## Access

You are not currently logged in.

Access JSTOR through your library or other institution:

## If You Use a Screen Reader

This content is available through Read Online (Free) program, which relies on page scans. Since scans are not currently available to screen readers, please contact JSTOR User Support for access. We'll provide a PDF copy for your screen reader.

# Degrees of Unsolvability of Continuous Functions

Joseph S. Miller
The Journal of Symbolic Logic
Vol. 69, No. 2 (Jun., 2004), pp. 555-584
Stable URL: http://www.jstor.org/stable/30041743
Page Count: 30
Preview not available

## Abstract

We show that the Turing degrees are not sufficient to measure the complexity of continuous functions on [0, 1]. Computability of continuous real functions is a standard notion from computable analysis. However, no satisfactory theory of degrees of continuous functions exists. We introduce the continuous degrees and prove that they are a proper extension of the Turing degrees and a proper substructure of the enumeration degrees. Call continuous degrees which are not Turing degrees non-total. Several fundamental results are proved: a continuous function with non-total degree has no least degree representation, settling a question asked by Pour-El and Lempp; every non-computable f $\epsilon \mathcal{C}[0, 1]$ computes a non-computable subset of $\mathbb{N}$; there is a non-total degree between Turing degrees $a _\eqslantless_{\tau}$ b iff b is a PA degree relative to a; $\mathcal{S} \subseteq 2^{\mathbb{N}}$ is a Scott set iff it is the collection of f-computable subsets of $\mathbb{N}$ for some f $\epsilon \mathcal{C}[O, 1]$ of non-total degree; and there are computably incomparable f, g $\epsilon \mathcal{C}[0, 1]$ which compute exactly the same subsets of $\mathbb{N}$. Proofs draw from classical analysis and constructive analysis as well as from computability theory.

• 555
• 556
• 557
• 558
• 559
• 560
• 561
• 562
• 563
• 564
• 565
• 566
• 567
• 568
• 569
• 570
• 571
• 572
• 573
• 574
• 575
• 576
• 577
• 578
• 579
• 580
• 581
• 582
• 583
• 584