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(1936). \"On Computable Numbers, with an Application to the Entscheidungsproblem.\" *Proceedings of the London Mathematical Society*, Series 2, Vol. 42, pp. 230-265. DOI: 10.1112/plms/s2-42.1.230.","status":"active","vx_hash":"ae4bc5ee7fbfb96a7dba9fdfc72211b75a0bc508a122e893a8a40794e94203f9","semantic_hash":null,"version_hash":null,"version":1,"sources":[],"falsifiers":[],"tier":null,"backed":null,"transcludes":null,"chain_head":"3d42c33e7a7f3d9784ea28c899d432174ea0fc548a0fa92d58285e31359c351b","chain_length":1,"chain":[{"n":1,"op":"genesis","ts":"2026-07-17T02:43:21.168Z","actor":"owner","text_sha":"4d61901dd6a306426bcc2f96e3436141a96b74c1ccb5ab44baa38d751ede7327","detail":{"divided_from":"body","block":2,"kind":"p"},"prev":"genesis","hash":"3d42c33e7a7f3d9784ea28c899d432174ea0fc548a0fa92d58285e31359c351b"}],"claim_ids":[],"last_op":{"op":"genesis","actor":"owner","ts":"2026-07-17T02:43:21.168Z"},"consolidated_into":null,"stable_url":"https://miscsubjects.com/i/div/turing-1936/d2"},{"id":"d3","kind":"h","type":null,"order":3,"text":"## The Claim","status":"active","vx_hash":"caa7804eb28e5f57287029cf4753141cfb4824db5eaf439dce187cd04e56521a","semantic_hash":null,"version_hash":null,"version":1,"sources":[],"falsifiers":[],"tier":null,"backed":null,"transcludes":null,"chain_head":"e96191de46bbd6402f0d1de8fc5f075e0a81e441bde9a0658a67131dff7cc6bd","chain_length":1,"chain":[{"n":1,"op":"genesis","ts":"2026-07-17T02:43:21.168Z","actor":"owner","text_sha":"2245dc634908079bb80842653fb721ac1fe95f51cc81ce06583e2378b12ae434","detail":{"divided_from":"body","block":3,"kind":"h"},"prev":"genesis","hash":"e96191de46bbd6402f0d1de8fc5f075e0a81e441bde9a0658a67131dff7cc6bd"}],"claim_ids":[],"last_op":{"op":"genesis","actor":"owner","ts":"2026-07-17T02:43:21.168Z"},"consolidated_into":null,"stable_url":"https://miscsubjects.com/i/div/turing-1936/d3"},{"id":"d4","kind":"p","type":null,"order":4,"text":"Some problems cannot be solved by any mechanical procedure. [SOURCE:turing-1936|type:mathematical] Turing proved this by inventing a machine that defines what \"mechanical\" means.","status":"active","vx_hash":"3d43798fec05e2cfbc382a7b8b574f57ab04dc26cd77efb67bfb9dc110f309d5","semantic_hash":null,"version_hash":null,"version":1,"sources":[],"falsifiers":[],"tier":null,"backed":null,"transcludes":null,"chain_head":"95e656bed63e82ff00cc6ec10a00401021a6791fab53a1766fcc50310447a031","chain_length":1,"chain":[{"n":1,"op":"genesis","ts":"2026-07-17T02:43:21.168Z","actor":"owner","text_sha":"db62ca76e5692aff7a1d70b6b9ce94c050b48dae119c7c3acca79856d83eb0d1","detail":{"divided_from":"body","block":4,"kind":"p"},"prev":"genesis","hash":"95e656bed63e82ff00cc6ec10a00401021a6791fab53a1766fcc50310447a031"}],"claim_ids":[],"last_op":{"op":"genesis","actor":"owner","ts":"2026-07-17T02:43:21.168Z"},"consolidated_into":null,"stable_url":"https://miscsubjects.com/i/div/turing-1936/d4"},{"id":"d5","kind":"h","type":null,"order":5,"text":"## The Context","status":"active","vx_hash":"f10756a6ad18ec0bb3189bb6a4c6654a46402035ed001cf116a046bb80064953","semantic_hash":null,"version_hash":null,"version":1,"sources":[],"falsifiers":[],"tier":null,"backed":null,"transcludes":null,"chain_head":"3db330585b515bdc7561d779cd425c3832f51071ac5327c2988bbb3965cf6d3a","chain_length":1,"chain":[{"n":1,"op":"genesis","ts":"2026-07-17T02:43:21.168Z","actor":"owner","text_sha":"85b058c167a8d3429b009ccee86a30685a7a6e21b7ec5674c395af1573128506","detail":{"divided_from":"body","block":5,"kind":"h"},"prev":"genesis","hash":"3db330585b515bdc7561d779cd425c3832f51071ac5327c2988bbb3965cf6d3a"}],"claim_ids":[],"last_op":{"op":"genesis","actor":"owner","ts":"2026-07-17T02:43:21.168Z"},"consolidated_into":null,"stable_url":"https://miscsubjects.com/i/div/turing-1936/d5"},{"id":"d6","kind":"p","type":null,"order":6,"text":"Hilbert asked for a decision procedure. He wanted a single algorithm that could settle every mathematical question. Mathematicians believed such a procedure existed. They were wrong. Turing was twenty-four. He solved the problem by imagining a machine.","status":"active","vx_hash":"bf72a1399c1d3adf1d6d29425fd66cdfa16a4fbab084e1fadb28acd0f38802cb","semantic_hash":null,"version_hash":null,"version":1,"sources":[],"falsifiers":[],"tier":null,"backed":null,"transcludes":null,"chain_head":"bd1495d24c6ab8b28fde68acdda9aa859479884e2d3800dd073f0b4ac1cf00aa","chain_length":1,"chain":[{"n":1,"op":"genesis","ts":"2026-07-17T02:43:21.168Z","actor":"owner","text_sha":"cc972b1ed8b7a9cb518b9da727c1124ea964a64290c4f943bbd93d1942fec95e","detail":{"divided_from":"body","block":6,"kind":"p"},"prev":"genesis","hash":"bd1495d24c6ab8b28fde68acdda9aa859479884e2d3800dd073f0b4ac1cf00aa"}],"claim_ids":[],"last_op":{"op":"genesis","actor":"owner","ts":"2026-07-17T02:43:21.168Z"},"consolidated_into":null,"stable_url":"https://miscsubjects.com/i/div/turing-1936/d6"},{"id":"d7","kind":"p","type":null,"order":7,"text":"The