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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 Internal Set Theory resolves the asymmetry between standard and non-standard reals by enriching the basic language of mathematics to distinguish between standard and non-standard real numbers and between internal and external 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.IST's 'standardness' predicate is not definable within ZFC, making it a metalinguistic annotation rather than a genuine enrichment of mathematical ontology.
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    • 2.Nelson's transfer principle preserves first-order truths but cannot eliminate the metatheoretic asymmetry, since 'standard' remains undefinable by any internal formula.
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    • 3.A distinction that cannot be expressed within the object language resolves no asymmetry but merely relocates it to the metalanguage, leaving the foundational problem intact.
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    Reason for 2 of 2
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    • 1.Hrbacek's critique demonstrates that IST's axiomatic constraints force every set to be either fully standard or non-standard in ways that conflict with classical analysis's continuity assumptions.
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    • 2.If the resolution of an asymmetry requires restricting which set-theoretic operations preserve standardness, the resulting framework imposes new asymmetries rather than eliminating the original one.
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    Reasons Against

    1 perspective
    Reason against
    ?
    • 1.Edward Nelson's Internal Set Theory enriches the basic language of mathematics to allow distinction between standard and non-standard real numbers.
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    • 2.Internal sets behave the same as standard sets of standard reals, including that bounded internal sets always have a least upper bound.
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    • 3.The set of all infinitesimals is an external set that cannot be defined within the language and thus does not necessarily have a least upper bound.
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