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    Recursive irrationals alone are insufficient to fill all ... — Carmelics
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    Home/Modality & Possibility
    HistoryEditSee Inverse

    Recursive irrationals alone are insufficient to fill all gaps in the real number line.

    Modality & Possibility
    ?Rate how convincing each reason is below to see the overall strength.
    1 reason for
    2 reasons against

    Reasons For

    1 perspective
    Reason for
    ?
    • 1.Adding recursive irrationals to the rationals still leaves gaps in the continuum.
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    • 2.Therefore, lawless irrationals must also be introduced to complete the mathematical continuum.
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    Reasons Against

    2 perspectives
    Reason against 1 of 2
    ?
    • 1.Brouwer's intuitionism holds that the continuum is a primitive intuition not constructed by adding elements to a discrete base.
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    • 2.If the continuum is foundationally prior to any enumeration of points, no supplementation argument—recursive or lawless—can 'complete' it.
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    • 3.Therefore the claim presupposes a set-theoretic atomism about the continuum that constructivists like Brouwer explicitly reject.
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    Reason against 2 of 2
    ?
    • 1.Wittgenstein argued that 'gaps in the number line' is a picture whose application requires surveyable, rule-governed mathematical practice.
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    • 2.Lawless sequences, lacking any governing rule, cannot be identified, applied, or meaningfully said to occupy positions that 'fill' anything.
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    • 3.If lawless irrationals are mathematically inert for any actual calculation, invoking them to complete the continuum is a grammatical fiction, not a mathematical result.
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    Related

    Adding recursive irrationals to the rationals still leaves gaps in the continuum...Brouwer's intuitionism holds that the continuum is a primitive intuition not con...If lawless irrationals are mathematically inert for any actual calculation, invo...If the continuum is foundationally prior to any enumeration of points, no supple...
    +4 moreShow less
    Lawless sequences, lacking any governing rule, cannot be identified, applied, or...Therefore the claim presupposes a set-theoretic atomism about the continuum that...Therefore, lawless irrationals must also be introduced to complete the mathemati...Wittgenstein argued that 'gaps in the number line' is a picture whose applicatio...

    Similar

    There must be gaps between the rational numbers that require filling w...84%Adding recursive irrationals to the rationals still leaves gaps in the...80%Lawless and pseudo-irrationals are needed for the mathematical continu...80%Therefore, lawless irrationals must also be introduced to complete the...79%

    Source

    AI-extracted1/3 agreementValid
    SEP: wittgenstein-mathematics
    View source passageHide passage
    The problem, as Wittgenstein sees it, is that mathematicians, especially foundationalists (e.g., set theorists), have sought to accommodate physical continuity by a theory that ‘describes’ the mathematical continuum (PR §171). When, for example, we think of continuous motion and the (mere) density of the rationals, we reason that if an object moves continuously from A to B, and it travels only the distances marked by “rational points”, then it must skip some distances (intervals, or points) not
    Extraction notes

    Validity: Extracted via Max plan + API grounding/validity checks

    Details

    Type
    claim
    Perspectives
    3 (1 for, 2 against)
    Edits
    1 edit