Computability and Logic

300 DECIDABILITY OF ARITHMETIC WITHOUT MULTIPLICATION
This in turn is coextensive with
(iii)
∃y(D jkm (y − jki)&ks < y & y < jt)
which is the sentence on the right, except for relettering the variable. For if x is as
in (ii), then y = jkx will be as in (iii); and conversely, if y is as in (iii), then since
jk divides y − jki, jk must divide y, which is to say that y will be of the form jkx
for some x, which x will then be as in (ii).
(30) At this point, the only occurrences of x are in sentences of the form
Replace this by the disjunction of
∃x(D m (x − i)&s < x & x < t).
D m (s + j − i)&s + j < t
for all j with 1 ≤ j ≤ m. This step is justified because, given two integers a and b,
there will be an integer strictly between them that leaves the same remainder as i
when divided by m if and only if one of a + 1,...,a + m is such an integer.
We now have eliminated x altogether, and have obtained a quantifier-free formula
coextensive with our original formula and involving no additional free variables, and
we are done.
Problems
24.1 Consider monadic logic without identity, and add to it a new quantifier
(Mx)(A(x) > B(x)), which is to be true if and only if there are more x such
that A(x) than there are x such that B(x). Call the result comparative logic.
Show how to define in terms of M:
(a) ∀ and ∃ (so that these can be officially dropped and treated as mere
abbreviations)
(b) ‘most x such that A(x) are such that B(x)’
24.2 Define a comparison to be a formula of the form (Mx)(A(x) >B(x)) where
A(x) and B(x) are quantifier-free. Show that any sentence is equivalent to a
truth-functional compound of comparisons (which then by relettering may be
taken all to involve the same variable x).
24.3 As with sets of sentences of first-order logic, a set of sentences of logic with
the quantifier M is (finitely) satisfiable if there is an interpretation (with a finite
domain) in which all sentences in the set come out true. Show that finite satisfiability
for finite sets of sentences of logic with the quantifier M is decidable.
(The same is true for satisfiability, but this involves more set theory than we
wish to presuppose.)
24.4 For present purposes, by an inequality is meant an expression of the form
a 1 x 1 +···+a m x m § b
where the x i are variables, the a i and b are (numerals for) specific rational
numbers, and § may be any of ,≥. A finite set of inequalities is coherent