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    The equational theory of the integers as a whole entails ... — Carmelics
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    The equational theory of the integers as a whole entails that the integers must be an infinite set.

    Truth & Knowledge
    ?Rate how convincing each reason is below to see the overall strength.
    1 reason for
    2 reasons against

    Reasons For

    1 perspective
    Reason for
    ?
    • 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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    Reasons Against

    2 perspectives
    Reason against 1 of 2
    ?
    • 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 against 2 of 2
    ?
    • 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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    Related

    Any finite model of the equational theory of the integers must satisfy some law ...Argument 1 conflates the equational theory of integers with a richer first-order...By Birkhoff's completeness theorem, equational theories characterize varieties c...If finite models satisfy the same universal equations as the integers, the equat...
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    The equational theory of the integers contains no law of the form x+x+...+x=0 fo...The equational theory of the integers, as a set of universally quantified equati...Therefore no finite model can satisfy the entire equational theory of the intege...Therefore the supporting argument proves only that no finite model is term-equiv...

    Similar

    No single equational property of the integers can establish that the i...91%Therefore no finite model can satisfy the entire equational theory of ...85%If a finite model satisfies all the same equational laws as the intege...85%Any finite model of the equational theory of the integers must satisfy...83%

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    (i) It could tell us to what extent the equational laws holding of the integers characterize the integers. Since the set \(\{0, 1\}\) of integers mod 2 under addition and negation satisfies all the laws that the integers do, we immediately see that no single equational property of the integers tells us that there are infinitely many integers. On the other hand any finite model of the equational theory of the integers necessarily satisfies some law that the integers don’t satisfy, in particular t
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    Details

    Type
    claim
    Perspectives
    3 (1 for, 2 against)
    Edits
    1 edit