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    Inverse View

    It is not the case that 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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    Reasons For

    1 perspective
    Reason for
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    • 1.The claim conflates 'not equationally forced' with 'equational theories don't entail infinitude,' but theories can entail infinitude non-equationally.
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    • 2.Finite groups existing in varieties doesn't prove infinitude isn't entailed; it only shows finiteness is consistent with equational axioms.
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    • 3.Some equational theories (e.g., for dense linear orders) do entail infinitude despite Birkhoff's structural characterization of varieties.
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    Reasons Against

    1 perspective
    Reason against
    ?
    • 1.Birkhoff's theorem establishes that varieties are exactly the classes closed under products, subalgebras, and homomorphic images.
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    • 2.Finite groups form varieties since they are closed under these operations, proving infinitude is not logically entailed by equational axioms.
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    • 3.If a finite structure satisfies an equational theory, that theory cannot logically force infinitude without additional non-equational constraints.
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