Two Applications of Logic to Mathematics

by Gaisi Takeuti

Overview

Using set theory in the first part of his book, and proof theory in the second, Gaisi Takeuti gives us two examples of how mathematical logic can be used to obtain results previously derived in less elegant fashion by other mathematical techniques, especially analysis. In Part One, he applies Scott- Solovay's Boolean-valued models of set theory to analysis by means of complete Boolean algebras of projections. In Part Two, he develops classical analysis including complex analysis in Peano's arithmetic, showing that any arithmetical theorem proved in analytic number theory is a theorem in Peano's arithmetic. In doing so, the author applies Gentzen's cut elimination theorem.

Although the results of Part One may be regarded as straightforward consequences of the spectral theorem in function analysis, the use of Boolean- valued models makes explicit and precise analogies used by analysts to lift results from ordinary analysis to operators on a Hilbert space. Essentially expository in nature, Part Two yields a general method for showing that analytic proofs of theorems in number theory can be replaced by elementary proofs.

Originally published in 1978.

The Princeton Legacy Library uses the latest print-on-demand technology to again make available previously out-of-print books from the distinguished backlist of Princeton University Press. These editions preserve the original texts of these important books while presenting them in durable paperback and hardcover editions. The goal of the Princeton Legacy Library is to vastly increase access to the rich scholarly heritage found in the thousands of books published by Princeton University Press since its founding in 1905.

Product Details

ISBN-13:	9780691638379
Publisher:	Princeton University Press
Publication date:	04/19/2016
Series:	Publications of the Mathematical Society of Japan , #1576
Pages:	148
Product dimensions:	6.10(w) x 9.30(h) x 0.90(d)

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Two Applications of Logic to Mathematics

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By Gaisi Takeuti

PRINCETON UNIVERSITY PRESS

Copyright © 1978 The Mathematical Society of Japan
All rights reserved.
ISBN: 978-0-691-08212-7

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CHAPTER 1

Boolean Valued Analysis Using Projection Algebras

§ 1. Hilbert space

A bounded operator P, of a Hilbert space, is called a projection if P is self-adjoint and P2=P. We will use the symbol I to denote the identity operator i. e. Ix=x and 0 to denote an operator defined by 0.x=0.

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

1. Both I and 0 are members of B and members of B are pairwise commutable.

2. If P1, and P2 are members of B, so are P1 [disjunction] P2, P1 [and] P2 and [??]P1, where P1 [disjunction] P2 = P1 + P2 - P1 x P2, P1 [and] P2 = P1 x P2, and — P1 = I - P1.

A Boolean algebra B of projections is said to be complete if B is not only complete as a Boolean algebra but also satisfies the following condition. If [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], then the range of P, denoted by R(P), is the closure of the linear space spanned by [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

From now on, let B be a complete Boolean algebra of projections. Let A be a self-adjoint operator and let A = ∫ λdEλ be its spectral decomposition. Then A is said to be in (B) if for every real λ, Eλ is a member of B.

Let A and B be self-adjoint and let A = ∫ λdEλ and B = ∫ λdE'λ be their spectral decompositions. Then A and B are said to be commutable if for every pair λ, λ' of reals

Eλ x E'λ' = E'λ' x Eλ

If A and B are bounded, then the commutativity of A and B is equivalent to A x B = B x A.

The following single fact is very useful in our work and will be used without mention.

Lemma 1.1.If {Aα} be a set of self-adjoint, pairwise commutable operators, then there exists a complete Boolean algebra of projections such that for every α, Aα is in (B).

Let A and B be commutable self-adjoint operators. It is usual to define A+B as the operator satisfying the conditions

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where D(A) denotes the domain of A. The operator A+B, defined in this way, has a unique closed extension. For our purposes we define A+B to be this unique closed extension. The operator A+B is also self-adjoint. In the same way, A x B is defined to be the unique closed extension of the operator which maps x, with and x [member of] D(B) and Bx [member of] D(A), to ABx. The operator A x B is also self-adjoint and A x B = B x A. Because of this definition, there is a possibility that A+B and/or A x B is defined on the whole Hilbert space, and therefore bounded, even if A and B are unbounded. In general, if the result of an operator 0(A, B) is not closed but has a unique closed extension, we define 0(A, B) to be the unique closed extension of the result.

