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Topic: Exercises in Robbin
Replies: 8   Last Post: Oct 10, 2017 1:47 PM

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Peter Percival

Posts: 2,623
Registered: 10/25/10
Re: Exercises in Robbin
Posted: Oct 4, 2017 6:31 AM
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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

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