most recent 30 from http://mathoverflow.net 2013-05-22T12:43:58Z http://mathoverflow.net/feeds/question/25100 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/25100/order-types-of-positive-reals David Eppstein 2010-05-18T07:30:37Z 2010-05-18T10:45:59Z

<p>Suppose one has a set $S$ of positive real numbers, such that the usual numerical ordering on $S$ is a well-ordering. Is it possible for $S$ to have any countable ordinal as its order type, or are the order types that can be formed in this way more restricted than that?</p>

http://mathoverflow.net/questions/25100/order-types-of-positive-reals/25101#25101 Robin Chapman 2010-05-18T07:36:04Z 2010-05-18T08:16:14Z

<p>Yes, one can have any countable ordering. Indeed any countable totally ordered set can be embedded in $\mathbb{Q}$. Write your ordered set as $ \lbrace a_1,a_2,\ldots \rbrace $ and define the embedding recursively: once you have placed $a_1,\ldots,a_{n-1}$ there will always be an interval to slot $a_n$ into.</p>

http://mathoverflow.net/questions/25100/order-types-of-positive-reals/25102#25102 gowers 2010-05-18T07:37:05Z 2010-05-18T07:37:05Z

<p>You can get any order type. Let's assume you can get all order types up to but not including alpha, using subsets of (0,1]. If alpha=beta + 1 then squash your representation of beta and add an extra point. If alpha is a limit ordinal, choose a sequence of ordinals that converges to alpha and put the first one into (0,1/2], the second into (1/2,3/4] etc. and the result will have order type alpha.</p>

http://mathoverflow.net/questions/25100/order-types-of-positive-reals/25103#25103 Pietro Majer 2010-05-18T07:42:52Z 2010-05-18T08:23:03Z

<p>To complete the picture (the obvious remaining part). If ${S\subset\mathbb R}$ is well ordered, then it is countable: indeed it has countable cofinality. Thus well-ordered subsets of <strong>R</strong> are <em>exactly</em> countable ordinals.</p>

http://mathoverflow.net/questions/25100/order-types-of-positive-reals/25109#25109 Andrew 2010-05-18T08:47:21Z 2010-05-18T08:47:21Z

<p>Using wellorderings of positive reals is actually the standard way to construct an Aronszajn tree.</p>