An operator N is said to be normal, if N = A+iB where A and B are self-adjoint and commutable. Also and N* = A - iB and NN* = N*N = A2 + B2. Furthermore, N is said to be in (B), if A and B are in (B). We define |N| to be [square root of (A2+B2)]. The operator |N| is self-adjoint.

Let A and B be self-adjoint and commutable. Then A ≤ B if and only if for every x [member of] D(A) [intersection] D(B), (Ax, x) ≤ (Bx, x).

§ 2. The model V(B)

In this section, we summarize the necessary back ground about the Boolean valued model V(B) of set theory. For detail, see § 13. and § 16. in [10], though +, x, -, Π, Σ are used there as Boolean operations in the place of [disjunction], [and], [??], inf, sup, and O and I are used in the place of 0 and I.

First we shall give a rough idea of V(B). Let D be a domain. A set of members in D is decided by assigning true or false to x [member of] A for every member x in D. A B-valued set of members in D is decided by assigning any element P in B to x [member of] A for every member x in D. We write [[x [member of] A]] = P if the assigned value of x [member of] A is P. The symbol '[x [member of] A] = P' is read as 'x [member of] A holds with probability P'. However, '[x [member of] A] = I' is read as 'x [member of] A holds' and '[x [member of] A] = 0' is read as 'x [member of] A does not hold'.

The universe V of all sets is obtained by starting with the empty set and by creating sets of sets, sets of sets of sets etc. The Boolean valued universe V(B) is obtained by starting with the empty set and by creating Boolean valued sets of Boolean valued sets, Boolean valued sets of Boolean valued sets of Boolean valued sets etc. By interpreting true by I and false by 0, there exists a natural embedding, denoted by [square root], of V into V(B) i. e.

[square root]: V -> V(B).

Let a [member of] V. The corresponding element in V(B) is denoted by [??].

Formally, we shall carry this out as follows. Let B be a complete Boolean algebra (of projections). For an ordinal α, we define V(B)α by transfinite induction as follows.

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where On is the class of all ordinal numbers.

It is easy to see that

1) if α is a limit, then [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

2) V(B)α+1 = {u | u: D(u) -> B and D(u) [subset of equal to] V(B)α},

3) if α ≤ β, then V(B)α [subset or equal to] V(B)β.

For u, v [member of] V(B), [u [member of] v] and [u = v] are defined as functions from V(B) x V(B) to B satisfying the following properties.

1. [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

2. [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where P1 =>P2 is [??] P1 [disjunction] P2 i. e.

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

In the following, we also use [and], [disjunction], [??], => as logical connections. Let φ be a formula in set theory, that is, let ψ be obtained by applying logical symbols to atomic formulas of the form u [member of] v or u = v. If φ does not contain any free variable and all the constants in φ are members in V(B), we define [φ] by the following rules.

1. [[??] φ] = [??] [φ] = I - [φ]

2. [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

3. [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

4. [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

5. [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

From this definition we have the following basic properties.

1. [u=u] = I

2. [u=v]=[v=u]

3. [u1=u2] x [u2=u3] ≤ [u1=u3]

4. [u=v] x [φ(u)] ≤ [φ(v)]

5. [u=v] = [[for all]x(x [member of] u iff x [member of] u)].

Since all the statements in analysis can be expressed in the language of set theory, we can assign a value in B to any statement in analysis by [].

Let M=R(P). Roughly the meaning of [φ]=P is that φ holds if we restrict our Boolean algebra B to a subspace M. In another words, the meaning of [φ]=P is that φ does not hold if we restrict our Boolean algebra B to a subspace M[perpendicular to].

