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::::Yes, it is true that the things called Godel numberings in computability theory, like the Godel numbering of the sentences of a formal language, cannot literally be "computable functions" unless their ___domain and range are included in the set of legal inputs/outputs of whatever model of computation we use. I used the word "corresponds" in this article precisely for that reason. Unfortunately the WP article on Godel numberings is currently in very bad shape. Feel free to edit either article if you would like to improve them.
::::And yes, the Church-Turing thesis is limited to functions whose ___domain and range are included in the set of possible inputs/outputs of the model of computation. For example, the successor operation on [[ordinal]]s, although in some sense effective, can't be computable because there is no model of computation that can represent all the ordinals as inputs. [[User:CMummert|CMummert]] · <small>[[User talk:CMummert|talk]]</small> 23:14, 28 February 2007 (UTC)
== Dumb Questions ==
QUOTING from the article:
"A set S of natural numbers is called recursively enumerable if there is a partial recursive (i.e. computable) function whose ___domain (co-range) is exactly S, meaning that the function is defined if and only if its input is a member of S."
I'm starting pretty far back in trying to comprehend this. May I please have answers to these questions:
* Is "partial recursive function" synonymous with "computable function"? (Partial recursive and computable are not defined or linked. Are they regarded as obvious in meaning?)
* I know the terms ___domain and range. But why "co-range"? Is there any good reason to introduce this alternate and confusing term here?
* The test of whether the set S is recursively enumerable relates to the "___domain" of functions. Does the "range" of functions have any relevance to the matter?
* After pondering my way through the introduction to the article, I was pleased to see that there is a section of "Examples". I eagerly went to it, and was deeply disappointed to see that it contains no (IMO) real examples of functions; only what I would call examples of types of functions. Are there any simple examples of what this article is about?
* Is the set of all integers that are multiples of 7 recursively enumerable?. If so, what function satisfies the test?
Thanks very much. [[User:Wanderer57|Wanderer57]] ([[User talk:Wanderer57|talk]]) 14:13, 8 August 2008 (UTC)
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