Two Applications of Logic to Mathematics

Gaisi Takeuti
Pages: 152
https://www.jstor.org/stable/j.ctt130hk3w

1. Front Matter
(pp. i-iv)
2. Preface
(pp. v-vi)
Gaisi Takeuti
(pp. vii-viii)
4. Introduction
(pp. 1-4)

Mathematical logic is reflection on mathematics. More specifically it is reflection on such questions as, What is logical validity? What is effective calculability? What is a set? What are the basic principles of the universe of sets. Apart from reflecting on these questions themselves, and proving relevant metatheorems, one also wants to know the chances of an effective use of various logical metatheorems in specific branches of mathematics. For example, one might say that Abraham Robinson’s non-standard analysis answers the question of how the completeness theorem can be put to good use in analysis, or that Ax-Kochen theory answers the...

5. Part I Boolean Valued Analysis
• Chapter 1 Boolean Valued Analysis Using Projection Algebras
(pp. 6-50)

A bounded operatorP, of a Hilbert space, is called a projection ifPis self-adjoint andP²=P. We will use the symbolIto denote the identity operator i. e.Ix=xand 0 to denote an operator defined by 0.x=0.

A setBof projections is called a Boolean algebra of projections, if it satisfies the following conditions.

1. BothIand 0 are members ofBand members ofBare pairwise commutable.

2. IfP1andP2are members ofB, so areP_{1}\vee P_{2}, P_{1}\wedge P_{2},and ⦢P1, whereP_{1}\vee P_{2}=P_{1}+P_{2}-P_{1}\cdot P_{2}, P_{1}\wedge P_{2}=P_{1}\cdot P_{2}, and ⦢P1=IP1.

A Boolean algebraBof projections is...

• Chapter 2 Boolean Valued Analysis Using Measure Algebras
(pp. 51-70)

As we pointed out earlier, Dana Scott started Boolean valued analysis [4]. He proposed using measure algebras. In this chapter we will study Boolean valued analysis using measure algebras.

Let (X, S, μ) be a measure space, that is, letSbe a Borel field of subsets ofX, and letμ:S—>[0, ∞] be aσ-finiteσ-additive measure. (Byσ-finite, we mean that there exists a sequenceX_{1},X_{2},X_{3},\cdots \inSsuch that for everyn μ(Xn) <∞ andX=\bigcup_{n}X_{n}.)

LetJ={BS|μ(B)=0} andB=S/J. ThenJis aσ-additive ideal ofSandBis a Booleanσ-algebra. The...

• References
(pp. 71-72)
6. Part II A Conservative Extension of Peano Arithmetic
• [Part II Introduction]
(pp. 73-76)

In my opinion there is a wide gap between logic and mathematics. Let me illustrate with two examples.

1. In recursive function theory, almost all interesting recursive functions are not primitive, but almost all recursive functions, found in mathematical practice, are primitive recursive*.

2. In logic, we can easily construct many arithmetical statements that are not provable in Peano arithmetic, but we hardly find any such statement in mathematical practice.*

One explanation of the second fact might be the following. When we learned to formalize mathematics, the fomalization itself was an important but difficult task. Naturally we chose a very strong system...

• Chapter 1 Real Analysis
(pp. 77-113)

We use the higher type language. The use of higher type language is very convenient since it is the natural language for analysis and we can take all the definitions in analysis as they are without any change. We talk about “types” rather than “orders”.

Definition 1.1.Finite types:0is a finite type (the basic one); if ι1, …,ιnare finite types, then ι=[ι1, …,ιn]is also; those are the only finite types. “Finite” will be omitted most of the time. Type[0]is also called type1.

Definition 1.2. Language.

1.Free and bound variables of each...

• Chapter 2 Complex Analysis
(pp. 114-135)

We will formulate the arithmetic of complex numbers in a system of finite types, which is a conservative extension of Peano Arithmetic. The rational complex numbers are objects of type 0.

Definition 1.1.The system T.

Symbols: N, r, 0, 1,i, +, ., -, ÷, =, <.

The logical system: The system of finite types defined in Chapter1.

Axioms:(1)The equality axiom Eq.

(2)Axioms of Peano Arithmetic relativized to N.

(3)MI, the mathematical induction relativized to N.

(4)Axioms on0, 1, +, ·, -, ÷,and<relativized to r.

(5)\forall x\exists !y\exists !z(r(y)\wedge r(z)\wedge x=y+iz)where x, y, z...

• References
(pp. 136-137)
7. Index
(pp. 138-138)
8. Back Matter
(pp. 139-139)