January 2011
Intermediate to advanced
192 pages
5h 34m
English
Content preview from Computability Theory
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130 Computability Theory
Proof. We use the fact that the union of countably many countable sets is countable. This
implies that the union of countably many degrees must be a countable subset of PN. But
the union of all degrees is PN, which is uncountable. a
6.3 Preordering Relations
Suppose that R is any binary relation on a set U (that is, R ⊆ U × U). As before, we can
write xRy to mean that hx, yi ∈ R.
Definition: R is a preordering on U if it is reflexive on U and is transitive.
Example 1: Turing reducibility ≤
T
is a preordering on PN.
Example 2: Many-one reducibility ≤
m
is a preordering on PN.
Example 3: One-one reducibility ≤
1
is a preordering on PN
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