Computability and Logic

PROBLEMS 317
(In general, there could be more than two as for each U, but the proof would be no
different.) Now let
H(x, u 1 , u 2 ,v 11 ,v 12 ,v 21 ,v 22 )
be the result of replacing any subformula of form U i (w) byG i (w, v i1 ,v i2 ). Then
T ={b: M |= H[b, s 1 , s 2 , a 11 , a 12 , a 21 , a 22 ]}
and is parametrically definable as required.
Problems
25.1 Show how the proof of the existence of averages can be formalized in P, in
the style of Chapter 16.
25.2 Show that there is a recursive relation ≺ on the natural numbers that is also
isomorphic to the order < K on the set K defined in the proof of Theorem 25.1.
25.3 Show that the successor function † associated with ≺ may also be taken to be
recursive.
25.4 Show that in an ∈-model that is not an ω-model, the upper domain cannot
contain all subsets of the lower domain.
The remaining problems outline the proof of the arithmetical Löwenheim–
Skolem theorem, and refer to the alternative proof of the model existence lemma
in section 13.5 and the problems following it.
25.5 Assuming Church’s thesis, explain why, if Ɣ is a recursive set of (code numbers
of) sentences in a recursive language, the set Ɣ* obtained by adding (the code
numbers of) the Henkin sentences to Ɣ is still recursive (assuming a suitable
coding of the language with the Henkin constants added).
25.6 Explain why, if is an arithmetical set of sentences, then the relation
i codes a finite set of sentences ,
j codes a sentence D,
and ∪ implies D
is also arithmetical.
25.7 Suppose Ɣ* is a set of sentences in a language L* and i 0 , i 1 , ...an enumeration
of all the sentences of L*, and suppose we form Ɣ # as the union of sets Ɣ n ,
where Ɣ 0 = Ɣ* and Ɣ n+1 = Ɣ n if Ɣ n implies ∼i n , while Ɣ n+1 = Ɣ n ∪{i n }
otherwise. Explain why, if Ɣ* is arithmetical, then Ɣ # is arithmetical.
25.8 Suppose we have a language with relation symbols and enumerably many
constants c 0 , c 1 ,..., but function symbols and identity are absent. Suppose
Ɣ # is arithmetical and has the closure properties required for the construction
of section 13.2. In that construction take as the element ci
M associated with
the constant c i the number i. Explain why the relation R M associated with
any relation symbol R will then be arithmetical.
25.9 Suppose we have a language with relation symbols and enumerably many
constants c 0 , c 1 ,..., but that function symbols are absent, though identity
may be present. Suppose Ɣ # is arithmetical has the closure properties required