Steeple #3: Gödel s Silver Blaze Theorem. Selmer Bringsjord Are Humans Rational? Dec RPI Troy NY USA
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- Edward Jan Borowski
- 4 lat temu
- Przeglądów:
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1 Steeple #3: Gödel s Silver Blaze Theorem Selmer Bringsjord Are Humans Rational? Dec RPI Troy NY USA
2 Gödels Great Theorems (OUP) by Selmer Bringsjord Introduction ( The Wager ) Brief Preliminaries (e.g. the propositional calculus) The Completeness Theorem The First Incompleteness Theorem The Second Incompleteness Theorem The Speedup Theorem The Continuum-Hypothesis Theorem The Time-Travel Theorem Gödel s God Theorem Could a Machine Match Gödel s Genius?
3 Gödels Great Theorems (OUP) by Selmer Bringsjord Introduction ( The Wager ) Brief Preliminaries (e.g. the propositional calculus) The Completeness Theorem The First Incompleteness Theorem The Second Incompleteness Theorem The Speedup Theorem The Continuum-Hypothesis Theorem The Time-Travel Theorem Gödel s God Theorem Could a Machine Match Gödel s Genius?
4 Gödels Great Theorems (OUP) by Selmer Bringsjord Introduction ( The Wager ) Brief Preliminaries (e.g. the propositional calculus) The Completeness Theorem The First Incompleteness Theorem The Second Incompleteness Theorem The Speedup Theorem The Continuum-Hypothesis Theorem The Time-Travel Theorem Gödel s God Theorem Could a Machine Match Gödel s Genius?
5 Gödels Great Theorems (OUP) by Selmer Bringsjord 3/4 Introduction ( The Wager ) Brief Preliminaries (e.g. the propositional calculus) The Completeness Theorem The First Incompleteness Theorem The Second Incompleteness Theorem The Speedup Theorem The Continuum-Hypothesis Theorem The Time-Travel Theorem Gödel s God Theorem Could a Machine Match Gödel s Genius?
6 Gödels Great Theorems (OUP) by Selmer Bringsjord 3/4 1 Introduction ( The Wager ) Brief Preliminaries (e.g. the propositional calculus) The Completeness Theorem The First Incompleteness Theorem The Second Incompleteness Theorem The Speedup Theorem The Continuum-Hypothesis Theorem The Time-Travel Theorem Gödel s God Theorem Could a Machine Match Gödel s Genius?
7 Gödels Great Theorems (OUP) by Selmer Bringsjord 3/4 1.1 Introduction ( The Wager ) Brief Preliminaries (e.g. the propositional calculus) The Completeness Theorem The First Incompleteness Theorem The Second Incompleteness Theorem The Speedup Theorem The Continuum-Hypothesis Theorem The Time-Travel Theorem Gödel s God Theorem Could a Machine Match Gödel s Genius?
8 ZFC (The Foundation of Mathematics)
9 ZFC (The Foundation of Mathematics) EXT: Two sets which contain the same elements are equal.
10 ZFC (The Foundation of Mathematics) EXT: Two sets which contain the same elements are equal. PAIR: Given two sets A and B, the pair set {A, B} exists.
11 ZFC (The Foundation of Mathematics) EXT: Two sets which contain the same elements are equal. PAIR: Given two sets A and B, the pair set {A, B} exists.
12 ZFC (The Foundation of Mathematics) EXT: Two sets which contain the same elements are equal. PAIR: Given two sets A and B, the pair set {A, B} exists. INF: There exists an infinite set, viz. {ø, {ø}, {ø, {ø}}, }.
13 ZFC (The Foundation of Mathematics) EXT: Two sets which contain the same elements are equal. PAIR: Given two sets A and B, the pair set {A, B} exists. INF: There exists an infinite set, viz. {ø, {ø}, {ø, {ø}}, }. AC: Given a set A of nonempty pairwise disjoint sets, there exists a set which contains exactly one element of each set in A.
