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
    An infinite (nonstandard) hyperreal exists — Carmelics
    Statements
    321,452
    Perspectives
    108,905
    Topics
    42
    Home/Modality & Possibility
    HistoryEditSee Inverse

    An infinite (nonstandard) hyperreal exists

    Modality & PossibilityProof of definition segments
    ?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.The saturation principle applies to the hyperreal line
      ?

      Think about whether this reason is strong or weak

    • 2.The saturation principle implies the existence of a hyperreal a such that a > n for every natural number n
      ?

      Think about whether this reason is strong or weak

    Reasons Against

    2 perspectives
    Reason against 1 of 2
    ?
    • 1.The saturation principle is only valid within a specific non-standard model constructed via ultrafilter; it is not a general logical truth.
      ?

      Think about whether this reason is strong or weak

    • 2.The existence of a non-principal ultrafilter—required to construct hyperreals—cannot be proven in ZF alone and requires the Boolean Prime Ideal Theorem.
      ?

      Think about whether this reason is strong or weak

    • 3.Mathematical existence claims that depend on axioms independent of ZF do not establish existence in any robust ontological sense.
      ?

      Think about whether this reason is strong or weak

    Reason against 2 of 2
    ?
    • 1.Wittgenstein's finitist critique holds that infinitary mathematical objects are not discovered but are artifacts of notational extension without determinate reference.
      ?

      Think about whether this reason is strong or weak

    • 2.If 'a > n for every natural number n' can only be verified schema-theoretically and never instancewise, the quantifier range presupposes what it purports to prove.
      ?

      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.

    Topics

    Modality & PossibilityProof of definition segments

    Connections

    1 topic

    Truth & Knowledge1 linked

    Related

    If 'a > n for every natural number n' can only be verified schema-theoretically ...Mathematical existence claims that depend on axioms independent of ZF do not est...The existence of a non-principal ultrafilter—required to construct hyperreals—ca...The saturation principle applies to the hyperreal line
    +3 moreShow less
    The saturation principle implies the existence of a hyperreal a such that a > n ...The saturation principle is only valid within a specific non-standard model cons...Wittgenstein's finitist critique holds that infinitary mathematical objects are ...

    Similar

    An infinitesimal hyperreal exists93%Nonstandard infinitesimal hyperreals exist in substantial number87%Nonstandard hyperreals must exist86%An infinite hyperreal a exists such that a > n for every natural numbe...84%

    Source

    AI-extracted1/3 agreementValid
    SEP: continuity
    View source passageHide passage
    Now suppose that the set \(\bbN\) of natural numbers is a member of \(U\). Then so is the set \(\Re\) of real numbers, since each real number may be identified with a set of natural numbers. \(\Re\) may be regarded as an ordered field, and the same is therefore true of its inflate \(\hat{\Re}\), since the latter has precisely the same first-order properties as \(\Re\). \(\hat{\Re}\) is called the hyperreal line, and its members hyperreals. A standard hyperreal is then just a real, to which we sh
    Extraction notes

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

    Details

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