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If M |= ϕ[s], we shall say that M satisfies ϕ on assignment s or that ϕ is true in M on assignment s. We will often write M ϕ[s] if it is not the case that M |= ϕ[s]. Also, if Γ is a set of formulas of L, we shall take M |= Γ[s] to mean that M |= γ[s] for every formula γ in Γ and say that M satisfies Γ on assignment s. Similarly, we shall take M Γ[s] to mean that M γ[s] for some formula γ in Γ. 1. The key clause is 5, which says that ∀ should be interpreted as “for all elements of the universe”.

The idea is that every element of the universe which Σ proves must exist is named, or “witnessed”, by a constant symbol in C. Note that if Σ ¬∃x ϕ, then Σ ∃x ϕ → ϕxc for any constant symbol c. 8. 11. Suppose Γ and Σ are sets of sentences of L, Γ ⊆ Σ, and C is a set of witnesses for Γ in L. Then C is a set of witnesses for Σ in L. 2. Let LO be the first-order language with a single 2place relation symbol, <, and countably many constant symbols, cq for each q ∈ Q. Let Σ include all the sentences (1) cp < cq , for every p, q ∈ Q such that p < q, (2) ∀x (¬x < x), (3) ∀x ∀y (x < y ∨ x = y ∨ y < x), (4) ∀x ∀y ∀z (x < y → (y < z → x < z)), (5) ∀x ∀y (x < y → ∃z (x < z ∧ z < y)), (6) ∀x ∃y (x < y), and (7) ∀x ∃y (y < x).

Observe that any first-order language L has countably many logical symbols. It may have uncountably many symbols if it has uncountably many non-logical symbols. Unless explicitly stated otherwise, we will 1It is possible to formalize almost all of mathematics in a single first-order language, like that of set theory or category theory. However, trying to actually do most mathematics in such a language is so hard as to be pointless. 2Specifically, to countable one-sorted first-order languages with equality.

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