By Dieter Probst, Peter Schuster

This booklet presents the reader with learn bobbing up from the Humboldt-Kolleg 'Proof' held in Bern in fall 2013, which accumulated best specialists actively concerned with the concept that 'proof' in philosophy, arithmetic and laptop technology. This quantity goals to do justice to the breadth and intensity of the topic and offers proper present conceptions and technical advances that includes 'proof' in these fields.

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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.