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    Sentences proved from first-order axioms are true in all ... — Carmelics
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    Sentences proved from first-order axioms are true in all models of those axioms, including countable models and non-standard models

    Proof of definition segmentsTruth & 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.First-order logic satisfies the Completeness Theorem
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    • 2.The Completeness Theorem uses the concept of a first-order structure, so provability implies truth across all such structures
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    Reasons Against

    2 perspectives
    Reason against 1 of 2
    ?
    • 1.The Löwenheim-Skolem theorem entails that first-order axioms for real analysis have countable models, yet 'true in all models' cannot capture intended mathematical truth about uncountable reals.
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    • 2.Skolem's paradox reveals that 'truth in all models' is model-relative: a sentence about uncountability can be true in the intended model yet satisfied in a countable model via different satisfaction relations.
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    • 3.Therefore, provability from first-order axioms tracks formal satisfiability across structures, not truth about the mathematical domain the axioms were designed to characterize.
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    Reason against 2 of 2
    ?
    • 1.Non-standard models of Peano arithmetic, as demonstrated by Thoralf Skolem in 1933, contain 'natural numbers' with no standard correlates, making first-order provability insufficient to exclude pathological interpretations.
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    • 2.If sentences proved from first-order arithmetic axioms are 'true' in non-standard models containing infinite natural numbers, then 'true in all models' conflates formal truth-in-a-structure with arithmetical truth about the natural numbers.
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    Truth & KnowledgeProof of definition segments

    Related

    First-order logic satisfies the Completeness TheoremIf sentences proved from first-order arithmetic axioms are 'true' in non-standar...Non-standard models of Peano arithmetic, as demonstrated by Thoralf Skolem in 19...Skolem's paradox reveals that 'truth in all models' is model-relative: a sentenc...
    +3 moreShow less
    The Completeness Theorem uses the concept of a first-order structure, so provabi...The Löwenheim-Skolem theorem entails that first-order axioms for real analysis h...Therefore, provability from first-order axioms tracks formal satisfiability acro...

    Similar

    By the completeness theorem for first-order logic, every consistent th...81%By the completeness theorem for first-order logic, any consistent theo...80%By the completeness theorem for first-order logic, every consistent th...80%The Downward Löwenheim-Skolem Theorem would require that any theory wi...78%

    Source

    AI-extracted1/3 agreementValid
    SEP: logic-higher-order
    View source passageHide passage
    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