5分で分る直観主義数学

 5分で分る直観主義数学

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Yoriyuki Yamagata

November 03, 2018
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  1. 5෼Ͱ෼Δ௚؍ओٛ਺ֶ ࢁܗ↳೭ !ZPSJZVLJ

  2. 5෼Ͱ෼Δ௚؍ओٛ਺ֶ • ʮؾ࣋ͪʯΛઆ໌͢Δ • ௚؍ओٛͷཱ৔ʹཱͬͨ਺ֶͷ۩ମతͳల։͸͠ͳ͍ • ௚؍ओٛͷܗࣜతͳެཧܥͳͲ΋঺հ͠ͳ͍

  3. ௚؍ओٛͱ͸ • ʮ௚ײओٛʯͰ͸͋Γ·ͤΜ • ී௨ͷ਺ֶΛ൷൑͢Δʮର߅਺ֶʯͰ͢ • ී௨ͷ਺ֶͷཱ৔ʹཱͬͯɺ௚؍ओٛ਺ֶͷମܥΛ ݚڀ͢Δਓ΋͍Δ • ௚؍ओٛͷཱ৔ʹཱͬͯɺී௨ͷ਺ֶͷମܥΛݚڀ

    ͢Δਓ΋͍Δ
  4. ূ໌͕جຊ֓೦ • ී௨ͷ਺ֶͰ͸ਖ਼͍͔ؒ͠ҧ͍ͬͯΔ͔͕໰୊ • ূ໌͸खஈ • ௚؍ओٛͰ͸ূ໌͕͋Δ͔൓ূ͕͋Δ͔͕໰୊ • ͲͪΒ΋ͳ͍ͱ͖͸ɺਖ਼͍͠ͱ͔ؒҧ͍ͬͯΔͱ͔ ݴ͏͜ͱࣗମ͕ແҙຯ

  5. ΰʔϧυόοϋͷ༧૝ 4Ҏ্ͷۮ਺͸2ͭͷૉ਺ͷ࿨ʹͳ͍ͬͯΔ

  6. ௚؍ओٛͷʮূ໌ʯͱ͸ • ۩ମతͳܭࢉʹΑΔͦͷ໋୊ͷݕূ • ΰʔϧυόοϋͷ༧૝ͷূ໌͸ʮ4Ҏ্ͷۮ਺Λ2ͭ ͷૉ਺ͷ࿨ʹॻ͖௚ͯ͠ɺ݁Ռ͕ਖ਼͍͜͠ͱͷূ໌ ͱҰॹʹग़ྗ͢ΔϓϩάϥϜʯ • ܗࣜ࿦ཧʹΑΔূ໌Ͱ͸ʢຊདྷ͸ʣ͋Γ·ͤΜ •

    ௚؍ओٛͷূ໌Λهड़͢Δܗࣜݴޠ͸͋Γ·͢
  7. ഉத཯͸ͳͥਖ਼͘͠ͳ͍͔ • ഉத཯ͱ͸ P·ͨ͸PͰ͸ͳ͍ • PΛΰʔϧυόοϋͷ༧૝ͱ͢Δ • ഉத཯͸ɺP͕ূ໌͞ΕΔ͔P͕൓ূ͞ΕΔ͔ͲͪΒ͔ Ͱ͋Δ͜ͱΛओு͢Δ •

    ·ͩɺP͸ূ໌΋൓ূ΋͞Ε͍ͯͳ͍
  8. • ήʔσϧͷ൱ఆʹΑΔղऍ • ࣗવ਺ͷཧ࿦ͷൣғͰ͸ɺݹయ਺ֶͰ͋Δఆཧ͕ಋ͔ ΕΔͱɺ௚؍ओٛ਺ֶͰͦͷήʔσϧͷ൱ఆʹΑΔղ ऍ͕ಋ͔ΕΔ ഉத཯͕ͳͯ͘΋ͦΜͳʹࠔΒͳ͍ A _ B

    7! ¬(¬A ^ ¬B) 9x.A(x) 7! ¬8x.¬A(x) <latexit sha1_base64="RQKTTLuC04w7BqbpraPkFw9Njs8=">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</latexit> <latexit sha1_base64="RQKTTLuC04w7BqbpraPkFw9Njs8=">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</latexit> <latexit sha1_base64="RQKTTLuC04w7BqbpraPkFw9Njs8=">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</latexit> <latexit sha1_base64="RQKTTLuC04w7BqbpraPkFw9Njs8=">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</latexit>
  9. ू߹࿦ͱ௚؍ओٛ਺ֶ • Per Martin-Löfͷ௚؍ओٛܕཧ࿦ʹΑΓɺߏ੒తͳू߹ ࿦CZF͕ղऍͰ͖ΔʢAczel, 1978) • CZFʹഉத཯ΛՃ͑ΔͱZFʹͳΔʢAczel, 1978ʣ •

    ௚؍ओ͕ٛͲ͜·Ͱू߹࿦ΛղऍͰ͖Δ͔ɺڵຯ͕͋ Δͱ͜Ζ
  10. VoevodskyͱUnivalent Foundation • ੈք͸ʮܕʯͷू·ΓͰ͋Δʢ௚؍ओٛܕཧ࿦ʣ • ʮܕʯ͸ۭؒɺʮಉҰੑʯ͸ͦͷۭؒͷதͷύε • ʮܕશମʯ΋ܕU • UͷதͷಉҰੑ͸ܕͷಉܕʢUnivalence

    Axiom) ͝ΊΜɺΘ͔ͬͯͳ͍
  11. ͓·͚ɿ਺ֶөըΧϑΣ • ਺ֶ͕ςʔϚͷөըΛݟ͓ͯ͠Ό΂Γ͢Δձ • ׬શʹҰൠ޲͚ɺઐ໳తͳ࿩͸͠·ͤΜ • ։࠵Ͱ͖Δ͔Ͳ͏͔͸Θ͔Γ·ͤΜ • ໊લ͸ߟ͍͑ͨʢΦγϟϨʹ͍ͨ͠ʣ •

    ࢀՃ͍ͨ͠ਓɾӡӦख఻ͬͯ͘ΕΔਓืूத