14 <latexit sha1_base64="qja0aq3m+erqlh9vrdltxyzesuc=">aaab8xicbvbns8nafhypx7v+vt16wsycp5kiomeif09swdzig8pm+9iu3wzc7kyoof/ciwdfvppvvplv3lq5aovawjdzhjtvgkrwbvz32ymtrk6tb5q3k1vbo7t71f2dto5txbdfyhgrtka1ci6xzbgr2eku0igq+bcmr3p/4qmv5rg8n5me/ygojq85o8zkj72imleqzlftfrxm1t0zydlxclkdas1+9as3ifkaotrmuk27npsyp6pkcczwwumlghpkxnsixusljvd72szxljxyzudcwnkndzmpvzcygmk9iqi7msfui14u/ud1uxne+hmxswpqsvlhysqiiul+phlwhcyiiswukw6zejaiijjjs6ryerzfk5dj+6zuuxxv7rzwucrqkmmrhmmpehabdbibjrsagyrneiu3rzsvzrvzmr8tocxoifyb8/kdt0uq7g==</latexit> <latexit sha1_base64="qja0aq3m+erqlh9vrdltxyzesuc=">aaab8xicbvbns8nafhypx7v+vt16wsycp5kiomeif09swdzig8pm+9iu3wzc7kyoof/ciwdfvppvvplv3lq5aovawjdzhjtvgkrwbvz32ymtrk6tb5q3k1vbo7t71f2dto5txbdfyhgrtka1ci6xzbgr2eku0igq+bcmr3p/4qmv5rg8n5me/ygojq85o8zkj72imleqzlftfrxm1t0zydlxclkdas1+9as3ifkaotrmuk27npsyp6pkcczwwumlghpkxnsixusljvd72szxljxyzudcwnkndzmpvzcygmk9iqi7msfui14u/ud1uxne+hmxswpqsvlhysqiiul+phlwhcyiiswukw6zejaiijjjs6ryerzfk5dj+6zuuxxv7rzwucrqkmmrhmmpehabdbibjrsagyrneiu3rzsvzrvzmr8tocxoifyb8/kdt0uq7g==</latexit> <latexit sha1_base64="qja0aq3m+erqlh9vrdltxyzesuc=">aaab8xicbvbns8nafhypx7v+vt16wsycp5kiomeif09swdzig8pm+9iu3wzc7kyoof/ciwdfvppvvplv3lq5aovawjdzhjtvgkrwbvz32ymtrk6tb5q3k1vbo7t71f2dto5txbdfyhgrtka1ci6xzbgr2eku0igq+bcmr3p/4qmv5rg8n5me/ygojq85o8zkj72imleqzlftfrxm1t0zydlxclkdas1+9as3ifkaotrmuk27npsyp6pkcczwwumlghpkxnsixusljvd72szxljxyzudcwnkndzmpvzcygmk9iqi7msfui14u/ud1uxne+hmxswpqsvlhysqiiul+phlwhcyiiswukw6zejaiijjjs6ryerzfk5dj+6zuuxxv7rzwucrqkmmrhmmpehabdbibjrsagyrneiu3rzsvzrvzmr8tocxoifyb8/kdt0uq7g==</latexit> <latexit sha1_base64="qja0aq3m+erqlh9vrdltxyzesuc=">aaab8xicbvbns8nafhypx7v+vt16wsycp5kiomeif09swdzig8pm+9iu3wzc7kyoof/ciwdfvppvvplv3lq5aovawjdzhjtvgkrwbvz32ymtrk6tb5q3k1vbo7t71f2dto5txbdfyhgrtka1ci6xzbgr2eku0igq+bcmr3p/4qmv5rg8n5me/ygojq85o8zkj72imleqzlftfrxm1t0zydlxclkdas1+9as3ifkaotrmuk27npsyp6pkcczwwumlghpkxnsixusljvd72szxljxyzudcwnkndzmpvzcygmk9iqi7msfui14u/ud1uxne+hmxswpqsvlhysqiiul+phlwhcyiiswukw6zejaiijjjs6ryerzfk5dj+6zuuxxv7rzwucrqkmmrhmmpehabdbibjrsagyrneiu3rzsvzrvzmr8tocxoifyb8/kdt0uq7g==</latexit> ZFC (The Foundation of Mathematics) EXT: Two sets which contain the same elements are equal. PAIR: Given two sets A and B, the pair set {A, B} exists. INF: There exists an infinite set, viz. {ø, {ø}, {ø, {ø}}, }. AC: Given a set A of nonempty pairwise disjoint sets, there exists a set which contains exactly one element of each set in A. N
15 Arithmetic is Part of All Things Sci/Eng/Tech! and courtesy of Gödel: We can t even prove all truths of arithmetic! Each circle is a larger part of the formal sciences. PA
16 Actually, the true kernel is set theory! Each circle is a larger part of the formal sciences. ZFC PA
17 Cantor s Theorem
18 Cantor s Theorem Cantor (1878):
