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    • CommentRowNumber1.
    • CommentAuthorDavidRoberts
    • CommentTimeJun 6th 2011
    • (edited Jun 6th 2011)

    In this fom post, 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:X2X, x<f(x,y)<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, 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?

    • CommentRowNumber2.
    • CommentAuthorTobyBartels
    • CommentTimeJun 7th 2011

    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.

    • CommentRowNumber3.
    • CommentAuthorDavidRoberts
    • CommentTimeJun 7th 2011

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