\displaystyle T \vdash \neg \sigma$?

The completion of the set $\displaystyle \{0=0, \neg 0=0\}$ under logical implication is unsatisfiable and inconsistent. It logically implies and derives any sentence.

Originally Posted by emakarov

The completion of the set $\displaystyle \{0=0, \neg 0=0\}$ under logical implication is unsatisfiable and inconsistent. It logically implies and derives any sentence.

What is the completion of the set $\displaystyle \{0=0, \neg 0=0\}$ under logical implication?

As you know, T is a theory iff T is a set of sentences such that for any sentence $\displaystyle \sigma$ of the language,

$\displaystyle T \models \sigma \rightarrow \sigma \in T$.

If it logically implies anything, is it the theory consisting of all the sentences of the language? If the theory consists of all the sentences of the language, is the empty structure the only model for it?

Thanks.

Originally Posted by logics

What is the completion of the set $\displaystyle \{0=0, \neg 0=0\}$ under logical implication?

It's the set of all sentences.

Originally Posted by logics

If it logically implies anything, is it the theory consisting of all the sentences of the language?

Yes.

Originally Posted by logics

If the theory consists of all the sentences of the language, is the empty structure the only model for it?

No; rather, no structure is a model of it. Also, the definition of structure in first-order logic usually has a requirement that the universe is nonempty. Otherwise, the formula $\displaystyle \forall x\,Px\to\exists x\,Px$ is not a tautology.