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    If a mathematical structure is internally consistent and ... — Carmelics
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    Challenges→Infinite mathematical objects, such as the completed set of all natural numbers and arbitrary irrational numbers represented by Dedekind cuts, do not exist.

    If a mathematical structure is internally consistent and theoretically indispensable, denying its existence requires a stricter criterion of existence that the finitist has not independently justified.

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    Key Terms

    Criterion of existence(what someone needs to provide in this debate)
    A standard or rule for deciding what counts as real and what doesn't.
    Internally consistent(as a requirement the framework meets)
    Free from contradictions within itself; the different parts don't contradict each other.
    Mathematical structure(what structural realists claim is what we actually know about reality)
    The abstract patterns, equations, and relationships that describe how something works, stripped of real-world details.
    Theoretically indispensable(describing how useful mathematical structures are)
    Something that appears necessary to use in order to explain or solve real problems, even if we're not sure it really exists.
    existence

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    (Kant's analysis in the Critique of Pure Reason as applied to the ontological argument)
    Not a real predicate or positive determination; it does not add to or enlarge the concept of a subject.
    finitist(naming the rival approach the axioms are said to rule out)
    A mathematical philosophy that accepts only finite objects and rejects the idea of actual infinity.

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    2 topics

    Truth & Knowledge1 linkedModality & Possibility1 linked

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    Infinite mathematical objects, such as the completed set of all natural numbers ...

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