| |||
|---|---|---|---|
| 78.3 | 117 | ||
| 78.4 | 122 | ||
| 78.5 | 90 | ||
| 78.6 | 107 | ||
| ¬zP0 | 108 | ||
| Induction 1 | 123 | ||
| The basic fact about the weak ancestral | 134,136 | ||
| The lemma | 141 | ||
| Induction 2 | 144 | ||
| (5′) | 145 | ||
| (4′) | α149 | ||
| (1′) | 150 | ||
| Induction 3 | 152 | ||
| (2) | 154 | ||
| (0′) | 155 | ||
IN SECOND-ORDER ARITHMETIC
The language (of second-order arithmetic) contains variables x, y, z, . . . over natural numbers; variables α, β, γ, . . . over sets of numbers; and variables ρ, Ϛ, . . . . over binary relations of numbers. (We do not need variables over n-place relations for n>2.) Its non-logical symbols are 0, s, +, ×, >. Terms t are built up out of 0, s, +, × as usual; the atomic formulas are t=t′, t>t′, αt, ρtt′; formulas are then built up as usual.
The axioms of second-order arithmetic are induction: (α0 ∧ ∀x(αx → αsx) → αx) [a single formula]; the recursion axioms for successor, plus and times and the definition of less-than:
0≠sx, sx=sy → x=y, x+0 = x, x+sy = s(x+y), x×0 = 0, and the comprehension axioms (which are axioms of standard second-order logic): ∃α∀x(αx ↔ A), ∃ρ∀x∀y(ρxy ↔ B), A a formula in which α is not free and
B a formula in which p is not free. Since we have + and ×, we could have dispensed with binary relation variables;
and since we have binary relation variables, we could have dispensed with + and ×
and set variables: J, = λxyιz(x2 +2xy+y2 +3x+y = 2z), is an onto pairing function. Thus if we have + and ×, we have J, and so we can replace ρtt′ by αJ (t,t′). And we
can define x+y = z and x×y = z from 0 and s using binary relation variables: x+y = z -424-
x×sy = (x×y)+x, x
Questia, a part of Gale, Cengage Learning. www.questia.com
Publication information:
Book title: The Philosophy of Mathematics Today.
Contributors: Matthias Schirn - Editor.
Publisher: Clarendon Press.
Place of publication: Oxford.
Publication year: 1998.
Page number: 424.
This material is protected by copyright and, with the exception of fair use, may not be further copied, distributed or transmitted in any form or by any means.
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