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    No number precedes zero — Carmelics
    Statements
    321,452
    Perspectives
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    Topics
    42
    Home/Truth & Knowledge
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    No number precedes zero

    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.Assume for reductio that some number n is such that Precedes(n, 0)
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    • 2.By the definition of predecessor, there exists a concept Q and an object c such that Qc holds, 0 equals the number of Q, and n equals the number of objects satisfying Q that are not identical to c
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    • 3.By the Lemma Concerning Zero, 0 equals the number of Q implies there is no x such that Qx
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    Reasons Against

    2 perspectives
    Reason against 1 of 2
    ?
    • 1.The Lemma Concerning Zero presupposes logicism's reduction of 'number' to extensions of concepts, which Russell's paradox showed to be formally inconsistent in Frege's original system.
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    • 2.If Basic Law V is invalid, the derivation of the Lemma Concerning Zero collapses, undermining the reductio before it reaches the contradiction in P4.
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    Reason against 2 of 2
    ?
    • 1.On a Meinongian ontology, non-existent or impossible objects can instantiate concepts, so a concept Q satisfied by a non-actual object c can coherently have zero actual instances.
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    • 2.If Qc can hold for a non-existent object c while the number of Q remains zero, then P2 and P3 are simultaneously satisfiable, dissolving the intended contradiction.
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    Topics

    Truth & KnowledgeProof of definition segments

    Connections

    1 topic

    Philosophy of Language1 linked

    Related

    Assume for reductio that some number n is such that Precedes(n, 0)But the existence of c satisfying Q (i.e., Qc) contradicts the conclusion that n...By the Lemma Concerning Zero, 0 equals the number of Q implies there is no x suc...By the definition of predecessor, there exists a concept Q and an object c such ...
    +4 moreShow less
    If Basic Law V is invalid, the derivation of the Lemma Concerning Zero collapses...If Qc can hold for a non-existent object c while the number of Q remains zero, t...On a Meinongian ontology, non-existent or impossible objects can instantiate con...The Lemma Concerning Zero presupposes logicism's reduction of 'number' to extens...

    Similar

    F(0) holds, because 0 is the initial feasible number (S1).76%Negation can be defined as A implying 0 = 1.75%From (S1), 0 denotes a feasible number, so F(0) holds.74%ABb = 0 (ABb is a contradiction)73%

    Source

    AI-extracted1/3 agreementValid
    SEP: frege-theorem
    View source passageHide passage
    Proof: Assume, for reductio, that some number, say \(n\), is such that \(\mathit{Precedes}(n,0)\). Then, by the definition of predecessor, it follows that there is a concept, say \(Q\) and an object, say \(c\), such that \(Qc \amp 0\eqclose \#Q \amp n\eqclose \#[\lambda z\, Qz \amp z\neq c]\). But by the Lemma Concerning Zero (above), \(0 = \#Q\) implies \(\neg\exists xQx\), which contradicts the fact that \(Qc\).
    Extraction notes

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

    Details

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