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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 PM's no-classes theory and ZF set theory cannot simply be compared in terms of their theorems

    ?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.Specker and others showed that the simple theory of types (TST) is equiconsistent with ZFC minus the axiom of infinity, establishing a precise proof-theoretic bridge.
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    • 2.Equiconsistency results constitute a rigorous basis for comparing theorems across differently-expressed systems, since they expose shared deductive commitments.
      ?

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    • 3.If PM's ramified types can be reduced to simple types for most mathematical purposes—as Church and others argued—then the equiconsistency bridge to ZF is substantive, not merely formal.
      ?

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    Reason for 2 of 2
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    • 1.Quine demonstrated in 'Set Theory and Its Logic' that type-theoretic systems can be systematically reinterpreted into first-order set theories via stratification.
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      Think about whether this reason is strong or weak

    • 2.Where systematic translation between formal systems is possible, theorem-by-theorem comparison becomes tractable despite surface syntactic differences.
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      Think about whether this reason is strong or weak

    Reasons Against

    1 perspective
    Reason against
    ?
    • 1.The sentences of PM are expressed in the theory of types, whereas ZF is expressed in first-order logic
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    • 2.The two theories have different axioms
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    • 3.The languages in which PM and ZF are expressed differ in logical power
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