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    Carmelics

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    LoyalLoyalJusticeJustice
    Made withinDC&Austin
    Statements
    321,452
    Perspectives
    108,905
    Topics
    42
    Home/Original/inverse
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    Inverse View

    It is not the case that The hyperreals and standard reals satisfy the transfer principle for first-order logical results, but behave differently for results about sets.

    ?Set your confidence on the premises below to see your aggregate.

    Reasons For

    2 perspectives
    Reason for 1 of 2
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    • 1.The set of infinitesimal hyperreals lacks a least upper bound because 'least upper bound' in NSA refers to an internal set, not an external one.
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    • 2.The completeness axiom transfers correctly to hyperreals when restricted to internal sets, making the supporting argument's P3 a category error.
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    • 3.Robinson's transfer principle was explicitly formulated to apply only to internal properties, so citing external sets as counterexamples misrepresents the theorem's scope.
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    Reason for 2 of 2
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    • 1.The claim conflates set-theoretic and model-theoretic senses of 'behave differently,' obscuring that hyperreals are elementarily equivalent to the reals.
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    • 2.Keisler and Robinson demonstrated that any first-order sentence true of the reals is true of the hyperreals, leaving no formal asymmetry at the level of truth.
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    Reasons Against

    1 perspective
    Reason against
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    • 1.For results stated in a first-order logical language, the hyperreals and the standard reals satisfy the transfer principle.
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    • 2.Every bounded set of standard reals has a least upper bound.
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    • 3.The set of infinitesimal hyperreals is bounded but has no least upper bound.
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