January 2011
Intermediate to advanced
192 pages
5h 34m
English
Content preview from Computability Theory
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6 Degrees of Unsolvability
6.1 Relative Computability
All the noncomputable sets are noncomputable, but some are more noncomputable
than others. In this chapter,
1
we want to make sense of this idea.
For example, suppose A and B are both noncomputable subsets of N. On the one
hand, we might be able to show that if, hypothetically speaking, we could somehow
decide membership in B, then we could decide membership in A. This would lead us
to the opinion that A is no more undecidable than B is.
On the other hand, we might be able to show that even if our Fairy Godmother gave
us an oracle so we could decide membership in B, there still would be no decision
procedure for A. This might lead us to the opinion that A is more undecidable than B
or else that their ...
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