5分で分る直観主義数学
by
Yoriyuki Yamagata
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5ͰΔ؍ओֶٛ ࢁܗ↳೭ !ZPSJZVLJ
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5ͰΔ؍ओֶٛ • ʮؾ࣋ͪʯΛઆ໌͢Δ • ؍ओٛͷཱʹֶཱͬͨͷ۩ମతͳల։͠ͳ͍ • ؍ओٛͷܗࣜతͳެཧܥͳͲհ͠ͳ͍
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؍ओٛͱ • ʮײओٛʯͰ͋Γ·ͤΜ • ී௨ͷֶΛ൷͢Δʮର߅ֶʯͰ͢ • ී௨ͷֶͷཱʹཱͬͯɺ؍ओֶٛͷମܥΛ ݚڀ͢Δਓ͍Δ • ؍ओٛͷཱʹཱͬͯɺී௨ͷֶͷମܥΛݚڀ ͢Δਓ͍Δ
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ূ໌͕جຊ֓೦ • ී௨ͷֶͰਖ਼͍͔ؒ͠ҧ͍ͬͯΔ͔͕ • ূ໌खஈ • ؍ओٛͰূ໌͕͋Δ͔ূ͕͋Δ͔͕ • ͲͪΒͳ͍ͱ͖ɺਖ਼͍͠ͱ͔ؒҧ͍ͬͯΔͱ͔ ݴ͏͜ͱࣗମ͕ແҙຯ
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ΰʔϧυόοϋͷ༧ 4Ҏ্ͷۮ2ͭͷૉͷʹͳ͍ͬͯΔ
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؍ओٛͷʮূ໌ʯͱ • ۩ମతͳܭࢉʹΑΔͦͷ໋ͷݕূ • ΰʔϧυόοϋͷ༧ͷূ໌ʮ4Ҏ্ͷۮΛ2ͭ ͷૉͷʹॻ͖ͯ͠ɺ݁Ռ͕ਖ਼͍͜͠ͱͷূ໌ ͱҰॹʹग़ྗ͢ΔϓϩάϥϜʯ • ܗࣜཧʹΑΔূ໌Ͱʢຊདྷʣ͋Γ·ͤΜ • ؍ओٛͷূ໌Λهड़͢Δܗࣜݴޠ͋Γ·͢
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ഉதͳͥਖ਼͘͠ͳ͍͔ • ഉதͱ P·ͨPͰͳ͍ • PΛΰʔϧυόοϋͷ༧ͱ͢Δ • ഉதɺP͕ূ໌͞ΕΔ͔P͕ূ͞ΕΔ͔ͲͪΒ͔ Ͱ͋Δ͜ͱΛओு͢Δ • ·ͩɺPূ໌ূ͞Ε͍ͯͳ͍
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• ήʔσϧͷ൱ఆʹΑΔղऍ • ࣗવͷཧͷൣғͰɺݹయֶͰ͋Δఆཧ͕ಋ͔ ΕΔͱɺ؍ओֶٛͰͦͷήʔσϧͷ൱ఆʹΑΔղ ऍ͕ಋ͔ΕΔ ഉத͕ͳͯͦ͘ΜͳʹࠔΒͳ͍ A _ B 7! ¬(¬A ^ ¬B) 9x.A(x) 7! ¬8x.¬A(x) 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ू߹ͱ؍ओֶٛ • Per Martin-Löfͷ؍ओٛܕཧʹΑΓɺߏతͳू߹ CZF͕ղऍͰ͖ΔʢAczel, 1978) • CZFʹഉதΛՃ͑ΔͱZFʹͳΔʢAczel, 1978ʣ • ؍ओ͕ٛͲ͜·Ͱू߹ΛղऍͰ͖Δ͔ɺڵຯ͕͋ Δͱ͜Ζ
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VoevodskyͱUnivalent Foundation • ੈքʮܕʯͷू·ΓͰ͋Δʢ؍ओٛܕཧʣ • ʮܕʯۭؒɺʮಉҰੑʯͦͷۭؒͷதͷύε • ʮܕશମʯܕU • UͷதͷಉҰੑܕͷಉܕʢUnivalence Axiom) ͝ΊΜɺΘ͔ͬͯͳ͍
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͓·͚ɿֶөըΧϑΣ • ֶ͕ςʔϚͷөըΛݟ͓ͯ͠ΌΓ͢Δձ • શʹҰൠ͚ɺઐతͳ͠·ͤΜ • ։࠵Ͱ͖Δ͔Ͳ͏͔Θ͔Γ·ͤΜ • ໊લߟ͍͑ͨʢΦγϟϨʹ͍ͨ͠ʣ • ࢀՃ͍ͨ͠ਓɾӡӦखͬͯ͘ΕΔਓืूத