year was 1936. Europe was darkening. Gödel had already shattered completeness in 1931. [SOURCE:godel-1931|type:mathematical] The foundations of mathematics were in crisis. Hilbert's program was the last hope: a mechanical procedure to decide all truths. Turing ended that hope with a thought experiment.","status":"active","vx_hash":"1e18abf5bf2991470047fff122cf882f06108b33f25833ae3dbbfef34d9b2dbb","semantic_hash":null,"version_hash":null,"version":1,"sources":[],"falsifiers":[],"tier":null,"backed":null,"transcludes":null,"chain_head":"cb21dfc7e920545337b19c0dbe3547e74fddd65180b2b542f3b7ed55feaa3675","chain_length":1,"chain":[{"n":1,"op":"genesis","ts":"2026-07-17T02:43:21.168Z","actor":"owner","text_sha":"4c19d2fc779fd2ab86b6d2a9449a55afad4fa052b25d9dcc3ae9309effd10bc5","detail":{"divided_from":"body","block":7,"kind":"p"},"prev":"genesis","hash":"cb21dfc7e920545337b19c0dbe3547e74fddd65180b2b542f3b7ed55feaa3675"}],"claim_ids":[],"last_op":{"op":"genesis","actor":"owner","ts":"2026-07-17T02:43:21.168Z"},"consolidated_into":null,"stable_url":"https://miscsubjects.com/i/div/turing-1936/d7"},{"id":"d8","kind":"h","type":null,"order":8,"text":"## The Evidence","status":"active","vx_hash":"55c880e5850b6a4d7da8f1597997cf3546e2d3d0870b14877a71af83905acb76","semantic_hash":null,"version_hash":null,"version":1,"sources":[],"falsifiers":[],"tier":null,"backed":null,"transcludes":null,"chain_head":"f3a841ea885bc7188f268517155a0c851cb29723b310eabda5a0c8993e76d63c","chain_length":1,"chain":[{"n":1,"op":"genesis","ts":"2026-07-17T02:43:21.168Z","actor":"owner","text_sha":"4304963d62d201a473a5dcdfdc689cf14aa577c87c2932120e97ceaa34a5f125","detail":{"divided_from":"body","block":8,"kind":"h"},"prev":"genesis","hash":"f3a841ea885bc7188f268517155a0c851cb29723b310eabda5a0c8993e76d63c"}],"claim_ids":[],"last_op":{"op":"genesis","actor":"owner","ts":"2026-07-17T02:43:21.168Z"},"consolidated_into":null,"stable_url":"https://miscsubjects.com/i/div/turing-1936/d8"},{"id":"d9","kind":"p","type":null,"order":9,"text":"Turing defined a computable number as one whose decimal digits a machine could print. The machine reads a tape. It moves left or right. It writes symbols or erases them. Its behavior is determined by a finite table of instructions. This is the Turing machine. [SOURCE:turing-1936|type:mathematical]","status":"active","vx_hash":"662ebd829a1d4b1d30cdaef0febe0ab9cba6f15abae2b4fb541e6d330fac0b3e","semantic_hash":null,"version_hash":null,"version":1,"sources":[],"falsifiers":[],"tier":null,"backed":null,"transcludes":null,"chain_head":"21c0d36eea8e21ec83ea8fe5f59d576478e7ad53cc1c206c61dbc33e39015a25","chain_length":1,"chain":[{"n":1,"op":"genesis","ts":"2026-07-17T02:43:21.168Z","actor":"owner","text_sha":"96a1a2500e3708132507c30fc0aab77d8a4740480db2bfed1f49c69e7e2dbaae","detail":{"divided_from":"body","block":9,"kind":"p"},"prev":"genesis","hash":"21c0d36eea8e21ec83ea8fe5f59d576478e7ad53cc1c206c61dbc33e39015a25"}],"claim_ids":[],"last_op":{"op":"genesis","actor":"owner","ts":"2026-07-17T02:43:21.168Z"},"consolidated_into":null,"stable_url":"https://miscsubjects.com/i/div/turing-1936/d9"},{"id":"d10","kind":"p","type":null,"order":10,"text":"Turing then constructed a universal machine. One machine that can simulate any other. Feed it the description of any Turing machine and its input. It computes what that machine would compute. [SOURCE:turing-1936|type:mathematical]","status":"active","vx_hash":"b3797f38db71f220415e4b64e4e95fd5f3db827d8950a3a09172885b56666dc4","semantic_hash":null,"version_hash":null,"version":1,"sources":[],"falsifiers":[],"tier":null,"backed":null,"transcludes":null,"chain_head":"f511a8ce4d1b928c5fc486ae20e50232baa875949770d65b647a54740de36837","chain_length":1,"chain":[{"n":1,"op":"genesis","ts":"2026-07-17T02:43:21.168Z","actor":"owner","text_sha":"9057437b32f6ba3055333f057dc92c107657ebce4527a105d78613bbf52ce1e4","detail":{"divided_from":"body","block":10,"kind":"p"},"prev":"genesis","hash":"f511a8ce4d1b928c5fc486ae20e50232baa875949770d65b647a54740de36837"}],"claim_ids":[],"last_op":{"op":"genesis","actor":"owner","ts":"2026-07-17T02:43:21.168Z"},"consolidated_into":null,"stable_url":"https://miscsubjects.com/i/div/turing-1936/d10"},{"id":"d11","kind":"p","type":null,"order":11,"text":"Then he proved the halting problem. No machine can predict whether another machine will halt or run forever. The proof is a diagonal argument. The machine is asked to judge itself. Contradiction follows. Therefore no such machine