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    Carmelics

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    Home/Original/inverse
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    Inverse View

    It is not the case that The hyperreals exist only relative to a non-constructive ultrafilter whose existence depends on the Axiom of Choice, making their ontological status model-relative rather than absolute.

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

    Reasons For

    1 perspective
    Reason for
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    • 1.AC is a standard axiom in modern mathematics; relying on it doesn't diminish ontological status any more than relying on the axiom of infinity for naturals.
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    • 2.Model-relativity is universal in mathematics—all structures are defined relative to formal systems, so hyperreals aren't uniquely dependent or deficient.
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    • 3.Hyperreals provide rigorous infinitesimal analysis with proven applications; pragmatic indispensability suggests genuine mathematical existence regardless of axiom choice.
      ?

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    Reasons Against

    1 perspective
    Reason against
    ?
    • 1.Ultrafilter existence requires AC, which is unprovable in ZF, so hyperreals lack absolute foundational grounding independent of set-theoretic choices.
      ?

      Think about whether this reason is strong or weak

    • 2.Different models of ZFC can have different ultrafilters, producing distinct hyperreal fields, demonstrating their ontological dependence on model structure.
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    • 3.Mathematical objects whose existence hinges on non-constructive principles have weaker claims to mind-independent reality than constructively definable entities.
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