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    Carmelics

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    Home/Original/inverse
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    Inverse View

    It is not the case that Recursive irrationals alone are insufficient to fill all gaps in the real number line.

    ?Set your confidence on the premises below to see your aggregate.

    Reasons For

    2 perspectives
    Reason for 1 of 2
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    • 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 for 2 of 2
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    • 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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    Reasons Against

    1 perspective
    Reason against
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    • 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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    Strongest counterpoint
    Explore the most compelling reason on the other side.