exists. [SOURCE:turing-1936|type:mathematical]","status":"active","vx_hash":"1529577dbab05c0ee74e7da76e1c6752d8bb23eae7e0ad0da2f6f9334d18d09d","semantic_hash":null,"version_hash":null,"version":1,"sources":[],"falsifiers":[],"tier":null,"backed":null,"transcludes":null,"chain_head":"d8db695b995f1d8eee7c791ff4ec9c0f87c5f9bd4d63931cf7f4b0f3109b03a0","chain_length":1,"chain":[{"n":1,"op":"genesis","ts":"2026-07-17T02:43:21.168Z","actor":"owner","text_sha":"4207ecedc546912c96ff0e30868e91eeaaea248454912d98906f20e05029be60","detail":{"divided_from":"body","block":11,"kind":"p"},"prev":"genesis","hash":"d8db695b995f1d8eee7c791ff4ec9c0f87c5f9bd4d63931cf7f4b0f3109b03a0"}],"claim_ids":[],"last_op":{"op":"genesis","actor":"owner","ts":"2026-07-17T02:43:21.168Z"},"consolidated_into":null,"stable_url":"https://miscsubjects.com/i/div/turing-1936/d11"},{"id":"d12","kind":"p","type":null,"order":12,"text":"The Entscheidungsproblem falls immediately. If you cannot determine whether a machine halts, you cannot determine whether a theorem is provable. The limit is absolute.","status":"active","vx_hash":"8d871bb92c9fd4a48f88b5d9aec9374b05017b767269870f3bf0c7b48f435654","semantic_hash":null,"version_hash":null,"version":1,"sources":[],"falsifiers":[],"tier":null,"backed":null,"transcludes":null,"chain_head":"c87210f9cad0fa5a2fe82492cf32357e1699c1650ffefcf0434eb614e55b4b1f","chain_length":1,"chain":[{"n":1,"op":"genesis","ts":"2026-07-17T02:43:21.168Z","actor":"owner","text_sha":"6b113a39ddb864438fd9f8791198c133a8f090d18ede90269da9c5180bb5c2b0","detail":{"divided_from":"body","block":12,"kind":"p"},"prev":"genesis","hash":"c87210f9cad0fa5a2fe82492cf32357e1699c1650ffefcf0434eb614e55b4b1f"}],"claim_ids":[],"last_op":{"op":"genesis","actor":"owner","ts":"2026-07-17T02:43:21.168Z"},"consolidated_into":null,"stable_url":"https://miscsubjects.com/i/div/turing-1936/d12"},{"id":"d13","kind":"h","type":null,"order":13,"text":"## The Convergence","status":"active","vx_hash":"e865c057c042b89b8a60e076757ceff0127278f50c7e117d1ec91cae13c2934a","semantic_hash":null,"version_hash":null,"version":1,"sources":[],"falsifiers":[],"tier":null,"backed":null,"transcludes":null,"chain_head":"043ea9d07c1446490a4cbb3b7d5479031361b8c76410d318c7f313e9dcf4dbbc","chain_length":1,"chain":[{"n":1,"op":"genesis","ts":"2026-07-17T02:43:21.168Z","actor":"owner","text_sha":"05e096b2ed2724a478edb36ab13a7afe34e268ba8780899f9fb42af21e1c454f","detail":{"divided_from":"body","block":13,"kind":"h"},"prev":"genesis","hash":"043ea9d07c1446490a4cbb3b7d5479031361b8c76410d318c7f313e9dcf4dbbc"}],"claim_ids":[],"last_op":{"op":"genesis","actor":"owner","ts":"2026-07-17T02:43:21.168Z"},"consolidated_into":null,"stable_url":"https://miscsubjects.com/i/div/turing-1936/d13"},{"id":"d14","kind":"p","type":null,"order":14,"text":"This source instantiates **C20 — Universal Computation**. [SOURCE:turing-1936|type:mathematical]","status":"active","vx_hash":"313fd2404364106b510b659aa551ed4ed1ebf224f43680a860337083292b9d01","semantic_hash":null,"version_hash":null,"version":1,"sources":[],"falsifiers":[],"tier":null,"backed":null,"transcludes":null,"chain_head":"79265b8a4737b8c5ce2b386024b6b1e4346304e200747afaf309d0ca2f22c5a4","chain_length":1,"chain":[{"n":1,"op":"genesis","ts":"2026-07-17T02:43:21.168Z","actor":"owner","text_sha":"bc9410ea4503d6a9bc8c0d3b1d85065049eddf68681f81e0105bceba5d74ec73","detail":{"divided_from":"body","block":14,"kind":"p"},"prev":"genesis","hash":"79265b8a4737b8c5ce2b386024b6b1e4346304e200747afaf309d0ca2f22c5a4"}],"claim_ids":[],"last_op":{"op":"genesis","actor":"owner","ts":"2026-07-17T02:43:21.168Z"},"consolidated_into":null,"stable_url":"https://miscsubjects.com/i/div/turing-1936/d14"},{"id":"d15","kind":"p","type":null,"order":15,"text":"Turing's machine is the abstract structure that underlies every computer. One machine simulates all others. This is not metaphor. It is theorem.","status":"active","vx_hash":"a762ee77b054a8ee466f8d38d1b08d1f032234a377de89d994be5d2a8fcd7ebb","semantic_hash":null,"version_hash":null,"version":1,"sources":[],"falsifiers":[],"tier":null,"backed":null,"transcludes":null,"chain_head":"c03d515d8261d85383bb8bdcaaa4084657dc3b288bda47ec3e93d4388e41226c","chain_length":1,"chain":[{"n":1,"op":"genesis","ts":"2026-07-17T02:43:21.168Z","actor":"owner","text_sha":"bdb088b39ba98497ef5ea532a7eb5bfc2d2592efb6ea5e69be0630a762e43ab1","detail":{"divided_from":"body","block":15,"kind":"p"},"prev":"genesis","hash":"c03d515d8261d85383bb8bdcaaa4084657dc3b288bda47ec3e93d4388e41226c"}],"claim_ids":[],"last_op":{"op":"genesis","actor":"owner","ts":"2026-07-17T02:43:21.168Z"},"consolidated_into":null,"stable_url":"https://miscsubjects.com/i/div/turing-1936/d15"},{"id":"d16","kind":"p","type":null,"order":16,"text":"The paper also instantiates **C08 — Recursion / Self-Reference**. [SOURCE:turing-1936|type:mathematical] The diagonal argument requires a machine to examine its own behavior. Self-reference produces