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    Carmelics

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    Made withinDC&Austin
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    Inverse View

    It is not the case that The equational theory of the integers as a whole entails that the integers must be an infinite set.

    ?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.Argument 1 conflates the equational theory of integers with a richer first-order theory; the torsion law x+x+...+x=0 is not an equation the integers violate but one absent from their equational theory.
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    • 2.By Birkhoff's completeness theorem, equational theories characterize varieties closed under products and homomorphic images, and finite groups appear in such varieties, undermining the claim that infinitude is entailed.
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    • 3.Therefore the supporting argument proves only that no finite model is term-equivalent to the integers, not that the equational theory itself forces infinitude.
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    Reason for 2 of 2
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    • 1.The equational theory of the integers, as a set of universally quantified equations, is satisfied by any abelian group, including finite cyclic groups like Z/nZ.
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    • 2.If finite models satisfy the same universal equations as the integers, the equational theory alone cannot entail infinitude without invoking non-equational axioms.
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    Reasons Against

    1 perspective
    Reason against
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    • 1.Any finite model of the equational theory of the integers must satisfy some law the integers do not satisfy — specifically, the law x+x+...+x=0 where the number of x's equals the size of the model.
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    • 2.The equational theory of the integers contains no law of the form x+x+...+x=0 for any fixed finite number of terms.
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    • 3.Therefore no finite model can satisfy the entire equational theory of the integers.
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