[FOM] Question about entailment between sentences belonging to different theories
addamo@wp.pl
addamo at wp.pl
Mon Jun 9 16:06:01 EDT 2003
Let theory T1 be the arithmetic of natural numbers with individual
variables: x, y, z, ...; S (symbol of successsor function); 0 (zero
constant); + (addition symbol); * (multiplication symbol); other first order
logic symbols ~ (negation), => (implication), A (for all), E (exists), =
(identity symbol);
Axioms:
N1. Ax ~(0=Sx),
N2. Ax,y Sx=Sy => x=y,
N3. Ax x + 0 = x,
N4. Ax,y x + Sy = S(x + y),
N5. Ax x * 0 = 0,
N6. Ax,y x * Sy = x*y + x,
N7. (phi(0) ^ Ax(phi(x) => phi(Sx))) => Ax phi(x). (axiom schema for each
phi).
Let theory T2 be the second order arithmetic of real numbers with primitive
concepts concernig individuals: +, *, < (less then), =, 0, 1; and variables
for individuals x,y,z, ..., and for sets X, Y, Z, ..., functions
F,G,H,.....; symbol # (x # X reads elemnt x belongs to the set X); symbol Q
for null set- .
Axioms:
R1-R15 axioms characterizing primitive concepts - addition, multiplication,
less then,
R16 ~ (X = Q) ^ ~(Y = Q) ^ Ax,y(x # X ^ y # Y => x < y) =>
Ez(Ax,y(x # X ^ y # Y => (x < z or x = z) ^ (z < y or z = y))).
R17 EX Ax(x # X <=> psi(x)) (schema of construction axioms)
R18 Ax(x # X <=> x # Y) => X = Y.
R19 axiom of constructions for functions,
R20 axiom of extensionality for functioms.
I hope this rough description will suffice to put more precise the question:
in which sense could we say 'R16 entails N7(phi)' or simply 'R16 entails
N7".
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