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    A sufficiently rich equational theory may indirectly forc... — Carmelics
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    Challenges→If finite models satisfy the same universal equations as the integers, the equational theory alone cannot entail infinitude without invoking non-equational axioms.

    A sufficiently rich equational theory may indirectly force infinitude by making finite models require exponentially complex representations, becoming practically indistinguishable.

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    Key Terms

    Exponentially complex(as used in computational and mathematical analysis)
    Something that becomes massively, dramatically harder to describe or calculate—so hard that even small increases in size require huge jumps in effort.
    Finite models(in formal logic)
    Specific, limited examples or setups in logic where you're only dealing with a restricted, countable set of things (like a small network of connected points).
    Infinitude(as used in mathematics and metaphysics)
    The quality or property of being infinite, without limit or end.
    Model (in logic)(in formal logic and semantics)
    An imaginary scenario or description of a possible world where certain statements are true or false—used to test whether logical arguments work.
    Practically indistinguishable

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    (as used in epistemology and philosophy of mind)
    So similar or hard to tell apart in real life that for practical purposes, they might as well be the same thing, even if technically they're different.
    equational theory(Used to characterize algebraic structures such as the integers under addition and negation.)
    The set of all equational laws (universally quantified equations) satisfied by a given algebraic structure.

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    If finite models satisfy the same universal equations as the integers, the equat...

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    If finite models satisfy the same universal equations as the integers, the equat...

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