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

 Messages: [ Previous | Next ]
 Peter Percival Posts: 2,623 Registered: 10/25/10
Re: Exercises in Robbin
Posted: Oct 4, 2017 6:31 AM

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

Date Subject Author
10/2/17 Peter Percival
10/2/17 William Elliot
10/4/17 Peter Percival
10/4/17 William Elliot
10/4/17 Peter Percival
10/9/17 Nick
10/10/17 Peter Percival
10/10/17 Nick
10/10/17 Peter Percival