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1. • CommentRowNumber1.
   • CommentAuthorDavidRoberts
   • CommentTimeJun 6th 2011
   • (edited Jun 6th 2011)
   • PermaLink
   Author: DavidRoberts
   Format: MarkdownItexIn [this fom post](http://www.cs.nyu.edu/pipermail/fom/2005-February/008773.html), Harvey Friedman characterises the unit interval and the real numbers as linearly ordered sets $X$ (with top and bottom in the case of [0,1], and no endpoints in the case of R) satisfying the LUB property and having a continuous 'betweeness' function, $f:X^2 \to X$, $x\lt f(x,y)\lt y$. The proof uses contradiction, but even apart from that I suspect it is highly non-constructive (although I'm happy to be proved any degree of wrong). The hypotheses are available in a topos (or with even less assumptions), with problems possibly arising with the LUB property, but I don't know the finer points of what could and couldn't be done with them. Any thoughts? ---- Edit: And further, in [this post](http://www.cs.nyu.edu/pipermail/fom/2005-February/008806.html), Friedman states the result: >An LUB field is an ordered field in which every bounded definable subset has a least upper bound. >THEOREM 1.1. The LUB fields are exactly the real closed fields. >We can strengthen Theorem 1.1 as follows. >THEOREM 1.2. Suppose F is an ordered field with the least upper bound property for all atomic formulas. Then F is a real closed field. perhaps none of this is surprising, given the characterisation of the reals as a complete ordered field, but is it an improvement to weaken the existence of a LUB to definable subsets/atomic formulae?

   In this fom post, Harvey Friedman characterises the unit interval and the real numbers as linearly ordered sets (with top and bottom in the case of [0,1], and no endpoints in the case of R) satisfying the LUB property and having a continuous ’betweeness’ function, , . The proof uses contradiction, but even apart from that I suspect it is highly non-constructive (although I’m happy to be proved any degree of wrong).

   The hypotheses are available in a topos (or with even less assumptions), with problems possibly arising with the LUB property, but I don’t know the finer points of what could and couldn’t be done with them.

   Any thoughts?

   ---

   Edit: And further, in this post, Friedman states the result:

   An LUB field is an ordered field in which every bounded definable subset has a least upper bound.

   THEOREM 1.1. The LUB fields are exactly the real closed fields.

   We can strengthen Theorem 1.1 as follows.

   THEOREM 1.2. Suppose F is an ordered field with the least upper bound property for all atomic formulas. Then F is a real closed field.

   perhaps none of this is surprising, given the characterisation of the reals as a complete ordered field, but is it an improvement to weaken the existence of a LUB to definable subsets/atomic formulae?

2. • CommentRowNumber2.
   • CommentAuthorTobyBartels
   • CommentTimeJun 7th 2011
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   Author: TobyBartels
   Format: MarkdownItexWell, the *difference* is that it gives you more models; the field of algebraic real numbers, for example, is real-closed, as is the field of algebraic expressions in one variable with real constants. I don't know by what standard to judge that this is an *improvement*.

   Well, the difference is that it gives you more models; the field of algebraic real numbers, for example, is real-closed, as is the field of algebraic expressions in one variable with real constants. I don’t know by what standard to judge that this is an improvement.

3. • CommentRowNumber3.
   • CommentAuthorDavidRoberts
   • CommentTimeJun 7th 2011
   • PermaLink
   Author: DavidRoberts
   Format: MarkdownItexWell, not an improvement, then. Poor choice of word, really. But I wonder what bifurcation would happen treating this internally?

   Well, not an improvement, then. Poor choice of word, really. But I wonder what bifurcation would happen treating this internally?