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    Made withinDC&Austin
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    Home/Original/inverse
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    Inverse View

    It is not the case that Any proof of a first-order theorem about the standard reals can be transferred to the hyperreals, and vice versa, sometimes greatly simplifying calculations and proofs

    ?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 transfer principle applies only to first-order sentences, but many foundational mathematical truths about the reals require second-order expressibility.
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    • 2.Properties like completeness (every bounded set has a least upper bound) are essentially second-order and fail in the hyperreals, limiting bidirectional transfer.
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    • 3.A proof strategy that essentially exploits completeness cannot be transferred hyperreal-to-real even if surface syntax appears first-order.
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    Reason for 2 of 2
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    • 1.Wittgenstein and later Skolem's paradox show that formal transferability between models does not preserve intended semantic content or mathematical meaning.
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    • 2.The hyperreals contain infinitesimals that are not standard reals, so transferred proofs may track structural isomorphisms rather than truths about the intended real-number system.
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    • 3.If the standard reals and hyperreals are non-isomorphic structures, proofs transferred between them establish facts about different mathematical objects, undermining the claim of genuine simplification.
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    Reasons Against

    1 perspective
    Reason against
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    • 1.Robinson's hyperreals satisfy a transfer principle
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    • 2.If statements are formulated entirely within a first-order language for the reals, then they are true of the standard reals if and only if they are true for the hyperreals
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    • 3.A proof valid in one system can be carried over to the other system under the transfer principle
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