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
    Regularity can be preserved for uncountable sample spaces... — Carmelics
    Statements
    321,452
    Perspectives
    108,905
    Topics
    42
    Home/Modality & Possibility
    HistoryEditSee Inverse

    Regularity can be preserved for uncountable sample spaces by using hyperreal-valued probability functions

    Modality & PossibilityTruth & Knowledge
    ?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.There exists a regular hyperreal-valued probability function for the dart throw at [0, 1]
      ?

      Think about whether this reason is strong or weak

    • 2.Each landing point in a hyperreal-valued assignment receives infinitesimal probability rather than 0
      ?

      Think about whether this reason is strong or weak

    Reasons Against

    2 perspectives
    Reason against 1 of 2
    ?
    • 1.Hyperreal-valued probability functions fail to satisfy countable additivity, undermining their status as genuine probability measures.
      ?

      Think about whether this reason is strong or weak

    • 2.Without countable additivity, hyperreal probabilities cannot ground standard probabilistic reasoning about infinite sequences of events.
      ?

      Think about whether this reason is strong or weak

    • 3.Kolmogorov's axioms, which require countable additivity, remain the only fully rigorous foundation for probability theory with empirical applicability.
      ?

      Think about whether this reason is strong or weak

    Reason against 2 of 2
    ?
    • 1.Assigning infinitesimal probabilities requires adopting non-standard analysis, which depends on the axiom of choice and yields non-constructive, unverifiable probability values.
      ?

      Think about whether this reason is strong or weak

    • 2.Non-uniqueness of hyperreal extensions means infinitely many incompatible regular hyperreal probability functions exist for the same sample space, making regularity indeterminate rather than preserved.
      ?

      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 & PossibilityTruth & Knowledge

    Related

    Assigning infinitesimal probabilities requires adopting non-standard analysis, w...Each landing point in a hyperreal-valued assignment receives infinitesimal proba...Hyperreal-valued probability functions fail to satisfy countable additivity, und...Kolmogorov's axioms, which require countable additivity, remain the only fully r...
    +3 moreShow less
    Non-uniqueness of hyperreal extensions means infinitely many incompatible regula...There exists a regular hyperreal-valued probability function for the dart throw ...Without countable additivity, hyperreal probabilities cannot ground standard pro...

    Similar

    Real-valued probability functions cannot assign positive probability t...74%Countably additive Kolmogorovian probability distributions must violat...73%There exists a regular hyperreal-valued probability function for the d...68%DCK allows probability distributions over value assignments arbitraril...67%

    Source

    AI-extracted1/3 agreementValid
    SEP: infinity
    View source passageHide passage
    We have seen a striking violation of regularity in de Finetti’s lottery: his assignment of 0 to each ticket. Regularity may be preserved here by countably additive probabilities, but at the expense of a uniform distribution—for example, \(\frac{1}{2}\) to ticket 1, \(\frac{1}{4}\), to ticket 2, \(\frac{1}{8}\) to ticket 3, and so on. It may be shown that if \(F\) is uncountable, a Kolmogorovian (real-valued) probability distribution must violate regularity. (See e.g. Hájek 2003b.) This has led
    Extraction notes

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

    Details

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