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    By the completeness theorem for first-order logic, any co... — Carmelics
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    Supports→There exists a consistent first-order model in which the exponential function is not total, i.e., a model M satisfying S¹₂ + ∃y¬∃z ε(2,y,z).

    By the completeness theorem for first-order logic, any consistent theory has a model.

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    Buss's theory S¹₂ also does not prove the totality of the exponential function.IΔ₀ does not prove the totality of the exponential function.The theory S¹₂ + ∃y¬∃z ε(2,y,z) is therefore proof-theoretically consistent.There exists a consistent first-order model in which the exponential function is...

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    [53] And from this it follows that such orders cannot be reached from below by a sorties-like sequence of feasible orders of growth \(O(1) \prec O(n) \prec O(n^2) \prec O(n^3) \ldots\) When analyzed according to the Cobham-Edmonds thesis, it hence appears that the ‘naive’ notion of feasible computability does not suffer from the sort of instability which Dummett takes to plague the notion of a feasibly constructible number. These observations point to another theme within the writings of some st

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