# The Philosophy of Mathematics Today

By Matthias Schirn | Go to book overview
 Proposition of this paper Proposition of Grundgesetze
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

APPENDIX 2. INTERPRETING FREGE ARITHMETIC
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,
x×sy = (x×y)+x, x

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-

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