The cardinality count of second-order sentences presupposes a fixed background set theory, but set-theoretic pluralism (Hamkins) allows different universes where 'countably many' is not absolute.

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Reasons For

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Reason for

1.Set-theoretic pluralism (Hamkins) demonstrates that cardinality is relative to model; what counts as 'countable' varies across universes.
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2.Second-order logic's semantics require fixing a background universe to determine which sets satisfy quantified formulas.
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3.If pluralism is true, claims about 'all second-order sentences' presuppose a particular universe rather than an absolute standard.
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Reasons Against

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Reason against

1.Cardinality comparisons (countable vs. uncountable) remain invariant across set-theoretic universes sharing the same ordinal structure.
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2.Second-order logic can be given semantics independent of any single background universe (category-theoretic or proof-theoretic approaches).
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3.Pluralism about set theory doesn't entail pluralism about logical consequence; countability is definable uniformly across models.
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Key Terms

Absolute (in logic/mathematics)(in mathematical philosophy): A property or fact that holds true in every possible mathematical universe or context, rather than being relative to a particular framework.
Background set theory(mathematical logic): The foundational rules and framework that mathematicians use to build all of mathematics—basically the starting assumptions everything else is built on.
Countably many(in logic and set theory): A technical term meaning 'infinite in size but no larger than the infinity of whole numbers'—roughly, something you could theoretically list out one by one forever.
Hamkins(as a philosopher of mathematics): Joel David Hamkins, a contemporary mathematician and philosopher who developed the idea that set theory allows for multiple equally valid 'universes' rather than a single absolute universe.
Second-order sentences(as used in formal logic): Logical statements that don't just talk about individual things, but also talk about properties and relationships themselves—like saying 'there exists a property that everything has' rather than just 'there exists a thing.'
Set theory(as used in mathematics): A branch of mathematics that studies collections of objects (called 'sets') and the rules for how they relate to each other.
Set-theoretic pluralism(in philosophy of mathematics): The philosophical view that there isn't just one 'correct' universe of sets, but multiple equally valid mathematical universes with different properties.
cardinality(Central to comparing infinite sets and establishing that no universal set exists.): A measure of the size of a set, indicating the number of elements it contains.

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Modality & Possibility1 linked Skepticism1 linked

Cardinality comparisons (countable vs. uncountable) remain invariant across set-...If pluralism is true, claims about 'all second-order sentences' presuppose a par...

Details

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