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46 KAM Mathematical Colloquium

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Prof. THEODORE A. SLAMAN

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Berkeley

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ASPECTS OF THE TURING JUMP

March 6, 2003
Lecture room in the 4th floor, Letenska 17, Praha
1
14:00 PM

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Abstract

The Turing jump is the function which maps $X$, a set of
natural numbers, to $X'$, the halting problem relative to $X$. $X'$
is the canonical example of an arithmetically definable set which is
not recursive in $X$.
We will discuss two aspects of this remarkable function and its
iterations. First, we will examine the thesis that the only robust
definable way to produce a set from $X$ which is strictly more
complicated than $X$ is to produce a set which is as least as
complicated as $X'$. Second, we will discuss how this canonical
aspect of the jump and of the double-jump leads to a proof that the
jump is definable in the partial order of the Turing degrees.