19 <latexit sha1_base64="0ffrjxdedwdvkuxps6ubww/vrwm=">aaacdxicbvdlssnafl3xwesr6tlnybxqpiqi6eqkblxjbfuanptjdniontyymqgl5afc+ctuxcji1r07/8zjglbbdwycoede7r3hjtityrk+jixfpewv1djaex1jc2vb3nltytawhdzjyeprcbgknaw0qzjitbmjin2x07y7vsr89j0vkoxbnzpe1phxmgaei1hpqw8e9nysrgtzpjfwc+66yu16jc7qz69vvqyalqpne7sgfsjq6jufvufiyp8ginaszde2iuukwchgoe3lvvjscjmxhtkupgh2qxss/jouhwllglxq6bcolku/oxlssznxxv2zbshnvuz8z+vgyjt3ehzesaibmq7yyo5uiljo0iajshsfaikjyhpxrezyykj0ggudgj178jxpndrsq2bfnlbql0ucjdiha6icdwdqh2toqbmipmatvmcr8wg8g2/g+7r0wsh69uapji9va6mbga==</latexit> <latexit sha1_base64="0ffrjxdedwdvkuxps6ubww/vrwm=">aaacdxicbvdlssnafl3xwesr6tlnybxqpiqi6eqkblxjbfuanptjdniontyymqgl5afc+ctuxcji1r07/8zjglbbdwycoede7r3hjtityrk+jixfpewv1djaex1jc2vb3nltytawhdzjyeprcbgknaw0qzjitbmjin2x07y7vsr89j0vkoxbnzpe1phxmgaei1hpqw8e9nysrgtzpjfwc+66yu16jc7qz69vvqyalqpne7sgfsjq6jufvufiyp8ginaszde2iuukwchgoe3lvvjscjmxhtkupgh2qxss/jouhwllglxq6bcolku/oxlssznxxv2zbshnvuz8z+vgyjt3ehzesaibmq7yyo5uiljo0iajshsfaikjyhpxrezyykj0ggudgj178jxpndrsq2bfnlbql0ucjdiha6icdwdqh2toqbmipmatvmcr8wg8g2/g+7r0wsh69uapji9va6mbga==</latexit> <latexit sha1_base64="0ffrjxdedwdvkuxps6ubww/vrwm=">aaacdxicbvdlssnafl3xwesr6tlnybxqpiqi6eqkblxjbfuanptjdniontyymqgl5afc+ctuxcji1r07/8zjglbbdwycoede7r3hjtityrk+jixfpewv1djaex1jc2vb3nltytawhdzjyeprcbgknaw0qzjitbmjin2x07y7vsr89j0vkoxbnzpe1phxmgaei1hpqw8e9nysrgtzpjfwc+66yu16jc7qz69vvqyalqpne7sgfsjq6jufvufiyp8ginaszde2iuukwchgoe3lvvjscjmxhtkupgh2qxss/jouhwllglxq6bcolku/oxlssznxxv2zbshnvuz8z+vgyjt3ehzesaibmq7yyo5uiljo0iajshsfaikjyhpxrezyykj0ggudgj178jxpndrsq2bfnlbql0ucjdiha6icdwdqh2toqbmipmatvmcr8wg8g2/g+7r0wsh69uapji9va6mbga==</latexit> <latexit sha1_base64="0ffrjxdedwdvkuxps6ubww/vrwm=">aaacdxicbvdlssnafl3xwesr6tlnybxqpiqi6eqkblxjbfuanptjdniontyymqgl5afc+ctuxcji1r07/8zjglbbdwycoede7r3hjtityrk+jixfpewv1djaex1jc2vb3nltytawhdzjyeprcbgknaw0qzjitbmjin2x07y7vsr89j0vkoxbnzpe1phxmgaei1hpqw8e9nysrgtzpjfwc+66yu16jc7qz69vvqyalqpne7sgfsjq6jufvufiyp8ginaszde2iuukwchgoe3lvvjscjmxhtkupgh2qxss/jouhwllglxq6bcolku/oxlssznxxv2zbshnvuz8z+vgyjt3ehzesaibmq7yyo5uiljo0iajshsfaikjyhpxrezyykj0ggudgj178jxpndrsq2bfnlbql0ucjdiha6icdwdqh2toqbmipmatvmcr8wg8g2/g+7r0wsh69uapji9va6mbga==</latexit> Cantor s Theorem Cantor (1878): P(N) > N
20 <latexit sha1_base64="0ffrjxdedwdvkuxps6ubww/vrwm=">aaacdxicbvdlssnafl3xwesr6tlnybxqpiqi6eqkblxjbfuanptjdniontyymqgl5afc+ctuxcji1r07/8zjglbbdwycoede7r3hjtityrk+jixfpewv1djaex1jc2vb3nltytawhdzjyeprcbgknaw0qzjitbmjin2x07y7vsr89j0vkoxbnzpe1phxmgaei1hpqw8e9nysrgtzpjfwc+66yu16jc7qz69vvqyalqpne7sgfsjq6jufvufiyp8ginaszde2iuukwchgoe3lvvjscjmxhtkupgh2qxss/jouhwllglxq6bcolku/oxlssznxxv2zbshnvuz8z+vgyjt3ehzesaibmq7yyo5uiljo0iajshsfaikjyhpxrezyykj0ggudgj178jxpndrsq2bfnlbql0ucjdiha6icdwdqh2toqbmipmatvmcr8wg8g2/g+7r0wsh69uapji9va6mbga==</latexit> <latexit sha1_base64="0ffrjxdedwdvkuxps6ubww/vrwm=">aaacdxicbvdlssnafl3xwesr6tlnybxqpiqi6eqkblxjbfuanptjdniontyymqgl5afc+ctuxcji1r07/8zjglbbdwycoede7r3hjtityrk+jixfpewv1djaex1jc2vb3nltytawhdzjyeprcbgknaw0qzjitbmjin2x07y7vsr89j0vkoxbnzpe1phxmgaei1hpqw8e9nysrgtzpjfwc+66yu16jc7qz69vvqyalqpne7sgfsjq6jufvufiyp8ginaszde2iuukwchgoe3lvvjscjmxhtkupgh2qxss/jouhwllglxq6bcolku/oxlssznxxv2zbshnvuz8z+vgyjt3ehzesaibmq7yyo5uiljo0iajshsfaikjyhpxrezyykj0ggudgj178jxpndrsq2bfnlbql0ucjdiha6icdwdqh2toqbmipmatvmcr8wg8g2/g+7r0wsh69uapji9va6mbga==</latexit> <latexit sha1_base64="0ffrjxdedwdvkuxps6ubww/vrwm=">aaacdxicbvdlssnafl3xwesr6tlnybxqpiqi6eqkblxjbfuanptjdniontyymqgl5afc+ctuxcji1r07/8zjglbbdwycoede7r3hjtityrk+jixfpewv1djaex1jc2vb3nltytawhdzjyeprcbgknaw0qzjitbmjin2x07y7vsr89j0vkoxbnzpe1phxmgaei1hpqw8e9nysrgtzpjfwc+66yu16jc7qz69vvqyalqpne7sgfsjq6jufvufiyp8ginaszde2iuukwchgoe3lvvjscjmxhtkupgh2qxss/jouhwllglxq6bcolku/oxlssznxxv2zbshnvuz8z+vgyjt3ehzesaibmq7yyo5uiljo0iajshsfaikjyhpxrezyykj0ggudgj178jxpndrsq2bfnlbql0ucjdiha6icdwdqh2toqbmipmatvmcr8wg8g2/g+7r0wsh69uapji9va6mbga==</latexit> <latexit