What is the relation between consistency and soundness in mathematical logic? Are consistency and soundness the same or some how related in mathematical logic?
 A: On notation, if $\Gamma$ is a  collection of formulas and $P$ is a formula, then:


*

*$\Gamma \vdash P$ means that, using the formulas in $\Gamma$, you can use the rules of inference to deduce $P$

*$\Gamma \models P$ means that, in every way to interpret logic so that the formulas in $\Gamma$ are true, $P$ is also true

*$\bot$ is a symbol denoting a contradictory proposition



Soundness is a property that first-order logic has: 

$ \Gamma \vdash P $ implies $\Gamma \models P $

Inconsistency is a property that a set of formulas might have.  "$\Gamma$ is inconsistent" means 

$\Gamma \vdash \bot$

A consistent set of formulas is one that is not inconsistent.
A: They are two separate concepts. Soundness is used to describe a logical argument, while consistency is used to describe a set of sentences.
A logical argument is sound if and only if the logical argument is valid and the premises are true.
A set of sentences is consistent if and only if it is possible for all the sentences to be true.
