# Automorphisms of the Lattice of Recursively Enumerable Sets by Peter Cholak

By Peter Cholak

This paintings explores the relationship among the lattice of recursively enumerable (r.e.) units and the r.e. Turing levels. Cholak provides a degree-theoretic procedure for developing either automorphisms of the lattice of r.e. units and isomorphisms among numerous substructures of the lattice. as well as supplying one other evidence of Soare's Extension Theorem, this method is used to turn out a suite of recent effects, together with: each non recursive r.e. set is automorphic to a excessive r.e. set; and for each non recursive r.e. set \$A\$ and for each excessive r.e. measure h there's an r.e. set \$B\$ in h such that \$A\$ and \$B\$ shape isomorphic valuable filters within the lattice of r.e. units.

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Additional resources for Automorphisms of the Lattice of Recursively Enumerable Sets

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However, the term "cause" has been assiduously avoided for several reasons. First, the word "cause" has a multitude of usages some of which cut across the present classification and obscure those distinctions being pointed out. Moreover, it would be most helpful at this stage to remain as neutral as possible with regard to various philosophical analyses of causation. F o r these reasons, then, we shall avoid the term "cause" until a later chapter when it can command our full attention. The central puzzle about genesis involves a question of change—change in state or condition or location.

But it suggests the sort of material one would expect if one asked Q 3 . " It is certainly not the meaning of the words themselves which is desired in Q 3 . The various usages of the maxim would be helpful and might be a part of the explanation, but one might be aware of the situations in which a maxim is used and still not know the meaning. Such is commonly the case when one uses a foreign language that is grasped only imperfectly. Moreover, it is conceivable that certain maxims no longer are used in any situations at all; they are obsolete.

It is only the answer to the second question, where a relationship between the subject and its presumed goal is elucidated, which can be called an explanation. The distinction, however, is not quite so straightforward as it seems, particularly with regard to human behavior. Consider the following example. Q 3 : Explain why John is pounding nails into that tire. A 3 : H e is earning money. " Answer 3 might well be considered unsatisfactory as an explanation because in asking Q 3 , we are probably also meaning to ask how the goal is to be achieved through such a peculiar activity.