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
    Nonstandard hyperreals must exist — Carmelics
    Statements
    321,452
    Perspectives
    108,905
    Topics
    42
    Home/Modality & Possibility
    HistoryEditSee Inverse

    Nonstandard hyperreals must exist

    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 set of real numbers is infinite
      ?

      Think about whether this reason is strong or weak

    • 2.The hyperreals include all standard reals plus additional elements entailed by the saturation principle
      ?

      Think about whether this reason is strong or weak

    Reasons Against

    2 perspectives
    Reason against 1 of 2
    ?
    • 1.Mathematical existence requires only formal consistency within an axiomatic system, not ontological necessity.
      ?

      Think about whether this reason is strong or weak

    • 2.The saturation principle is a stipulated axiom of NSA, not a discovered truth about mathematical reality.
      ?

      Think about whether this reason is strong or weak

    • 3.What can be consistently postulated need not thereby be said to 'must exist' in any robust sense.
      ?

      Think about whether this reason is strong or weak

    Reason against 2 of 2
    ?
    • 1.Benacerraf's challenge shows that mathematical Platonism requires an unexplained epistemic access to abstract objects.
      ?

      Think about whether this reason is strong or weak

    • 2.If hyperreals 'must exist,' their necessity presupposes a Platonist ontology that remains philosophically contested.
      ?

      Think about whether this reason is strong or weak

    • 3.Robinson himself characterized NSA as a formal tool, not a metaphysical revelation about necessary mathematical entities.
      ?

      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

    Benacerraf's challenge shows that mathematical Platonism requires an unexplained...If hyperreals 'must exist,' their necessity presupposes a Platonist ontology tha...Mathematical existence requires only formal consistency within an axiomatic syst...Robinson himself characterized NSA as a formal tool, not a metaphysical revelati...
    +4 moreShow less
    The hyperreals include all standard reals plus additional elements entailed by t...The saturation principle is a stipulated axiom of NSA, not a discovered truth ab...The set of real numbers is infiniteWhat can be consistently postulated need not thereby be said to 'must exist' in ...

    Similar

    An infinitesimal hyperreal exists87%An infinite (nonstandard) hyperreal exists86%Nonstandard infinitesimal hyperreals exist in substantial number84%The hyperreals include all standard reals plus additional elements ent...80%

    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