Order types of positive reals - MathOverflow 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 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 Answer by Robin Chapman for Order types of positive reals 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 Answer by gowers for Order types of positive reals 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 Answer by Pietro Majer for Order types of positive reals 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 Answer by Andrew for Order types of positive reals 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>