sha1_base64="0ffrjxdedwdvkuxps6ubww/vrwm=">aaacdxicbvdlssnafl3xwesr6tlnybxqpiqi6eqkblxjbfuanptjdniontyymqgl5afc+ctuxcji1r07/8zjglbbdwycoede7r3hjtityrk+jixfpewv1djaex1jc2vb3nltytawhdzjyeprcbgknaw0qzjitbmjin2x07y7vsr89j0vkoxbnzpe1phxmgaei1hpqw8e9nysrgtzpjfwc+66yu16jc7qz69vvqyalqpne7sgfsjq6jufvufiyp8ginaszde2iuukwchgoe3lvvjscjmxhtkupgh2qxss/jouhwllglxq6bcolku/oxlssznxxv2zbshnvuz8z+vgyjt3ehzesaibmq7yyo5uiljo0iajshsfaikjyhpxrezyykj0ggudgj178jxpndrsq2bfnlbql0ucjdiha6icdwdqh2toqbmipmatvmcr8wg8g2/g+7r0wsh69uapji9va6mbga==</latexit> Cantor s Theorem Cantor (1878): P(N) > N The power set of the natural numbers is larger than the natural numbers!
21 <latexit sha1_base64="0ffrjxdedwdvkuxps6ubww/vrwm=">aaacdxicbvdlssnafl3xwesr6tlnybxqpiqi6eqkblxjbfuanptjdniontyymqgl5afc+ctuxcji1r07/8zjglbbdwycoede7r3hjtityrk+jixfpewv1djaex1jc2vb3nltytawhdzjyeprcbgknaw0qzjitbmjin2x07y7vsr89j0vkoxbnzpe1phxmgaei1hpqw8e9nysrgtzpjfwc+66yu16jc7qz69vvqyalqpne7sgfsjq6jufvufiyp8ginaszde2iuukwchgoe3lvvjscjmxhtkupgh2qxss/jouhwllglxq6bcolku/oxlssznxxv2zbshnvuz8z+vgyjt3ehzesaibmq7yyo5uiljo0iajshsfaikjyhpxrezyykj0ggudgj178jxpndrsq2bfnlbql0ucjdiha6icdwdqh2toqbmipmatvmcr8wg8g2/g+7r0wsh69uapji9va6mbga==</latexit> <latexit sha1_base64="0ffrjxdedwdvkuxps6ubww/vrwm=">aaacdxicbvdlssnafl3xwesr6tlnybxqpiqi6eqkblxjbfuanptjdniontyymqgl5afc+ctuxcji1r07/8zjglbbdwycoede7r3hjtityrk+jixfpewv1djaex1jc2vb3nltytawhdzjyeprcbgknaw0qzjitbmjin2x07y7vsr89j0vkoxbnzpe1phxmgaei1hpqw8e9nysrgtzpjfwc+66yu16jc7qz69vvqyalqpne7sgfsjq6jufvufiyp8ginaszde2iuukwchgoe3lvvjscjmxhtkupgh2qxss/jouhwllglxq6bcolku/oxlssznxxv2zbshnvuz8z+vgyjt3ehzesaibmq7yyo5uiljo0iajshsfaikjyhpxrezyykj0ggudgj178jxpndrsq2bfnlbql0ucjdiha6icdwdqh2toqbmipmatvmcr8wg8g2/g+7r0wsh69uapji9va6mbga==</latexit> <latexit sha1_base64="0ffrjxdedwdvkuxps6ubww/vrwm=">aaacdxicbvdlssnafl3xwesr6tlnybxqpiqi6eqkblxjbfuanptjdniontyymqgl5afc+ctuxcji1r07/8zjglbbdwycoede7r3hjtityrk+jixfpewv1djaex1jc2vb3nltytawhdzjyeprcbgknaw0qzjitbmjin2x07y7vsr89j0vkoxbnzpe1phxmgaei1hpqw8e9nysrgtzpjfwc+66yu16jc7qz69vvqyalqpne7sgfsjq6jufvufiyp8ginaszde2iuukwchgoe3lvvjscjmxhtkupgh2qxss/jouhwllglxq6bcolku/oxlssznxxv2zbshnvuz8z+vgyjt3ehzesaibmq7yyo5uiljo0iajshsfaikjyhpxrezyykj0ggudgj178jxpndrsq2bfnlbql0ucjdiha6icdwdqh2toqbmipmatvmcr8wg8g2/g+7r0wsh69uapji9va6mbga==</latexit> <latexit sha1_base64="0ffrjxdedwdvkuxps6ubww/vrwm=">aaacdxicbvdlssnafl3xwesr6tlnybxqpiqi6eqkblxjbfuanptjdniontyymqgl5afc+ctuxcji1r07/8zjglbbdwycoede7r3hjtityrk+jixfpewv1djaex1jc2vb3nltytawhdzjyeprcbgknaw0qzjitbmjin2x07y7vsr89j0vkoxbnzpe1phxmgaei1hpqw8e9nysrgtzpjfwc+66yu16jc7qz69vvqyalqpne7sgfsjq6jufvufiyp8ginaszde2iuukwchgoe3lvvjscjmxhtkupgh2qxss/jouhwllglxq6bcolku/oxlssznxxv2zbshnvuz8z+vgyjt3ehzesaibmq7yyo5uiljo0iajshsfaikjyhpxrezyykj0ggudgj178jxpndrsq2bfnlbql0ucjdiha6icdwdqh2toqbmipmatvmcr8wg8g2/g+7r0wsh69uapji9va6mbga==</latexit> Cantor s Theorem Cantor (1878): P(N) > N The power set of the natural numbers is larger than the natural numbers! How do we know this???!???