The theorem, which is the base of our work, is the following.

Theorem 2.1.The Boolean valued universe V(B)is a model of ZF set theory with the axiom of choice. This set theory we denote by ZFC. In other words, if φ is a theorem of ZFC, then [φ]=I i.c. φ holds in V(B).

Since every theorem in contemporary mathematics is a theorem in ZFC, we can express this in the following way. If φ is a theorem, then φ holds in V(B). Moreover, from the proof of the Theorem 2. 1., we have the following corollary.

Corollary 2. 2.If φ is a theorems then 'φ holds in V(B)' is also a theorem.

Since conditional is often used in mathematics, the following equivalence is useful in practice

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

The following lemma which is Theorem 13. 13. in [10] is also useful.

Lemma 2. 3.For u [member of] V(B),

1. [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

2. [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Now we define the embedding [square root]: V ->V(B) precisely, by transfinite induction as follows.

For y [member of] V,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

i. e. [??] is a constant function such that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and its value is constantly I.

Obviously the following proposition holds.

Proposition 2. 4. (Theorem 13. 17. in [10]). For x, y [member of] V,

1. [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

2. [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

In order to develop analysis in set theory, first the natural numbers are constructed from ?S, which is also denoted by 0, the integers are constructed as pairs of natural numbers, the rational numbers are constructed as pairs of integers, and finally, the real numbers are constructed by Dedekind's cuts of rational numbers. Since V(B) satisfies ZFC, we adopt the same definition of the natural numbers, integers, rational numbers and real numbers so that all the theorems in analysis also hold in V(B). Let us denote the set of all natural numbers by ω, the set of all rational numbers by Q, the set of all real numbers by R and the set of all complex numbers by C. The first natural question is what are ω Q, R, and C in V(B). First two are answered as follows, (cf. pp. 129-130 in [10]).

1. Let n be a natural number. Then n in V(B) is [??]. More precisely there exists a formula φ(x) which define n in the sense that [there exists]!xφ(x) [and] φ(n) is provable. Then [there exists]!xφ(x) [and] φ([??]) holds in V(B). In particular 0, in V(B), is [??], that is, [??] is the empty set in V(B). In V(B), ω is [??].

Moreover [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], in V(B), are (n+m)[square root] and (nxm)[square root] respectively. Similarly [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

2. Let r be a rational number. Then r, in V(B), is [??]. Moreover [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], hold in V(B). And, [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Finally Q, in V(B), is Q. We often write n, r, ω and Q in the place of [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], and [??]. Note, in passing, that R, in V(B), may not be R. In many cases, we can prove that [[??]=R]=0.

A subset {{Pα} of B is called a partition of unity if the following conditions are satisfied.

1. If α ≠ β, then Pα x Pβ = 0.

2. [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where [summation over α] Pα is defined in the strong topology. Let {Pα} be a partition of unity and let {uα} be a subset of V(B). Then there exists an element u of such that

[for all]α [u=uα] ≥ Pα.

Moreover if there exists another u' such that [for all]α(u'=uα] ≥ Pα), then [u=u']= I, i. e. u=u' holds in V(B). We denote this u by [summation over α] uαPα which is determined uniquely in the sense of equality in V(B). (cf. Theorem 6. 9. and Corollary 16. 3. in [10]).

(Continues...)

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Table of Contents

• Frontmatter, pg. i
• Preface, pg. v
• Contents, pg. vii
• Introduction, pg. 1
• Introduction, pg. 5
• Chapter 1. Boolean Valued Analysis Using Projection Algebras, pg. 6
• Chapter 2. Boolean Valued Analysis Using Measure Algebras, pg. 51
• References, pg. 71
• Introduction, pg. 73
• Chapter 1. Real Analysis, pg. 77
• Chapter 2. Complex Analysis, pg. 114
• References, pg. 136
• Index, pg. 138