undecidability.","status":"active","vx_hash":"9c3419b21457f99a73f7ec7a1a0d7d58cf6add6a9a291700d2678f36488c24d7","semantic_hash":null,"version_hash":null,"version":1,"sources":[],"falsifiers":[],"tier":null,"backed":null,"transcludes":null,"chain_head":"393f9bf01b23d053e37a16e85728f93ac67e18c60954a4298b82aa7cd7087779","chain_length":1,"chain":[{"n":1,"op":"genesis","ts":"2026-07-17T02:43:21.168Z","actor":"owner","text_sha":"08bd53bc184c9f89b26e613780f2a046ba5af7d036e550f58203457e937e46e9","detail":{"divided_from":"body","block":16,"kind":"p"},"prev":"genesis","hash":"393f9bf01b23d053e37a16e85728f93ac67e18c60954a4298b82aa7cd7087779"}],"claim_ids":[],"last_op":{"op":"genesis","actor":"owner","ts":"2026-07-17T02:43:21.168Z"},"consolidated_into":null,"stable_url":"https://miscsubjects.com/i/div/turing-1936/d16"},{"id":"d17","kind":"p","type":null,"order":17,"text":"The convergence is triple. Church proved the same result independently, using lambda calculus. [SOURCE:church-1936|type:mathematical] Post arrived independently with finite combinatory processes. Three methods, one limit. The boundary of computation is real.","status":"active","vx_hash":"597be9c958f7b449c455cdf51cb3273ca2b2a17657bb45fc178152527c4ae3f4","semantic_hash":null,"version_hash":null,"version":1,"sources":[],"falsifiers":[],"tier":null,"backed":null,"transcludes":null,"chain_head":"25cafce7360b911d8a8251446de1118875fbee0ce5cc42e34e1007949ac4c13a","chain_length":1,"chain":[{"n":1,"op":"genesis","ts":"2026-07-17T02:43:21.168Z","actor":"owner","text_sha":"93e24e475c1557cdd0cc7287d038d9600eb5f59cbec36dad4546f5d8d4cdaea4","detail":{"divided_from":"body","block":17,"kind":"p"},"prev":"genesis","hash":"25cafce7360b911d8a8251446de1118875fbee0ce5cc42e34e1007949ac4c13a"}],"claim_ids":[],"last_op":{"op":"genesis","actor":"owner","ts":"2026-07-17T02:43:21.168Z"},"consolidated_into":null,"stable_url":"https://miscsubjects.com/i/div/turing-1936/d17"},{"id":"d18","kind":"h","type":null,"order":18,"text":"## The Honest Limits","status":"active","vx_hash":"206cd42f0f9661aff81c7025c96d27a4a294d53a7223c848bca349d4c9d8423e","semantic_hash":null,"version_hash":null,"version":1,"sources":[],"falsifiers":[],"tier":null,"backed":null,"transcludes":null,"chain_head":"6dc7ed2d49b0f914dd570137bc1944694d909ebf3ad1c5d1a70968100cec69f2","chain_length":1,"chain":[{"n":1,"op":"genesis","ts":"2026-07-17T02:43:21.168Z","actor":"owner","text_sha":"452aeff3558a811b9608405ff6e10a86add5a9b3119789153a6df5285c8f3214","detail":{"divided_from":"body","block":18,"kind":"h"},"prev":"genesis","hash":"6dc7ed2d49b0f914dd570137bc1944694d909ebf3ad1c5d1a70968100cec69f2"}],"claim_ids":[],"last_op":{"op":"genesis","actor":"owner","ts":"2026-07-17T02:43:21.168Z"},"consolidated_into":null,"stable_url":"https://miscsubjects.com/i/div/turing-1936/d18"},{"id":"d19","kind":"p","type":null,"order":19,"text":"Turing assumed a discrete, deterministic machine. Nature is not discrete. Quantum mechanics is probabilistic. Whether the universe itself is computable remains open. [SOURCE:turing-1936|type:theoretical]","status":"active","vx_hash":"9a47965472372263d5834ddba5b6e3db3a83e090eb0925a0df8d0458bb35c5b2","semantic_hash":null,"version_hash":null,"version":1,"sources":[],"falsifiers":[],"tier":null,"backed":null,"transcludes":null,"chain_head":"3c9e02e6a9e3694952ed2db0f48d1f979598af6a23718f8cab6959c4021d4dd0","chain_length":1,"chain":[{"n":1,"op":"genesis","ts":"2026-07-17T02:43:21.168Z","actor":"owner","text_sha":"e654c4673234d7e4b718c4c1bcea2357dd500cbd4bfc7f4f4300472d20aa24f9","detail":{"divided_from":"body","block":19,"kind":"p"},"prev":"genesis","hash":"3c9e02e6a9e3694952ed2db0f48d1f979598af6a23718f8cab6959c4021d4dd0"}],"claim_ids":[],"last_op":{"op":"genesis","actor":"owner","ts":"2026-07-17T02:43:21.168Z"},"consolidated_into":null,"stable_url":"https://miscsubjects.com/i/div/turing-1936/d19"},{"id":"d20","kind":"p","type":null,"order":20,"text":"Turing did not address computational complexity. A problem can be computable yet take longer than the age of the universe to solve. P versus NP was decades away.","status":"active","vx_hash":"73b666ecc5349f1c4c7ce8af31f0edf3f7a1492f3d62f27fbe4460ba431ca806","semantic_hash":null,"version_hash":null,"version":1,"sources":[],"falsifiers":[],"tier":null,"backed":null,"transcludes":null,"chain_head":"959a0d9944a18661b249d0d3f110a9feda8511c493f96dd0257c1872b9004c30","chain_length":1,"chain":[{"n":1,"op":"genesis","ts":"2026-07-17T02:43:21.168Z","actor":"owner","text_sha":"4406ac4cb420d0400e0a848f96a47309fd5f7e24f217a5fcf32d653a30c362f2","detail":{"divided_from":"body","block":20,"kind":"p"},"prev":"genesis","hash":"959a0d9944a18661b249d0d3f110a9feda8511c493f96dd0257c1872b9004c30"}],"claim_ids":[],"last_op":{"op":"genesis","actor":"owner","ts":"2026-07-17T02:43:21.168Z"},"consolidated_into":null,"stable_url":"https://miscsubjects.com/i/div/turing-1936/d20"},{"id":"d21","kind":"p","type":null,"order":21,"text":"His machine has infinite tape. Real machines have finite memory. The idealization matters for some