Below are some basic properties of a maximally consistent set :
Suppose , then by 1. Then is not consistent (since is maximal), which means , or , or .
If , then by 1, so by 2, and therefore by 1 again.
If , then by 3., so that is a deduction of from , showing that is not consistent.
If is a theorem, then , so that by 1. If and , then is a deduction of from , so by 1.
This is true for any consistent set.
Suppose . If , then since is closed under modus ponens. Conversely, suppose implies . This means that . Then by the deduction theorem, and therefore by 1.
Suppose , then by modus ponens on theorems and , we get , since is a logic by 5. Conversely, suppose , then by modus ponens twice on theorem , we get by 5.
any complete consistent theory is maximally consistent.
any consistent set satisfying the condition in 3 above is maximally consistent.
Suppose is complete consistent. Let be a consistent superset of . is also complete. If , then , so since is consistent. But then since is a superset of , which means since is complete. But then since is deductively closed, which is a contradiction. Hence is maximal.
Next, suppose is consistent satisfying the condition: either or for any wff . Suppose is a consistent superset of . If , then by assumption, which means since is a superset of . But then both and are deducible from , contradicting the assumption that is consistent. Therefore, is not a proper superset of , or . ∎
In the converse of 2, we require that be a theory, for there are complete consistent sets that are not deductively closed. One such an example is the set of all propositional variables: it can be shown that for every wff , exactly one of or holds.
So far, none of the above actually tell us that a maximally consistent set exists. However, by Zorn’s lemma, it is not hard to see that such a set does exist. For more detail, see here (http://planetmath.org/LindenbaumsLemma).
|Date of creation||2013-03-22 19:35:13|
|Last modified on||2013-03-22 19:35:13|
|Last modified by||CWoo (3771)|