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    Hilbert's distinction between ideal and real mathematics ... — Carmelics
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    Challenges→The theory PA^F, though technically inconsistent, is practically consistent because any proof of a contradiction in PA^F must be infeasibly long.

    Hilbert's distinction between ideal and real mathematics shows that relaxing consistency standards undermines the foundational role formal systems are meant to play.

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    Reasons For

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    Reason for
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    • 1.Formal systems serve foundational purposes by guaranteeing we can't derive contradictions, which consistency standards directly protect.
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    • 2.Relaxing consistency allows contradictions, making it impossible to use formal systems for reliable deduction or mathematical certainty.
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    • 3.Hilbert's ideal/real distinction shows real mathematics needs consistency to avoid trivializing all propositions via explosion.
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    Reasons Against

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    Reason against
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    • 1.Paraconsistent logics tolerate contradictions without trivializing all theorems, suggesting consistency isn't strictly necessary for foundations.
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    • 2.Many formal systems already contain inconsistencies (e.g., Frege's system) yet contributed substantively before their contradictions were discovered.
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    • 3.The foundational role of formal systems could be pragmatic utility rather than absolute consistency—usefulness doesn't require perfect coherence.
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    Related

    Formal systems serve foundational purposes by guaranteeing we can't derive contr...Hilbert's ideal/real distinction shows real mathematics needs consistency to avo...Many formal systems already contain inconsistencies (e.g., Frege's system) yet c...Paraconsistent logics tolerate contradictions without trivializing all theorems,...
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    Relaxing consistency allows contradictions, making it impossible to use formal s...The foundational role of formal systems could be pragmatic utility rather than a...The theory PA^F, though technically inconsistent, is practically consistent beca...

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