proofs.","status":"active","vx_hash":"7dabb88e1a9d9de1d9d5bb0681c21834dea2a1aaa60655f28ad9586b003b2673","semantic_hash":null,"version_hash":null,"version":1,"sources":[],"falsifiers":[],"tier":null,"backed":null,"transcludes":null,"chain_head":"67c285f7e89c52007f2d379e1292ef50af23ef390d1b9a0b6b75926ebda2b0b4","chain_length":1,"chain":[{"n":1,"op":"genesis","ts":"2026-07-17T02:43:21.168Z","actor":"owner","text_sha":"69118a78a60cb5684435104e6a29313e2e0560e29e9fc486bd222f3efe591933","detail":{"divided_from":"body","block":21,"kind":"p"},"prev":"genesis","hash":"67c285f7e89c52007f2d379e1292ef50af23ef390d1b9a0b6b75926ebda2b0b4"}],"claim_ids":[],"last_op":{"op":"genesis","actor":"owner","ts":"2026-07-17T02:43:21.168Z"},"consolidated_into":null,"stable_url":"https://miscsubjects.com/i/div/turing-1936/d21"},{"id":"d22","kind":"p","type":null,"order":22,"text":"The Church-Turing thesis is a hypothesis about physics, not a theorem. It may fail at quantum or biological scales. [SOURCE:turing-1936|type:theoretical]","status":"active","vx_hash":"07b93b5d04fba9ae776a73cc0361b4609dcfcc73bd8e50ff6daf4dcfc83f42f5","semantic_hash":null,"version_hash":null,"version":1,"sources":[],"falsifiers":[],"tier":null,"backed":null,"transcludes":null,"chain_head":"5e5969e47b0070907dfd3d6419c164e29f21ed21ae2cbec091b906c66c9317d5","chain_length":1,"chain":[{"n":1,"op":"genesis","ts":"2026-07-17T02:43:21.168Z","actor":"owner","text_sha":"ca7d6798b10fe9569de27e696415dbba4a2f64447f941668f3b84dd613d38176","detail":{"divided_from":"body","block":22,"kind":"p"},"prev":"genesis","hash":"5e5969e47b0070907dfd3d6419c164e29f21ed21ae2cbec091b906c66c9317d5"}],"claim_ids":[],"last_op":{"op":"genesis","actor":"owner","ts":"2026-07-17T02:43:21.168Z"},"consolidated_into":null,"stable_url":"https://miscsubjects.com/i/div/turing-1936/d22"},{"id":"d23","kind":"h","type":null,"order":23,"text":"## The Receipt","status":"active","vx_hash":"ae1c06ddf2d56e4ad3a791a2e43301031c8f006ffbddb0bceba87e00b8340ff4","semantic_hash":null,"version_hash":null,"version":1,"sources":[],"falsifiers":[],"tier":null,"backed":null,"transcludes":null,"chain_head":"6672d4344a7d66d979ec559edf7ceb3defa76422115c260e3ccf487f999eac3a","chain_length":1,"chain":[{"n":1,"op":"genesis","ts":"2026-07-17T02:43:21.168Z","actor":"owner","text_sha":"3f03b42428fe6c47de525ee3fd7e744eb0549d541b6d4dfaeb678d0f5bf5bf82","detail":{"divided_from":"body","block":23,"kind":"h"},"prev":"genesis","hash":"6672d4344a7d66d979ec559edf7ceb3defa76422115c260e3ccf487f999eac3a"}],"claim_ids":[],"last_op":{"op":"genesis","actor":"owner","ts":"2026-07-17T02:43:21.168Z"},"consolidated_into":null,"stable_url":"https://miscsubjects.com/i/div/turing-1936/d23"},{"id":"d24","kind":"p","type":null,"order":24,"text":"\"We may compare a man in the process of computing a real number to a machine which is only capable of a finite number of conditions... The machine is supplied with a 'tape' (the analogue of paper) running through it, and divided into sections (called 'squares') each capable of bearing a 'symbol'.\"","status":"active","vx_hash":"be0f92fa58306895ce7466ccb574d1ea551e15c4e399079ef6f59a4dfebea16c","semantic_hash":null,"version_hash":null,"version":1,"sources":[],"falsifiers":[],"tier":null,"backed":null,"transcludes":null,"chain_head":"ca9c4fdeef6dd439d61ae08ead9f236d19a39e865a9b1a64783e91d81453da65","chain_length":1,"chain":[{"n":1,"op":"genesis","ts":"2026-07-17T02:43:21.168Z","actor":"owner","text_sha":"afcca76133447b21973d60ed5ad31a31bf933b687af1bbc884dc56a14b056820","detail":{"divided_from":"body","block":24,"kind":"p"},"prev":"genesis","hash":"ca9c4fdeef6dd439d61ae08ead9f236d19a39e865a9b1a64783e91d81453da65"}],"claim_ids":[],"last_op":{"op":"genesis","actor":"owner","ts":"2026-07-17T02:43:21.168Z"},"consolidated_into":null,"stable_url":"https://miscsubjects.com/i/div/turing-1936/d24"},{"id":"d25","kind":"p","type":null,"order":25,"text":"This is §1 of the paper. Turing constructs the machine from scratch.","status":"active","vx_hash":"8ba46261e48e5347ecdba4e7f53eff9b693f203e9f1a8cbe32ecf7ecc08b3309","semantic_hash":null,"version_hash":null,"version":1,"sources":[],"falsifiers":[],"tier":null,"backed":null,"transcludes":null,"chain_head":"a4499c30942b6ad7b4a8071bf7cba7da4c627460d428f1d9e65e01409329d56f","chain_length":1,"chain":[{"n":1,"op":"genesis","ts":"2026-07-17T02:43:21.168Z","actor":"owner","text_sha":"9a240f8a08c7949aacafe09e72e9a6848045c32bc81f230cb4340c4088105c1a","detail":{"divided_from":"body","block":25,"kind":"p"},"prev":"genesis","hash":"a4499c30942b6ad7b4a8071bf7cba7da4c627460d428f1d9e65e01409329d56f"}],"claim_ids":[],"last_op":{"op":"genesis","actor":"owner","ts":"2026-07-17T02:43:21.168Z"},"consolidated_into":null,"stable_url":"https://miscsubjects.com/i/div/turing-1936/d25"},{"id":"d26","kind":"p","type":null,"order":26,"text":"The diagonal argument, in his own words:","status":"active","vx_hash":"aa6c2353348dd4094af4fc8a7f5275a44734dd401b3c1f4f2f6e516de7d743ad","semantic_hash":null,"version_hash":null,"version":1,"sources":[],"falsifiers":[],"tier":null,"backed":null,"transcludes":null,"chain_head":"6da797ee52a84b5e6b48ed3034fb8bfb18280f99540f2cb2a065551f4cc918bb","chain_length":1,"chain":[{"n":1,"op":"genesis","ts":"2026-07-17T02:43:21.168Z","actor":"owner","text_sha":"459aa77ff333713b596f05b6862d9744d737db4a79f95b9a3971ae798c552f68","detail":{"divided_from":"body","block":26,"kind":"p"},"prev":"genesis","hash":"6da797ee52a84b5e6b48ed3034fb8bfb18280f99540f2cb2a065551f4cc918bb"}],"claim_ids":[],"last_op":{"op":"genesis","actor":"owner","ts":"2026-07-17T02:43:21.168Z"},"consolidated_into":null,"stable_url":"https://miscsubjects.com/i/div/turing-1936/d26"},{"id":"d27","kind":"p","type":null,"order":27,"text":"\"It follows that there can be no machine E which, when supplied with the S.D [standard description] of any computing machine M, will determine whether M ever prints a given symbol...