Skip to content
Carmelics
TopicsThinkersChangesContributorsLoading account…

    Carmelics

    A reasoning platform. Break down any belief into clear reasons, explore both sides, and weigh the evidence honestly.

    Navigate

    • Topics
    • Search
    • Recent Changes
    • Contribute
    • How It Works
    • Glossary
    • Thinkers
    • Contributors
    • About
    • Statistics
    • Terms
    • Privacy

    Database

    Statements
    —
    Perspectives
    —
    Topics
    —

    Press ? for keyboard shortcuts

    LoyalLoyalJusticeJustice
    Made withinDC&Austin
    Statements
    321,452
    Perspectives
    108,905
    Topics
    42
    The completeness axiom transfers correctly to hyperreals ... — Carmelics
    Home
    HistoryEditSee Inverse

    Part of a larger discussion

    Challenges→The hyperreals and standard reals satisfy the transfer principle for first-order logical results, but behave differently for results about sets.

    The completeness axiom transfers correctly to hyperreals when restricted to internal sets, making the supporting argument's P3 a category error.

    ?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.Internal sets in hyperreal theory form a proper subcategory with distinct closure properties, making unrestricted completeness inapplicable to them.
      ?

      Think about whether this reason is strong or weak

    • 2.Applying standard real-number axioms to hyperreals without restriction commits a type error by conflating different mathematical structures.
      ?

      Think about whether this reason is strong or weak

    • 3.The completeness axiom's meaning shifts when restricted to internal sets, so comparing restricted and unrestricted versions equivocates on P3.
      ?

      Think about whether this reason is strong or weak

    Reasons Against

    1 perspective
    Reason against
    ?
    • 1.Internal sets satisfy completeness by construction in standard hyperreal models, so restriction doesn't create a category error but clarification.
      ?

      Think about whether this reason is strong or weak

    • 2.Many legitimate mathematical arguments apply axioms to proper subcategories without committing category errors—restriction is standard practice.
      ?

      Think about whether this reason is strong or weak

    • 3.P3's claim can be evaluated without category-error diagnosis; disagreement about completeness's scope doesn't establish logical miscategorization.
      ?

      Think about whether this reason is strong or weak

    Sign in or register to share your perspective on this statement.

    Next step

    Based on where you are in your exploration

    Strongest counterpoint
    Explore the most compelling reason on the other side.

    Connections

    2 topics

    Truth & Knowledge1 linkedModality & Possibility1 linked

    Related

    Applying standard real-number axioms to hyperreals without restriction commits a...Internal sets in hyperreal theory form a proper subcategory with distinct closur...Internal sets satisfy completeness by construction in standard hyperreal models,...Many legitimate mathematical arguments apply axioms to proper subcategories with...
    +3 moreShow less
    P3's claim can be evaluated without category-error diagnosis; disagreement about...The completeness axiom's meaning shifts when restricted to internal sets, so com...The hyperreals and standard reals satisfy the transfer principle for first-order...

    Details

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