By W. W. Rouse Ball

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**Example text**

1) There are no reals z with 0 <∞ z <∞ 0 . (2) There are sets of reals A with 0 <∞ A <∞ 0 . Indeed, there are incomparable semi-decidable sets of reals A ⊥∞ B. The degrees of the accidentally writable reals are linearly ordered and in fact form a well-ordered hierarchy of order type ζ + 1, which corresponds also to their order of earliest appearance on any computation. In other work, Welch [28] found minimality in the inﬁnite time Turing degrees. Hamkins and Seabold [15] analyzed one-tape versus multi-tape inﬁnite time Turing machines, and Benedikt L¨owe [23] observed the connection between inﬁnite time Turing machines and revision theories of truth.

Eﬀective Categoricity. It is often possible to produce two classically computable structures which are isomorphic, but for which the isomorphism is not witnessed by a computable function. Any theory of eﬀective mathematics must take account of this phenomenon. 1. A computable structure M is said to be computably categorical if and only if for any computable structure N M there is a computable function f : N → M. The number of equivalence classes under computable isomorphism contained in an isomorphism type is called its computable dimension.

We will consider here reduction functions which are computable by an inﬁnite time Turing machine (see [2] for a more complete exposition). Thus, for any two equivalence relations E, F on R, we say that E is inﬁnite time computably reducible to F, written E ≤c F, if there is an inﬁnite time computable function f (freely allowing real parameters) satisfying Equation (1). Similarly, we say that E is eventually reducible to F, written E ≤e F, if there is an eventually computable function f satisfying Equation (1).