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

    Philosophy of LanguageTruth & Knowledge
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    1 reason for
    2 reasons against

    Reasons For

    1 perspective
    Reason for
    ?
    • 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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    Reasons Against

    2 perspectives
    Reason against 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 against 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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    • 2.Where systematic translation between formal systems is possible, theorem-by-theorem comparison becomes tractable despite surface syntactic differences.
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    Related

    Equiconsistency results constitute a rigorous basis for comparing theorems acros...If PM's ramified types can be reduced to simple types for most mathematical purp...Quine demonstrated in 'Set Theory and Its Logic' that type-theoretic systems can...Specker and others showed that the simple theory of types (TST) is equiconsisten...
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    The languages in which PM and ZF are expressed differ in logical powerThe sentences of PM are expressed in the theory of types, whereas ZF is expresse...The two theories have different axiomsWhere systematic translation between formal systems is possible, theorem-by-theo...

    Similar

    The theorem asserting the non-existence of the set of non-self-membere...80%ZFC is a non-structural set theory78%Gödel and Cohen demonstrated the mathematical incompleteness of ZFC se...75%Set theory entails that if set A and set B have different members, the...75%

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    SEP: principia-mathematica
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    Strictly as presented in PM, however, the no-classes theory differs significantly from ZF. The sentences of the PM theory are expressed in the theory of types, as opposed to the first order theory of ZF. ZF and PM cannot simply be compared in terms of their theorems. Not only are there different axioms in the two theories, but the very languages in which they are expressed differ in logical power. If we follow Gödel and Boolos, however, the two are seen to be based on the same intuitive basis, a
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    Details

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    Perspectives
    3 (1 for, 2 against)
    Edits
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