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

    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.

    ?Rate how convincing each reason is below to see the overall strength.
    1 reason for
    1 reason against

    Reasons For

    1 perspective
    Reason for
    ?
    • 1.Universal equations in ℤ (like a+b=b+a) involve only +, -, and =, which are definable in any abelian group structure.
      ?

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    • 2.Any abelian group homomorphism from ℤ to another abelian group preserves all universally quantified equations satisfied by ℤ.
      ?

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    • 3.Z/nZ is an abelian group, so it inherits all universal equations true in ℤ by the preservation properties of group homomorphisms.
      ?

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    Reasons Against

    1 perspective
    Reason against
    ?
    • 1.ℤ is infinite with no torsion; Z/nZ has torsion (nx=0 for all x). Not all equational properties transfer between structures with different torsion properties.
      ?

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    • 2.Equations like 'x+x+...+x (n times) = 0' are satisfied in Z/nZ but false in ℤ, showing the equational theories genuinely differ.
      ?

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    • 3.Equational theory includes all universally quantified statements derivable from axioms; ℤ and Z/nZ satisfy different first-order consequences.
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    Related

    Any abelian group homomorphism from ℤ to another abelian group preserves all uni...Equational theory includes all universally quantified statements derivable from ...Equations like 'x+x+...+x (n times) = 0' are satisfied in Z/nZ but false in ℤ, s...The equational theory of the integers as a whole entails that the integers must ...
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    Universal equations in ℤ (like a+b=b+a) involve only +, -, and =, which are defi...Z/nZ is an abelian group, so it inherits all universal equations true in ℤ by th...ℤ is infinite with no torsion; Z/nZ has torsion (nx=0 for all x). Not all equati...

    Details

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