I hit reply instead of reply all. Sorry Richard.<br><br><div class="gmail_quote">On Fri, Jun 22, 2012 at 4:35 PM, <span dir="ltr"><<a href="mailto:ok@cs.otago.ac.nz" target="_blank">ok@cs.otago.ac.nz</a>></span> wrote:<br>
<blockquote class="gmail_quote" style="margin:0 0 0 .8ex;border-left:1px #ccc solid;padding-left:1ex"><div class="im"><br>
> An ordering does not typically induce a computable enumeration. For<br>
> example, there are infinitely many rationals between any pair of<br>
> rationals.<br>
<br>
</div>I didn't say it was odd that Ords weren't Enums,<br>
I said that it's odd that Enums aren't Ords.<br></blockquote><div><br></div><div><div class="im" style><div>You said:</div><div><br></div><div>"It always struck me as odd that Enum doesn't extend Ord."</div>
<div><br></div></div><div style>Ambiguous, at best.</div><div style><br></div><div style>In any case, the order induced by the enumeration of the rationals is not compatible with the magnitude/sign order either. This is typically true. </div>
</div><div> </div><blockquote class="gmail_quote" style="margin:0 0 0 .8ex;border-left:1px #ccc solid;padding-left:1ex">
It makes little or no sense to make treat rationals as Enums;<br>
the intent of enums is generally to provide *exhaustive*<br>
enumeration of a *discrete* set.<br>
</blockquote></div><br><div><div style>I don't see that in the documentation anywhere, and "enumerable" is synonymous with countable/computable (depending on the context). The usual topology on the structure has nothing to do with its enumerability.</div>
<div style><br></div><div style>The set of pairs of positive integers is "discrete" under the obvious product topology. Indeed, the "standard" enumeration is (almost) formally equivalent to that of the positive rationals. The only difference is we aren't skipping over the diagonal. </div>
<div style><br></div><div style>Enumerating the rationals is straight-forward.</div><br class="Apple-interchange-newline"><span style><a href="http://www.cs.ox.ac.uk/jeremy">http://www.cs.ox.ac.uk/jeremy</a>.</span><span style>gibbons/publications/</span><span style>rationals.pdf</span></div>