22 Continuum Hypothesis
23 <latexit sha1_base64="vnn0r0rhh4kwzdi0dpx0mfbwa30=">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</latexit> <latexit sha1_base64="vnn0r0rhh4kwzdi0dpx0mfbwa30=">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</latexit> <latexit sha1_base64="vnn0r0rhh4kwzdi0dpx0mfbwa30=">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</latexit> <latexit sha1_base64="vnn0r0rhh4kwzdi0dpx0mfbwa30=">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</latexit> Continuum Hypothesis CH : 8S[(S R ^ Fin(S))! (S N _ S R)]
24 <latexit sha1_base64="vnn0r0rhh4kwzdi0dpx0mfbwa30=">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</latexit> <latexit sha1_base64="vnn0r0rhh4kwzdi0dpx0mfbwa30=">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</latexit> <latexit sha1_base64="vnn0r0rhh4kwzdi0dpx0mfbwa30=">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</latexit> <latexit sha1_base64="vnn0r0rhh4kwzdi0dpx0mfbwa30=">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</latexit> Continuum Hypothesis CH : 8S[(S R ^ Fin(S))! (S N _ S R)] Every infinite subset of the reals is either the same size as the natural numbers or the same size as the reals.
25 Generalized Continuum Hypothesis
26 <latexit sha1_base64="6uwlt4bihhlumhuct3nrbnmqq+c=">aaachnicbvdlsgnbejz1bxxfpxozdeiecbui6eleqtxgyjsqhda76y1dzmewmv4xlpksl/6kfw+kcj70b5zehntyp6kqmq6umjhcou9/ehotu9mzs3pzuyxfpewv/oraldwp4vdlwmptc5kfkrruuacewmkaxage67b7otcvb8fyodul9hjoxqyjrcq4q0e18vunonr32zk2fjyvr4wkhbii1alspq3s0ebm8iyzmzx7xco2pakvuitf8ev+cog4ceagqeztbuxfg23n0xgucsmsrqd+gs2mgrrcqj/xsc0kjhdzb+ookhadbwbd9/p0yzftgrmikvzih+zpjyzf1vbi0dkhue1fbud+p9vtja6bmvbjiqd496eolrq1hxrf28iar9lzghejxfbkb5hhhf2jovdc8pflcxc1wwr8uncxvzg+gduxrzbijimsgbyqy3joyqrkolknj+szvhgp3pp36r19wye80c46+txexxco/qeb</latexit> <latexit sha1_base64="6uwlt4bihhlumhuct3nrbnmqq+c=">aaachnicbvdlsgnbejz1bxxfpxozdeiecbui6eleqtxgyjsqhda76y1dzmewmv4xlpksl/6kfw+kcj70b5zehntyp6kqmq6umjhcou9/ehotu9mzs3pzuyxfpewv/oraldwp4vdlwmptc5kfkrruuacewmkaxage67b7otcvb8fyodul9hjoxqyjrcq4q0e18vunonr32zk2fjyvr4wkhbii1alspq3s0ebm8iyzmzx7xco2pakvuitf8ev+cog4ceagqeztbuxfg23n0xgucsmsrqd+gs2mgrrcqj/xsc0kjhdzb+ookhadbwbd9/p0yzftgrmikvzih+zpjyzf1vbi0dkhue1fbud+p9vtja6bmvbjiqd496eolrq1hxrf28iar9lzghejxfbkb5hhhf2jovdc8pflcxc1wwr8uncxvzg+gduxrzbijimsgbyqy3joyqrkolknj+szvhgp3pp36r19wye80c46+txexxco/qeb</latexit> <latexit sha1_base64="6uwlt4bihhlumhuct3nrbnmqq+c=">aaachnicbvdlsgnbejz1bxxfpxozdeiecbui6eleqtxgyjsqhda76y1dzmewmv4xlpksl/6kfw+kcj70b5zehntyp6kqmq6umjhcou9/ehotu9mzs3pzuyxfpewv/oraldwp4vdlwmptc5kfkrruuacewmkaxage67b7otcvb8fyodul9hjoxqyjrcq4q0e18vunonr32zk2fjyvr4wkhbii1alspq3s0ebm8iyzmzx7xco2pakvuitf8ev+cog4ceagqeztbuxfg23n0xgucsmsrqd+gs2mgrrcqj/xsc0kjhdzb+ookhadbwbd9/p0yzftgrmikvzih+zpjyzf1vbi0dkhue1fbud+p9vtja6bmvbjiqd496eolrq1hxrf28iar9lzghejxfbkb5hhhf2jovdc8pflcxc1wwr8uncxvzg+gduxrzbijimsgbyqy3joyqrkolknj+szvhgp3pp36r19wye80c46+txexxco/qeb</latexit> <latexit