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    Specker and others showed that the simple theory of types... — Carmelics
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    Challenges→PM's no-classes theory and ZF set theory cannot simply be compared in terms of their theorems

    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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    Reasons For

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    Reason for
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    • 1.Equiconsistency results precisely calibrate proof-theoretic strength, enabling meaningful comparison between foundational systems otherwise difficult to relate.
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    • 2.TST's predicative structure avoiding impredicative set formation aligns naturally with ZFC-Infinity's finite-universe constraints, justifying their equivalence.
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    • 3.Specker's result demonstrates type theory can serve as rigorous alternative foundation without infinity, supporting pluralism about foundational choices.
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    Reasons Against

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    Reason against
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    • 1.Equiconsistency alone doesn't establish philosophical equivalence; systems can have equal strength while modeling mathematical reality quite differently.
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    • 2.TST and ZFC-Infinity differ substantively in expressiveness and naturalness for ordinary mathematics; their proof-theoretic parity obscures practical divergence.
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    • 3.The result's significance may be overstated: both systems are weak fragments; their equivalence tells us little about standard mathematics with infinity.
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    Related

    Equiconsistency alone doesn't establish philosophical equivalence; systems can h...Equiconsistency results precisely calibrate proof-theoretic strength, enabling m...PM's no-classes theory and ZF set theory cannot simply be compared in terms of t...Specker's result demonstrates type theory can serve as rigorous alternative foun...
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    TST and ZFC-Infinity differ substantively in expressiveness and naturalness for ...TST's predicative structure avoiding impredicative set formation aligns naturall...The result's significance may be overstated: both systems are weak fragments; th...

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