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    The cardinal number of the power set of the set of space-... — Carmelics
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    The cardinal number of the power set of the set of space-time points (at least ℵ₂) is the largest cardinal.

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    1 reason for
    2 reasons against

    Reasons For

    1 perspective
    Reason for
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    • 1.A philosopher with positivistic or naturalistic leanings should posit transfinite cardinals of only those sizes required by prevailing scientific theories.
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    • 2.The prevailing theory of the space-time continuum in physics is committed to the existence of infinite sets which have the cardinality of the set of space-time points (at least ℵ₁).
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    • 3.This scientific theory is also committed to the existence of the power set of the set of space-time points, needed to accommodate continuous and discontinuous linear paths.
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    Reasons Against

    2 perspectives
    Reason against 1 of 2
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    • 1.Scientific theories are revisable, and set-theoretic commitments of current physics cannot constrain what cardinalities exist independently of theory.
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    • 2.Large cardinal axioms (inaccessible, Mahlo, measurable cardinals) are mathematically consistent extensions of ZFC that posit sets far exceeding the power set of spacetime points.
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    • 3.A naturalist who defers to mathematics as a science must acknowledge that mathematical practice itself demands cardinals beyond any fixed bound.
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    Reason against 2 of 2
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    • 1.Quine-Putnam indispensability arguments license only those mathematical entities indispensable to best science, but Maddy's objections show actual scientific practice does not uniformly quantify over all posited mathematical objects.
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    • 2.The inference from 'physics requires no set larger than P(spacetime)' to 'no larger cardinal exists' commits a modal fallacy, conflating epistemic and ontological limitations.
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    Related

    A naturalist who defers to mathematics as a science must acknowledge that mathem...A philosopher with positivistic or naturalistic leanings should posit transfinit...Large cardinal axioms (inaccessible, Mahlo, measurable cardinals) are mathematic...Quine-Putnam indispensability arguments license only those mathematical entities...
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    Scientific theories are revisable, and set-theoretic commitments of current phys...Scientific theories do not appear to require the existence of any set larger tha...The inference from 'physics requires no set larger than P(spacetime)' to 'no lar...The prevailing theory of the space-time continuum in physics is committed to the...This scientific theory is also committed to the existence of the power set of th...

    Similar

    There is no largest transfinite cardinal number.81%The prevailing theory of the space-time continuum in physics is commit...77%Although there are infinitely many transfinite cardinal numbers, there...72%In ZF and ZFC, the totality of transfinite cardinal numbers does not q...71%

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    The founder of transfinite arithmetic, Georg Cantor (1845–1918), is also a founding father of set theory. He famously proved that the set of real numbers has a larger cardinal number than the set of natural numbers; the set of reals has the same cardinality as the power set (the set of all subsets) of the set of naturals. Cantor further argued that \(\aleph_0\) is the first (and smallest) transfinite cardinal number in an infinite series of increasingly larger transfinite cardinals, \(\aleph_0,\) \(\aleph_1,\) \(\aleph_2\), and so on. But note that the numerical subscripts of these alephs do n...

    Details

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