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How can a theory, $T$ (a set of sentences in $L_{PA}$) which is empty imply something?

Is it stated and assumed trivially that it implies a sentence such as $\phi(x): \forall x : x=x$ is implied by $T$. I don't see how this is the case.

Thanks in advance

Asaf Karagila
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mrhappysmile
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    All the valid sentences in the language of f-o arithmetic are implied by the empty set (of sentences); in addition to Asaf's example, you can consider any instance of $\forall x \varphi \to \varphi^t_x$, like $\forall x (x+S(0)=S(x)) \to (0+S(0)=S(0))$. – Mauro ALLEGRANZA May 07 '15 at 12:44

1 Answers1

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You can look at your inference rules and logical axioms, stating that $x=x$, and that $\varphi\to\forall x\varphi$.

Or you can look at the completeness theorem, and note that in any model of the empty theory, $\forall x(x=x)$ is true, so it must prove that.

Asaf Karagila
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