There are statements for natural number x, like followings
m: "x is even natural number"
n: "x+1 is odd number and x>1"
l: "x is positive integer multiple of two"
m, n, l has same boolean value for x. If m(x) is true, n(x), l(x) is also true. m(x) is false, n(x), l(x) is also false.
Let's define this statements are "equivalent for x." Hence n, l is equivalent for x with m.
q: "x is natural number and 3.5 < x < 10"
p: "x is integer and 3 < x < 19/2"
We can say "q is equivalent for x with p"
Now, let set M is "all Gödel numbers of the statements that are equivalent for x with statement m"
Set M is infinite set of natural numbers that is dependent for m.
My question: For given m (a statement of x), set M is recursively enumerable?