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I've trying to solve this for a while.

Problem

Let $L$ be a first order language with the predicate symbol $p$. Let S be the extension obtained by adding to $K_L$ the open $wf. p(x, y)$. Is S consistent?

Definitions

In my bibliography there is not definition of openness, but closeness. It is: A $wf. A$ of $L$ is said to be closed if no variable occurs free in $A$.

$K$ is a consistent first order system.

The book I'm using is: Logic for mathematicians, Hamilton.

My thoughts

Well, I think that it is consistent because a model can be created for $S$.

My question

In reality, what it is really puzzling me is the open word. What is the difference of creating an extension by adding an open or a closed $wf.$?

Thanks in advance.

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    @AndréNicolas, thank you very much for your comment. I've modified the post$a$bit in order to make my question cleaner. – 2012-12-14

1 Answers 1

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The usual convention is that if we put the formula $\varphi(x,y)$ is a system of axioms, that is equivalent to using $\forall x\forall y\varphi(x,y)$. Just shorter, saves trees.

Recall for example that one of the axioms used in the definition of group is $x(yz)=(xy)z$. There is implicit universal quantification of the three variable symbols.