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    Arguing for the consistency of a set of axioms by observi... — Carmelics
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    Home/Skepticism
    HistoryEditSee Inverse

    Arguing for the consistency of a set of axioms by observing that the intended structure itself satisfies those axioms begs the question

    SkepticismTruth & 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.The trivial approach to showing θ_M is consistent is to note that the structure M satisfies θ_M
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    • 2.This is circular because the existence of M is precisely what is at issue
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    Reasons Against

    2 perspectives
    Reason against 1 of 2
    ?
    • 1.Gödel's own practice shows that semantic consistency arguments via intended models have independent epistemic value even absent formal proof.
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    • 2.The charge of circularity conflates ontological presupposition with logical circularity: assuming M exists is not the same inference-step as concluding θ_M is consistent.
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    • 3.Hilbert's distinction between contentual and formal reasoning permits intended-model arguments as pre-formal evidence that guides, rather than replaces, rigorous consistency proofs.
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    Reason against 2 of 2
    ?
    • 1.Parsons and Shapiro argue that the iterative conception of sets provides independent, non-circular grounds for accepting the cumulative hierarchy V before any axiomatization is formulated.
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    • 2.If the intended structure M is apprehended through a conceptually prior informal understanding, invoking M is not question-begging but rather an appeal to the evidential basis that motivated the axioms in the first place.
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    Related

    Gödel's own practice shows that semantic consistency arguments via intended mode...Hilbert's distinction between contentual and formal reasoning permits intended-m...If the intended structure M is apprehended through a conceptually prior informal...Parsons and Shapiro argue that the iterative conception of sets provides indepen...
    +3 moreShow less
    The charge of circularity conflates ontological presupposition with logical circ...The trivial approach to showing θ_M is consistent is to note that the structure ...This is circular because the existence of M is precisely what is at issue

    Similar

    The trivial approach to showing θ_M is consistent is to note that the ...81%If the axioms are true, it follows that propositions exist and have th...79%Henkin's theorem establishes that every consistent set of formulas has...78%The consistency statement Con(F) must genuinely express that F is cons...78%

    Source

    AI-extracted1/3 agreementValid
    SEP: logic-higher-order
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    Let \(c,d\in (a,b)\) such that \(f(c)<0\) and \(f(d)>0\). Without loss of generality, \(c<d\). Let \(X=\{e\in(a,b) : f(e)<0\}\). Since we have relation variables for subsets of the domain, we can think of X simply as a value of such a relation variable. , \(X=\{e : e\notin X\}\)) and then we should not be able to claim that it exists. However, in this case the Comprehension Axiom Schema implies that X exists. Clearly, \(X\ne\emptyset\) and X is bounded from above by d. One of the sec
    Extraction notes

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

    Details

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