sha1_base64="6uwlt4bihhlumhuct3nrbnmqq+c=">aaachnicbvdlsgnbejz1bxxfpxozdeiecbui6eleqtxgyjsqhda76y1dzmewmv4xlpksl/6kfw+kcj70b5zehntyp6kqmq6umjhcou9/ehotu9mzs3pzuyxfpewv/oraldwp4vdlwmptc5kfkrruuacewmkaxage67b7otcvb8fyodul9hjoxqyjrcq4q0e18vunonr32zk2fjyvr4wkhbii1alspq3s0ebm8iyzmzx7xco2pakvuitf8ev+cog4ceagqeztbuxfg23n0xgucsmsrqd+gs2mgrrcqj/xsc0kjhdzb+ookhadbwbd9/p0yzftgrmikvzih+zpjyzf1vbi0dkhue1fbud+p9vtja6bmvbjiqd496eolrq1hxrf28iar9lzghejxfbkb5hhhf2jovdc8pflcxc1wwr8uncxvzg+gduxrzbijimsgbyqy3joyqrkolknj+szvhgp3pp36r19wye80c46+txexxco/qeb</latexit> Generalized Continuum Hypothesis For every infinite set S, P(S) >S.
27 <latexit sha1_base64="6uwlt4bihhlumhuct3nrbnmqq+c=">aaachnicbvdlsgnbejz1bxxfpxozdeiecbui6eleqtxgyjsqhda76y1dzmewmv4xlpksl/6kfw+kcj70b5zehntyp6kqmq6umjhcou9/ehotu9mzs3pzuyxfpewv/oraldwp4vdlwmptc5kfkrruuacewmkaxage67b7otcvb8fyodul9hjoxqyjrcq4q0e18vunonr32zk2fjyvr4wkhbii1alspq3s0ebm8iyzmzx7xco2pakvuitf8ev+cog4ceagqeztbuxfg23n0xgucsmsrqd+gs2mgrrcqj/xsc0kjhdzb+ookhadbwbd9/p0yzftgrmikvzih+zpjyzf1vbi0dkhue1fbud+p9vtja6bmvbjiqd496eolrq1hxrf28iar9lzghejxfbkb5hhhf2jovdc8pflcxc1wwr8uncxvzg+gduxrzbijimsgbyqy3joyqrkolknj+szvhgp3pp36r19wye80c46+txexxco/qeb</latexit> <latexit sha1_base64="6uwlt4bihhlumhuct3nrbnmqq+c=">aaachnicbvdlsgnbejz1bxxfpxozdeiecbui6eleqtxgyjsqhda76y1dzmewmv4xlpksl/6kfw+kcj70b5zehntyp6kqmq6umjhcou9/ehotu9mzs3pzuyxfpewv/oraldwp4vdlwmptc5kfkrruuacewmkaxage67b7otcvb8fyodul9hjoxqyjrcq4q0e18vunonr32zk2fjyvr4wkhbii1alspq3s0ebm8iyzmzx7xco2pakvuitf8ev+cog4ceagqeztbuxfg23n0xgucsmsrqd+gs2mgrrcqj/xsc0kjhdzb+ookhadbwbd9/p0yzftgrmikvzih+zpjyzf1vbi0dkhue1fbud+p9vtja6bmvbjiqd496eolrq1hxrf28iar9lzghejxfbkb5hhhf2jovdc8pflcxc1wwr8uncxvzg+gduxrzbijimsgbyqy3joyqrkolknj+szvhgp3pp36r19wye80c46+txexxco/qeb</latexit> <latexit sha1_base64="6uwlt4bihhlumhuct3nrbnmqq+c=">aaachnicbvdlsgnbejz1bxxfpxozdeiecbui6eleqtxgyjsqhda76y1dzmewmv4xlpksl/6kfw+kcj70b5zehntyp6kqmq6umjhcou9/ehotu9mzs3pzuyxfpewv/oraldwp4vdlwmptc5kfkrruuacewmkaxage67b7otcvb8fyodul9hjoxqyjrcq4q0e18vunonr32zk2fjyvr4wkhbii1alspq3s0ebm8iyzmzx7xco2pakvuitf8ev+cog4ceagqeztbuxfg23n0xgucsmsrqd+gs2mgrrcqj/xsc0kjhdzb+ookhadbwbd9/p0yzftgrmikvzih+zpjyzf1vbi0dkhue1fbud+p9vtja6bmvbjiqd496eolrq1hxrf28iar9lzghejxfbkb5hhhf2jovdc8pflcxc1wwr8uncxvzg+gduxrzbijimsgbyqy3joyqrkolknj+szvhgp3pp36r19wye80c46+txexxco/qeb</latexit> <latexit sha1_base64="6uwlt4bihhlumhuct3nrbnmqq+c=">aaachnicbvdlsgnbejz1bxxfpxozdeiecbui6eleqtxgyjsqhda76y1dzmewmv4xlpksl/6kfw+kcj70b5zehntyp6kqmq6umjhcou9/ehotu9mzs3pzuyxfpewv/oraldwp4vdlwmptc5kfkrruuacewmkaxage67b7otcvb8fyodul9hjoxqyjrcq4q0e18vunonr32zk2fjyvr4wkhbii1alspq3s0ebm8iyzmzx7xco2pakvuitf8ev+cog4ceagqeztbuxfg23n0xgucsmsrqd+gs2mgrrcqj/xsc0kjhdzb+ookhadbwbd9/p0yzftgrmikvzih+zpjyzf1vbi0dkhue1fbud+p9vtja6bmvbjiqd496eolrq1hxrf28iar9lzghejxfbkb5hhhf2jovdc8pflcxc1wwr8uncxvzg+gduxrzbijimsgbyqy3joyqrkolknj+szvhgp3pp36r19wye80c46+txexxco/qeb</latexit> Generalized Continuum Hypothesis For every infinite set S, P(S) >S. Generalized Continuum Hypothesis (GCH): There s no set (size-wise) between S and P(S).
