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08-formal

# 08-formal - Formalization Readings None A book consulted in...

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Formalization Readings: None. A book consulted in the preparation of these slides states, “This book, like almost every other modern mathematics book, develops its subject matter assuming a knowledge of elementary set theory. This assumption is often not justified.” The notation and conventions of arithmetic have been with us from childhood; those of set theory creep into mathematics in high school, and become important in university (which is why they were introduced as early as Math 135). But a gap remains between our high-level understanding of these notions and the formal systems of proof introduced in this course. 1

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To write formulas in formal logic that express ideas from mathematical proof, we definitely have to add symbols to our language such as + and . But this is not enough, because our semantics of predicate logic allows arbitrary interpretations to be assigned to these symbols. We need a way to build in rules that restrict interpretations. This is done by means of the idea of axioms introduced in the discussion of alternate systems of proof for propositional logic. Once we define a set of axioms, we can talk about the theory of those axioms, which is the set of all formulas provable from them. Our goal, then, is to define axioms for arithmetic and set theory. As a short warmup, we will define axioms for group theory. 2
Group theory A group is a set with one distinguished element called the identity, and two operations defined on elements of the set, one unary and one binary. We denote the identity by e , the unary operation applied to x by x * , and the binary operation applied to x and y by x y . The unary operation is supposed to represent an inverse, and the binary operation is supposed to be associative. We need to say these things in the axioms if they are to hold in any theorems of the theory that follows from those axioms. 3

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A1 : x y z ( x ( y z ) = ( x y ) z ) A2 : x ( x e = x ) A3 : x ( x x * = e ) Group theory is the set of all sentences φ such that Γ φ , where Γ = { A 1 , A 2 , A 3 } . It is a rich and useful theory, as explored in PMath 336/346, and we will not go very far into it. In fact, we will prove just one theorem, the right-cancellation law, which states that if x z = y z , then x = y . 4
Here is a mathematical proof from the axioms. Theorem : If x z = y z , then x = y . Proof : x z = y z assumption ( x z ) z * = ( y z ) z * z * to both sides x ( z z * ) = y ( z z * ) A1 x e = y e A3 x = y A2 5

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To formalize the statement of the theorem, we must recognize that the free occurrences of x , y , and z really represent implicit universal quantification. φ = x y z ( x z = y z x = y ) The mathematical proof is actually quite close to a formal proof of Γ φ using natural deduction. Rather than give the whole proof, we will sketch how to obtain it.
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