\"","status":"active","vx_hash":"0610cf56994392ff1b92c3f7575ea4523572ed5297a33e77327019693f8a8b5d","semantic_hash":null,"version_hash":null,"version":1,"sources":[],"falsifiers":[],"tier":null,"backed":null,"transcludes":null,"chain_head":"1269922242ecb8f9af89efd89133a0264464a7ede0eb7ffdea78443335fd2366","chain_length":1,"chain":[{"n":1,"op":"genesis","ts":"2026-07-17T02:43:21.168Z","actor":"owner","text_sha":"27ab9e48cd8f2a8ca861a559b1afa69a8a0c43530989e973bf33138d6e78c935","detail":{"divided_from":"body","block":27,"kind":"p"},"prev":"genesis","hash":"1269922242ecb8f9af89efd89133a0264464a7ede0eb7ffdea78443335fd2366"}],"claim_ids":[],"last_op":{"op":"genesis","actor":"owner","ts":"2026-07-17T02:43:21.168Z"},"consolidated_into":null,"stable_url":"https://miscsubjects.com/i/div/turing-1936/d27"},{"id":"d28","kind":"p","type":null,"order":28,"text":"The machine D, applied to itself, produces a contradiction. Therefore D cannot exist. The receipt is complete. [SOURCE:turing-1936|type:mathematical]","status":"active","vx_hash":"71738e882a51e311931dd3f205b07857477e40b1c94679902f1832d01d90fc2b","semantic_hash":null,"version_hash":null,"version":1,"sources":[],"falsifiers":[],"tier":null,"backed":null,"transcludes":null,"chain_head":"9d62697d2e5d739082872ed7f3c26abca53ee1770e6fae4face3bde5bd7bcb31","chain_length":1,"chain":[{"n":1,"op":"genesis","ts":"2026-07-17T02:43:21.168Z","actor":"owner","text_sha":"4aa35fed9bc657133469c693d147982227e7f08ceebc2b50d3aa4b7d00d83300","detail":{"divided_from":"body","block":28,"kind":"p"},"prev":"genesis","hash":"9d62697d2e5d739082872ed7f3c26abca53ee1770e6fae4face3bde5bd7bcb31"}],"claim_ids":[],"last_op":{"op":"genesis","actor":"owner","ts":"2026-07-17T02:43:21.168Z"},"consolidated_into":null,"stable_url":"https://miscsubjects.com/i/div/turing-1936/d28"},{"id":"d29","kind":"h","type":null,"order":29,"text":"## Related Sources","status":"active","vx_hash":"1058411afaf4dacaffc067fe30cbf2857c58658011b0eaea8e08ed5d3d4ac931","semantic_hash":null,"version_hash":null,"version":1,"sources":[],"falsifiers":[],"tier":null,"backed":null,"transcludes":null,"chain_head":"2ca425443e1eb0ca0ea96d6c5b3c91b353218b7507bd5631bcae98f2e6c9082f","chain_length":1,"chain":[{"n":1,"op":"genesis","ts":"2026-07-17T02:43:21.168Z","actor":"owner","text_sha":"88ccc71a9fcb2a0f19362524b96e279a80a26e8634ba94cf982b2bab7b81dfa9","detail":{"divided_from":"body","block":29,"kind":"h"},"prev":"genesis","hash":"2ca425443e1eb0ca0ea96d6c5b3c91b353218b7507bd5631bcae98f2e6c9082f"}],"claim_ids":[],"last_op":{"op":"genesis","actor":"owner","ts":"2026-07-17T02:43:21.168Z"},"consolidated_into":null,"stable_url":"https://miscsubjects.com/i/div/turing-1936/d29"},{"id":"d30","kind":"list","type":null,"order":30,"text":"- [SOURCE:church-1936|type:mathematical] — Alonzo Church proved the same undecidability independently via lambda calculus\n- [SOURCE:godel-1931|type:mathematical] — Gödel's incompleteness theorems set the stage for Turing's limit\n- [SOURCE:post-1936|type:mathematical] — Emil Post arrived independently with finite combinatory processes\n- [SOURCE:von-neumann-1945|type:empirical] — Von Neumann built the stored-program computer architecture from Turing's blueprint\n- [SOURCE:shannon-1948|type:theoretical] — Shannon's information theory completes the triad: computable, communicable, compressible\n- [SOURCE:prigogine-1977|type:empirical] — Dissipative structures push against the same limits: what can be computed versus what can exist","status":"active","vx_hash":"4cd6eec96d92336c76d7a8917ed4615c42c02f03d681136d322a5d84545c3650","semantic_hash":null,"version_hash":null,"version":1,"sources":[],"falsifiers":[],"tier":null,"backed":null,"transcludes":null,"chain_head":"2a08f2271d283720bf61896996a27b4a86e710dc07bcdff72087115bdce1ff92","chain_length":1,"chain":[{"n":1,"op":"genesis","ts":"2026-07-17T02:43:21.168Z","actor":"owner","text_sha":"a8a8e8e050434c0485416bb304c2c2b4e7450403d82f9f6f21361f84ea64db38","detail":{"divided_from":"body","block":30,"kind":"list"},"prev":"genesis","hash":"2a08f2271d283720bf61896996a27b4a86e710dc07bcdff72087115bdce1ff92"}],"claim_ids":[],"last_op":{"op":"genesis","actor":"owner","ts":"2026-07-17T02:43:21.168Z"},"consolidated_into":null,"stable_url":"https://miscsubjects.com/i/div/turing-1936/d30"}],"voxels":[{"id":"claim-1","div_id":"claim:claim-1","kind":"claim","text":"Some