28 Shorthand History, Moral Hilbert s #1 (1900): very plausible theorem : CH
29 Shorthand History, Moral Hilbert s #1 (1900): very plausible theorem : CH
30 Shorthand History, Moral Hilbert s #1 (1900): very plausible theorem : CH Gödel (1938):
31 Shorthand History, Moral Hilbert s #1 (1900): very plausible theorem : CH Gödel (1938): ZFC CH
32 Shorthand History, Moral Hilbert s #1 (1900): very plausible theorem : CH Gödel (1938): ZFC CH Cohen (1963): ZFC CH
33 Shorthand History, Moral Hilbert s #1 (1900): very plausible theorem : CH Gödel (1938): ZFC CH Cohen (1963): ZFC CH ADR-based approach to enabling an AI to discover and prove Gödel s First Incompleteness Theorem:
34 Shorthand History, Moral Hilbert s #1 (1900): very plausible theorem : CH Gödel (1938): ZFC CH Cohen (1963): ZFC CH ADR-based approach to enabling an AI to discover and prove Gödel s First Incompleteness Theorem: Liar FPT/GI
35 Shorthand History, Moral Hilbert s #1 (1900): very plausible theorem : CH Gödel (1938): ZFC CH Cohen (1963): ZFC CH ADR-based approach to enabling an AI to discover and prove Gödel s First Incompleteness Theorem:
36 Shorthand History, Moral Hilbert s #1 (1900): very plausible theorem : CH Gödel (1938): ZFC CH Cohen (1963): ZFC CH ADR-based approach to enabling an AI to discover and prove Gödel s First Incompleteness Theorem: Won t work on this theorem of Gödel s!
37 Fleshing Out a Bit
38 Fleshing Out a Bit 1900 Hilbert:... a very plausible theorem, which nevertheless, in spite of the most strenuous efforts, no one has succeeded in proving....
39 Fleshing Out a Bit 1900 Hilbert:... a very plausible theorem, which nevertheless, in spite of the most strenuous efforts, no one has succeeded in proving.... Continuum Hypothesis: There s no set of a size between the integers and the reals.
40 Fleshing Out a Bit 1900 Hilbert:... a very plausible theorem, which nevertheless, in spite of the most strenuous efforts, no one has succeeded in proving.... Continuum Hypothesis: There s no set of a size between the integers and the reals Gödel: If math is consistent, you can t prove that CH is false.
41 Fleshing Out a Bit 1900 Hilbert:... a very plausible theorem, which nevertheless, in spite of the most strenuous efforts, no one has succeeded in proving.... Continuum Hypothesis: There s no set of a size between the integers and the reals Gödel: If math is consistent, you can t prove that CH is false.
42 Fleshing Out a Bit 1900 Hilbert:... a very plausible theorem, which nevertheless, in spite of the most strenuous efforts, no one has succeeded in proving.... Continuum Hypothesis: There s no set of a size between the integers and the reals Gödel: If math is consistent, you can t prove that CH is false.
43 Fleshing Out a Bit 1900 Hilbert:... a very plausible theorem, which nevertheless, in spite of the most strenuous efforts, no one has succeeded in proving.... Continuum Hypothesis: There s no set of a size between the integers and the reals Gödel: If math is consistent, you can t prove that CH is false.
44 Fleshing Out a Bit 1900 Hilbert:... a very plausible theorem, which nevertheless, in spite of the most strenuous efforts, no one has succeeded in proving.... Continuum Hypothesis: There s no set of a size between the integers and the reals Gödel: If math is consistent, you can t prove that CH is false. (1962 Cohen: If math is consistent, you can t prove that CH is true.)
45 Fleshing Out a Bit 1900 Hilbert:... a very plausible theorem, which nevertheless, in spite of the most strenuous efforts, no one has succeeded in proving.... Continuum Hypothesis: There s no set of a size between the integers and the reals Gödel: If math is consistent, you can t prove that CH is false. (1962 Cohen: If math is consistent, you can t prove that CH is true.)
