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• A statement is true in absolute if it can be proven formally from the axioms.
• A statement is true in a model ifit holds for every element , using the interpretation of the formulas inside the model, it is a valid statement about those interpretations.

Assuming your set of axioms is consistent (which is equivalent to the existence of a model), then
$\qquad$ truth in absolute $\Rightarrow$ truth in any model.

Conversely, if a statement is not true in absolute, then there exists a model in which it is false.

Let's take an example to illustrate all this. Let $P$ be a property of integer numbers, and let's assume that you want to know whether the formula $\exists n\in \mathbb Z : P(n)$ is true. Three situations can occur:
You're able to find $n\in \mathbb Z$ such that $P(n)$.
You're able to prove that $\not\exists n\in \mathbb Z : P(n)$
Neither of the above.
In the latter case, there will exist a model $\tilde{\mathbb Z}$ of the integers (it's going to be some ring, probably much bigger than $\mathbb Z$, and that satisfies all the axioms that "characterize" $\mathbb Z$) that contains an element $n\in \tilde {\mathbb Z}$ satisgying $P$.

$\qquad$ truth in absolute $\Rightarrow$ truth in any model.