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The definition is:

"A set of formula's $\Sigma$ axiomizes $Th(M)$ if for all sentences $\phi$ the following applies: $\phi \in Th(M) \Leftrightarrow \Sigma \models \phi$"

Where $Th(M) = \{ \phi | \phi$ is a sentence and $M \models \phi \}$ and $M$ is a model.

What I'm reading than says the following:

According to the definition, a set of sentences $\Sigma$ axiomizes the theory $Th(M)$ if for all sentences $\phi$ the following applies: $\phi \in Th(M) \Leftrightarrow \Sigma \models \phi$

With a given set $\Sigma$ en model $M$, the implication from left to right is usually hard to proof.

My question is: why is this hard to proof? Don't you check if the given sentence is true in the model et voila; you know that it follows from $\Sigma$?

Garth Marenghi
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  • I guess you readt something else this is just a tautology what is the difference between $ M $ and $ \Sigma $ ? – Willemien Mar 23 '14 at 19:19

1 Answers1

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Let $M$, for example, be the complex numbers under addition and multiplication. Let $\Sigma$ be the theory of algebraically closed fields of characteristic $0$.

For this example, the implication from left to right says that the theory of algebraically closed fields of characteristic $0$ is complete. This is a non-trivial result.

André Nicolas
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