Peter Percival wrote: > If anyone has solutions to exercises 11 to 15 on pages 169-70 of Joel W. > Robbin's /Mathematical logic a first course/, W.A. Benjamin, inc., 1969 > and would like to share them with me I'll be very grateful. > > At the moment I'm struggling to understand > > $\mathfrak M_Q\vDash T(n,a)$ iff $n=1,2,\dots$ and $a$ is an > $n$-tuple of natural numbers > > The only terms of Q seem to be 0 and the individual variables, so I > don't even know what $n$ (in the formula $T(n,a)$) is. > My thoughts so far. The terms that n (etc) denote have to be defined. Among the definitions of the denotations of predicate letters is
$\mathfrak M_Q\vDash S(a,b)$ iff $a$ and $b$ are natural numbers and $b = a+1$
so let N(x) be defined to be EyS(x,y) and read as "x is a natural number". Also let there be an axiom
Ux(N(x) -> E!yS(x,y))
and define that y to be 0' when x is 0, 0'' when x is 0', and so on. Also let 0', 0'', ... be 1, 2, ...
-- Do, as a concession to my poor wits, Lord Darlington, just explain to me what you really mean. I think I had better not, Duchess. Nowadays to be intelligible is to be found out. -- Oscar Wilde, Lady Windermere's Fan