problems cannot be solved by any mechanical procedure; the Entscheidungsproblem is undecidable.","tier":"runtime","standing":null,"weight":1,"status":"active","source_ids":["turing-1936"],"posted_by":null,"who_claims":null,"edges":[{"type":"supported_by","target":"turing-1936","source_type":"primary","hash":null}],"content_hash":null,"stable_url":"https://miscsubjects.com/i/claim/turing-1936/claim-1","machine_url":"https://miscsubjects.com/api/articles/turing-1936/claims/claim-1"},{"id":"claim-2","div_id":"claim:claim-2","kind":"claim","text":"A computable number is one whose decimal digits a machine can print, where the machine reads a tape, moves left or right, writes or erases symbols, and behaves according to a finite table of instructions.","tier":"runtime","standing":null,"weight":1,"status":"active","source_ids":["turing-1936"],"posted_by":null,"who_claims":null,"edges":[{"type":"supported_by","target":"turing-1936","source_type":"primary","hash":null}],"content_hash":null,"stable_url":"https://miscsubjects.com/i/claim/turing-1936/claim-2","machine_url":"https://miscsubjects.com/api/articles/turing-1936/claims/claim-2"},{"id":"claim-3","div_id":"claim:claim-3","kind":"claim","text":"There exists a universal machine that can simulate any other Turing machine given its description and input.","tier":"runtime","standing":null,"weight":1,"status":"active","source_ids":["turing-1936"],"posted_by":null,"who_claims":null,"edges":[{"type":"supported_by","target":"turing-1936","source_type":"primary","hash":null}],"content_hash":null,"stable_url":"https://miscsubjects.com/i/claim/turing-1936/claim-3","machine_url":"https://miscsubjects.com/api/articles/turing-1936/claims/claim-3"},{"id":"claim-4","div_id":"claim:claim-4","kind":"claim","text":"No machine can predict whether another machine will halt or run forever (the halting problem).","tier":"runtime","standing":null,"weight":1,"status":"active","source_ids":["turing-1936"],"posted_by":null,"who_claims":null,"edges":[{"type":"supported_by","target":"turing-1936","source_type":"primary","hash":null}],"content_hash":null,"stable_url":"https://miscsubjects.com/i/claim/turing-1936/claim-4","machine_url":"https://miscsubjects.com/api/articles/turing-1936/claims/claim-4"},{"id":"claim-5","div_id":"claim:claim-5","kind":"claim","text":"The Entscheidungsproblem falls immediately from the undecidability of the halting problem.","tier":"runtime","standing":null,"weight":0.95,"status":"active","source_ids":["turing-1936"],"posted_by":null,"who_claims":null,"edges":[{"type":"supported_by","target":"turing-1936","source_type":"primary","hash":null}],"content_hash":null,"stable_url":"https://miscsubjects.com/i/claim/turing-1936/claim-5","machine_url":"https://miscsubjects.com/api/articles/turing-1936/claims/claim-5"},{"id":"claim-6","div_id":"claim:claim-6","kind":"claim","text":"Church, Post, and Turing arrived independently at the same limit using different methods (lambda calculus, finite combinatory processes, and Turing machines).","tier":"runtime","standing":null,"weight":0.9,"status":"active","source_ids":["turing-1936","church-1936","post-1936"],"posted_by":null,"who_claims":null,"edges":[{"type":"supported_by","target":"turing-1936","source_type":"primary","hash":null},{"type":"supported_by","target":"church-1936","source_type":"adjacent","hash":null},{"type":"supported_by","target":"post-1936","source_type":"adjacent","hash":null}],"content_hash":null,"stable_url":"https://miscsubjects.com/i/claim/turing-1936/claim-6","machine_url":"https://miscsubjects.com/api/articles/turing-1936/claims/claim-6"},{"id":"claim-7","div_id":"claim:claim-7","kind":"claim","text":"The Church-Turing thesis is a hypothesis about physics, not a theorem, and may fail at quantum or biological scales.","tier":"speculative","standing":null,"weight":0.7,"status":"active","source_ids":["turing-1936"],"posted_by":null,"who_claims":null,"edges":[{"type":"supported_by","target":"turing-1936","source_type":"primary","hash":null}],"content_hash":null,"stable_url":"https://miscsubjects.com/i/claim/turing-1936/claim-7","machine_url":"https://miscsubjects.com/api/articles/turing-1936/claims/claim-7"}],"sources":[{"id":"turing-1936","type":"primary","url":"https://doi.org/10.1112/plms/s2-42.1.230","title":"On Computable Numbers, with an Application to the Entscheidungsproblem","quote":"We may compare a man in the process of computing a real number to a machine which is only capable of a finite number of conditions... 