I am trying to assess whether a set of formulas is complete in predicate logic. The language at hand only has a one-ary predicate S. I am proving whether a set is complete by showing the following:
- Γ is consistent
- for every φ ∈ L ,it holds that either ∆⊢nφ or ∆⊢n¬φ.
The set of formulas is {∃xSx, ∃x¬Sx}
How do I go about proving whether the set is complete? Intuitively I would say the set is not complete, since when we have a model with D = {1}, it would lead to a contradiction?