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In his notes "Gödel without (too many) tears" Peter Smith (2014), defines negation completeness in the following way:

A theory $T$ is negation complete if it formally decides every closed well-formed formulaof its language, i.e. for every sentence $A$, $T ⊢ A$ or $T ⊢ ¬A$

Does negation completeness as defined above correspond to the notion of maximality?

Here is the definition of maximality given by some lecture notes I'm studying:

A theory $T$ is maximal $iff$, for every formula $A$ of the language, either $A\in T \quad\vee\; \sim A \in T$

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user405159
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    The quick answer is "yes". See https://en.wikipedia.org/wiki/Completeness_(logic) for information about various notions of completeness that all coincide in the context of classical first-order logic. If that doesn't help you to understand the lecture notes you are studying, then you'll have to supply some more information. – Rob Arthan Mar 21 '17 at 22:37
  • The quick answer is enough for me, as I'm just trying to understand if I get the main concepts. – user405159 Mar 21 '17 at 22:40
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    Whenever you have a minute, please take a look at how to format mathematics on Math SE. Furthermore, I suggest that you bookmark the following very useful MathJax guide for quick reference. Cheers! –  Mar 21 '17 at 22:51

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Short answer "yes".

Longer answer: Well, it strictly speaking will depend exactly what you count as a "theory". We are interested in formally axiomatized theories here when looking at Gödel's theorems, so what we care about is that we have some axioms to play with and some logical apparatus. Suppose I give you e.g. some axioms $\Sigma$ for Peano Arithmetic, and specify the relevant deductive apparatus, defining a consequence relation $\vdash$. But what's the "theory" so defined?

One natural line is to think of the theory as the pair $(\Sigma, \vdash)$ comprising the axioms $\Sigma$ plus the consequence relation. Another line is to think of a theory as comprising $\Theta$ the set of consequences of those axioms via that logical apparatus.

If you think of a theory $T$ the first way, then $A \in T$ and $T \vdash A$ don't come to the same. If you think of a theory the second way, then they do. Nothing much is going to hang on this, as everything we want to say about axioms, logics, consequences, negation-completeness can still be said whichever way we choose to pin down the word "theory".

Peter Smith
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