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    Real numbers like √2 are defined by Dedekind cuts, which ... — Carmelics
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    Challenges→A genuine number must measure in and of itself

    Real numbers like √2 are defined by Dedekind cuts, which partition the rationals without requiring any independent measuring capacity.

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    1 reason for
    1 reason against

    Reasons For

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    Reason for
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    • 1.Dedekind cuts are purely relational structures over rationals, requiring no reference to physical measurement or geometric intuition.
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    • 2.This construction explains why √2 exists mathematically without presupposing it as a pre-existing entity independent of rational order.
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    • 3.The approach unifies real numbers within set theory, avoiding mysterious 'gaps' that seem to require extra-mathematical grounding.
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    Reasons Against

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    • 1.Dedekind cuts define reals *in terms of* rationals, but this doesn't explain why rationals themselves have the ordering properties cuts depend on.
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    • 2.Claiming Dedekind cuts require no 'measuring capacity' conflates mathematical definition with metaphysical existence—definition alone doesn't establish what √2 *is*.
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    • 3.The construction is circular: we justify reals by appeal to rational structure, yet the entire framework presupposes intuitions about infinite ordered sets.
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    Related

    A genuine number must measure in and of itselfClaiming Dedekind cuts require no 'measuring capacity' conflates mathematical de...Dedekind cuts are purely relational structures over rationals, requiring no refe...Dedekind cuts define reals *in terms of* rationals, but this doesn't explain why...
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    The approach unifies real numbers within set theory, avoiding mysterious 'gaps' ...The construction is circular: we justify reals by appeal to rational structure, ...This construction explains why √2 exists mathematically without presupposing it ...

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    2 (1 for, 1 against)
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