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    Lawless sequences, lacking any rule of extension, cannot ... — Carmelics
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    Challenges→The term 'irrational number' should be extended to include lawless and pseudo-irrationals

    Lawless sequences, lacking any rule of extension, cannot be the subject of proofs or calculations in the way rule-governed irrationals can.

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    1 reason for
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    Reasons For

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    Reason for
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    • 1.Proofs require recursive procedures or algorithms; lawless sequences have no computable rule, so formal derivation becomes impossible.
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    • 2.Algebraic irrationals (like √2) satisfy polynomial equations enabling systematic calculation; lawless sequences lack any defining property.
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    • 3.Mathematical discourse presupposes shared, reproducible methods; lawless sequences resist intersubjective verification or communication.
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    Reasons Against

    1 perspective
    Reason against
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    • 1.Non-constructive objects (e.g., Dedekind cuts, real numbers via completeness axiom) are subject to rigorous proofs despite lacking explicit rules.
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    • 2.A sequence's unprovability about one property doesn't prevent all proofs about it; lawless sequences have provable features (e.g., boundedness assumptions).
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    • 3.The distinction between 'rule-governed' and 'lawless' conflates computability with mathematical legitimacy; both can be axiomatically characterized.
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    Proof of definition segments1 linkedPhilosophy of Language1 linked

    Related

    A sequence's unprovability about one property doesn't prevent all proofs about i...Algebraic irrationals (like √2) satisfy polynomial equations enabling systematic...Mathematical discourse presupposes shared, reproducible methods; lawless sequenc...Non-constructive objects (e.g., Dedekind cuts, real numbers via completeness axi...
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    Proofs require recursive procedures or algorithms; lawless sequences have no com...The distinction between 'rule-governed' and 'lawless' conflates computability wi...The term 'irrational number' should be extended to include lawless and pseudo-ir...

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