46 Fleshing Out a Bit 1900 Hilbert:... a very plausible theorem, which nevertheless, in spite of the most strenuous efforts, no one has succeeded in proving.... Continuum Hypothesis: There s no set of a size between the integers and the reals Gödel: If math is consistent, you can t prove that CH is false. (1962 Cohen: If math is consistent, you can t prove that CH is true.) Silver Blaze
47 Fleshing Out a Bit 1900 Hilbert:... a very plausible theorem, which nevertheless, in spite of the most strenuous efforts, no one has succeeded in proving.... Continuum Hypothesis: There s no set of a size between the integers and the reals Gödel: If math is consistent, you can t prove that CH is false. (1962 Cohen: If math is consistent, you can t prove that CH is true.) Silver Blaze
48 Fleshing Out a Bit 1900 Hilbert:... a very plausible theorem, which nevertheless, in spite of the most strenuous efforts, no one has succeeded in proving.... Continuum Hypothesis: There s no set of a size between the integers and the reals Gödel: If math is consistent, you can t prove that CH is false. (1962 Cohen: If math is consistent, you can t prove that CH is true.) Silver Blaze Straker dead from a savage blow. Inspector Gregory: Fitzroy Simpson definitely guilty, and we arrested him!
49 Fleshing Out a Bit 1900 Hilbert:... a very plausible theorem, which nevertheless, in spite of the most strenuous efforts, no one has succeeded in proving.... Continuum Hypothesis: There s no set of a size between the integers and the reals Gödel: If math is consistent, you can t prove that CH is false. (1962 Cohen: If math is consistent, you can t prove that CH is true.) Silver Blaze Straker dead from a savage blow. Inspector Gregory: Fitzroy Simpson definitely guilty, and we arrested him! Holmes: But I ve created a scenario in which all your clues/ evidence are true, but Fitzroy is perfectly innocent
50 Fleshing Out a Bit 1900 Hilbert:... a very plausible theorem, which nevertheless, in spite of the most strenuous efforts, no one has succeeded in proving.... Continuum Hypothesis: There s no set of a size between the integers and the reals Gödel: If math is consistent, you can t prove that CH is false. (1962 Cohen: If math is consistent, you can t prove that CH is true.) Silver Blaze Straker dead from a savage blow. Inspector Gregory: Fitzroy Simpson definitely guilty, and we arrested him! Holmes: But I ve created a scenario in which all your clues/ evidence are true, but Fitzroy is perfectly innocent Gödel: I ve created a scenario where all of ZFC is true, but so is the Continuum Hypothesis!
51 Fleshing Out a Bit 1900 Hilbert:... a very plausible theorem, which nevertheless, in spite of the most strenuous efforts, no one has succeeded in proving.... Continuum Hypothesis: There s no set of a size between the integers and the reals Gödel: If math is consistent, you can t prove that CH is false. (1962 Cohen: If math is consistent, you can t prove that CH is true.) Silver Blaze Straker dead from a savage blow. Inspector Gregory: Fitzroy Simpson definitely guilty, and we arrested him! Holmes: But I ve created a scenario in which all your clues/ evidence are true, but Fitzroy is perfectly innocent Gödel: I ve created a scenario where all of ZFC is true, but so is the Continuum Hypothesis!
52 Fleshing Out a Bit 1900 Hilbert:... a very plausible theorem, which nevertheless, in spite of the most strenuous efforts, no one has succeeded in proving.... Continuum Hypothesis: There s no set of a size between the integers and the reals Gödel: If math is consistent, you can t prove that CH is false. (1962 Cohen: If math is consistent, you can t prove that CH is true.) Silver Blaze Straker dead from a savage blow. Inspector Gregory: Fitzroy Simpson definitely guilty, and we arrested him! Holmes: But I ve created a scenario in which all your clues/ evidence are true, but Fitzroy is perfectly innocent Gödel: I ve created a scenario where all of ZFC is true, but so is the Continuum Hypothesis!
53 Facts F
54 Scenario G (in honor of Inspector Gregory)
55 Gödel s Proof Proof-Sketch: GCH is equivalent to the statement that there are at most c* subsets of c, where c is any cardinal number, and c* is the cardinal after c (i.e., P(c) c*). Gödel showed that there are at most c* constructible subsets of c; hence if all all sets are constructible, GCH follows. Gödel created a scenario in which all sets are constructible, and showed that this scenario is consistent with ZFC (assuming, as we are, that ZFC is consistent). Therefore GCH can never be disproved from ZFC! QED
56 Gödel s Proof Proof-Sketch: GCH is equivalent to the statement that there are at most c* subsets of c, where c is any cardinal number, and c* is the cardinal after c (i.e., P(c) c*). Gödel showed that there are at most c* constructible subsets of c; hence if all all sets are constructible, GCH follows. Gödel created a scenario in which all sets are constructible, and showed that this scenario is consistent with ZFC (assuming, as we are, that ZFC is consistent). Therefore GCH can never be disproved from ZFC! QED (Of course, in Silver Blaze, Sherlock did proceed to show that his scenario was true. The analogue to that doesn t hold in the the Gödel story.)
57 Main Claim
58 Main Claim
59 And Supporting Main Claim
60 Re Papers structure clarity writing cogency scholarship n1 n2 n3 n4 n5
61
62 slutten
Weronika Mysliwiec, klasa 8W, rok szkolny 2018/2019
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