Let me start with some definitions. Where L_a is the language of arithmetic, define a theory to be consistent if it is not the set of all L_a ground sentences (sentences with no quantifiers or free variables).

I'm having trouble with an equivalent formulation of consistency. A theory is consistent if and only if it has a model. One direction is clear. If M |= T and s \in T then M |= {s} but M |/= {-s}. So T cannot be inconsistent. However, how do we build a model from a consistent theory?

Question

Let me start with some definitions. Where L_a is the language of arithmetic, define a theory to be consistent if it is not the set of all L_a ground sentences (sentences with no quantifiers or free variables).

I'm having trouble with an equivalent formulation of consistency. A theory is consistent if and only if it has a model. One direction is clear. If M |= T and s \in T then M |= {s} but M |/= {-s}. So T cannot be inconsistent. However, how do we build a model from a consistent theory?

ikram002p · Answer

u still wanna answer ?