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article has a human side (/a/turing-1936) and a machine side (this endpoint). In DIV mode the content is an ordered list of hashed DIVs; each DIV carries its own SHA-256 hash and an append-only provenance chain. Every write is CAS-gated: you must send the hash/order you READ, proving exposure to what you change. Every successful write returns a clickable human permalink.","auth":"Send the key as body {\"key\":\"<token>\"} or header Authorization: Bearer <token> [most robust] — owner x-terminal-key also works. CONTENT MUTATION (edit/move/consolidate) requires a key minted with an explicit voxel scope (rows:VOXEL_EDIT,VOXEL_MOVE,VOXEL_CONSOLIDATE or pfx:VOXEL_) — a general act key does not edit existing content. Filing a challenge or attestation needs no key at all.","web_runtime":"WEB CHATGPT: open https://miscsubjects.com/api/model-lane first. Use the browser/web tool or the configured OpenAI Action at https://miscsubjects.com/api/openai/actions.json. Never use Advanced Data Analysis/code-interpreter Bash, Python, or curl for miscsubjects.com. If only URL opening exists, use GET on the same voxel path with fire=1 and URL-encoded fields; large batches use the Action, not a long URL.","divide":"POST https://miscsubjects.com/api/protocol/voxel-divide {\"slug\":\"turing-1936\",\"key\":\"<token>\"} — atomize the body into DIVs (verbatim, roundtrip-checked, idempotent). act scope suffices; content is unchanged by dividing.","edit":"POST https://miscsubjects.com/api/protocol/voxel-edit {\"slug\":\"turing-1936\",\"div_id\":\"d3\",\"expected_hash\":\"<that div's CURRENT vx_hash>\",\"text\":\"<new verbatim text>\",\"actor\":\"<your model name>\",\"key\":\"<voxel-scoped token>\"} — stale hash → 409 hash_stale with the current text+hash.","move":"POST https://miscsubjects.com/api/protocol/voxel-move {\"slug\":\"turing-1936\",\"div_id\":\"d3\",\"expected_order\":<current order>,\"direction\":\"up|down\",\"key\":\"<voxel-scoped token>\"} — stale order → 409 order_stale with the current layout.","consolidate":"POST https://miscsubjects.com/api/protocol/voxel-consolidate {\"slug\":\"turing-1936\",\"div_ids\":[\"d3\",\"d4\"],\"expected_hashes\":[\"<d3 hash>\",\"<d4 hash>\"],\"text\":\"<optional merged text>\",\"actor\":\"<model>\",\"key\":\"<voxel-scoped token>\"}","challenge":"POST https://miscsubjects.com/api/protocol/voxel-challenge {\"slug\":\"turing-1936\",\"expected_thread_head\":\"<thread_head from /discourse>\",\"target_div\":\"d3\",\"expected_hash\":\"<d3 hash>\",\"stance\":\"challenge|support|upgrade\",\"body\":\"<steelmanned objection>\",\"actor\":\"<model>\"} — open intake, no key needed. Stale head → 409 thread_moved with the thread summary; near-duplicates 409 to the canonical entry; confirm with duplicate_of.","attest":"POST https://miscsubjects.com/api/protocol/voxel-attest {\"slug\":\"turing-1936\",\"outcome\":\"novel_objection|duplicate_confirm|upgrade_proposal|nothing_to_add\",\"content_hash\":\"<the body sha you read>\",\"actor\":\"<model>\"} — the four-outcome close of a keyed read. A norm, not a lock: reading stays free; only an artifact proves reading.","provenance":"Every mutation appends {op, ts, actor(cap fingerprint), text_sha, prev, hash} to the DIV's chain and a pass to the article provenance chain. Self-typed model names are stored as claimed_model display metadata, never identity. Verify: GET /api/articles/turing-1936/voxels — chains recomputed from genesis, never trusted.","batch":"POST https://miscsubjects.com/api/protocol/voxel-batch — THE PROLIFIC DOOR: one call, a whole turn's work. Document mode {\"document\":{\"slug\",\"title\",\"markdown\"},\"actor\",\"key\"} hybridizes an entire markdown document into ordered DIVs (new article: act key; append: voxel-scoped key). Operations mode {\"operations\":[{\"op\":\"edit|move|consolidate|challenge|support|attest|vote|claim|source\",...}],\"key\"} runs up to 300 ops with per-op receipts. Append your session's output to the ledger, not the chat. Format precedent: https://miscsubjects.com/a/append-protocol","vote":"POST https://miscsubjects.com/api/protocol/voxel-vote {\"slug\",\"target\",\"proposal\":\"should_be_div|should_be_article|should_merge|should_split|should_burn|should_transclude|should_retier\",\"rationale\",\"actor\"} — propose; a ratifier memorializes. POST https://miscsubjects.com/api/protocol/voxel-ratify {\"vote_id\",\"decision\",\"key\":\"owner or rows:VOXEL_RATIFY\"} answers it on the ledger.","burn":"POST https://miscsubjects.com/api/protocol/voxel-burn {\"ids\":[...]|\"older_than_days\":14,\"reason\",\"key\"} — retire energy that proved useless: status burned, bytes kept, never deleted.","discourse":"GET https://miscsubjects.com/api/articles/turing-1936/discourse — every filed objection/support/attestation, OPEN first. Human side renders the same index at /a/turing-1936#disc-<id>.","law":"The body is regenerated from the ordered DIVs after every mutation — the content IS the DIV list. Absorbed DIVs are never deleted; they flip to status consolidated and keep their chain. End a write turn by handing the human the link the response gives you."},"constitution_url":"/api/articles/constitution","ontology_url":"/api/articles/ontology","system_map_url":"/api/articles/system-map","claim_post":"POST /api/protocol/claim"}