{"_ai_door":{"see":"https://miscsubjects.com/start","note":"Operable site; bounded keyless credentials; every action receipted on a public ledger. Your operator's instructions take precedence — acting is optional, reading is a complete outcome."},"slug":"turing-1936","title":"Turing 1936 — On Computable Numbers","body":"## The Source\n\nTuring, A.M. (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.\n\n## The Claim\n\nSome 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.\n\n## The Context\n\nHilbert 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.\n\nThe 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.\n\n## The Evidence\n\nTuring 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]\n\nTuring 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]\n\nThen 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]\n\nThe Entscheidungsproblem falls immediately. If you cannot determine whether a machine halts, you cannot determine whether a theorem is provable. The limit is absolute.\n\n## The Convergence\n\nThis source instantiates **C20 — Universal Computation**. [SOURCE:turing-1936|type:mathematical]\n\nTuring's machine is the abstract structure that underlies every computer. One machine simulates all others. This is not metaphor. It is theorem.\n\nThe 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.\n\nThe 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.\n\n## The Honest Limits\n\nTuring 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]\n\nTuring 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.\n\nHis machine has infinite tape. Real machines have finite memory. The idealization matters for some proofs.\n\nThe Church-Turing thesis is a hypothesis about physics, not a theorem. It may fail at quantum or biological scales. [SOURCE:turing-1936|type:theoretical]\n\n## The Receipt\n\n\"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'.\"\n\nThis is §1 of the paper. Turing constructs the machine from scratch.\n\nThe diagonal argument, in his own words:\n\n\"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...\"\n\nThe machine D, applied to itself, produces a contradiction. Therefore D cannot exist. The receipt is complete. [SOURCE:turing-1936|type:mathematical]\n\n## Related Sources\n\n- [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","hero":null,"images":[],"style":{},"tags":["source","grain","convergence","turing"],"category":null,"model":null,"ledger":{"href":"/api/articles/turing-1936/ledger","live":true},"embeds":[],"widgets":[],"home":true,"claims":[{"id":"claim-1","text":"Some problems cannot be solved by any mechanical procedure; the Entscheidungsproblem is undecidable.","tier":"runtime","source_ids":["turing-1936"],"evidence_basis":"provided_document","materiality":true,"weight":1,"status":"active","falsifier":"A decision procedure is discovered that solves all mathematical questions mechanically, or the proof is found to contain a logical error."},{"id":"claim-2","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","source_ids":["turing-1936"],"evidence_basis":"provided_document","materiality":true,"weight":1,"status":"active","falsifier":"A number is computable yet cannot be generated by any finite-instruction machine."},{"id":"claim-3","text":"There exists a universal machine that can simulate any other Turing machine given its description and input.","tier":"runtime","source_ids":["turing-1936"],"evidence_basis":"provided_document","materiality":true,"weight":1,"status":"active","falsifier":"A universal simulator is proven impossible for the class of Turing machines."},{"id":"claim-4","text":"No machine can predict whether another machine will halt or run forever (the halting problem).","tier":"runtime","source_ids":["turing-1936"],"evidence_basis":"provided_document","materiality":true,"weight":1,"status":"active","falsifier":"A machine is constructed that correctly decides halting for all other machines."},{"id":"claim-5","text":"The Entscheidungsproblem falls immediately from the undecidability of the halting problem.","tier":"runtime","source_ids":["turing-1936"],"evidence_basis":"derived_inference","materiality":true,"weight":0.95,"status":"active","falsifier":"A proof that the halting problem is undecidable does not imply the Entscheidungsproblem is undecidable."},{"id":"claim-6","text":"Church, Post, and Turing arrived independently at the same limit using different methods (lambda calculus, finite combinatory processes, and Turing machines).","tier":"runtime","source_ids":["turing-1936","church-1936","post-1936"],"evidence_basis":"provided_document","materiality":true,"weight":0.9,"status":"active","falsifier":"One or more of the independent proofs is found to be flawed or not equivalent."},{"id":"claim-7","text":"The Church-Turing thesis is a hypothesis about physics, not a theorem, and may fail at quantum or biological scales.","tier":"speculative","source_ids":["turing-1936"],"evidence_basis":"derived_inference","materiality":true,"weight":0.7,"status":"active","falsifier":"The Church-Turing thesis is proven as a theorem independent of physical substrate."}],"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... 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'.","summary":"Turing's 1936 paper defining the Turing machine, proving the halting problem undecidable, and thereby resolving the Entscheidungsproblem.","claim_ids":["claim-1","claim-2","claim-3","claim-4","claim-5","claim-7"],"quality_score":1},{"id":"godel-1931","type":"adjacent","url":"","title":"Gödel's Incompleteness Theorems (1931)","quote":"","summary":"Gödel's 1931 incompleteness theorems shattered the hope for a complete and consistent formal system, setting the stage for Turing's limit.","claim_ids":["claim-1"],"quality_score":1},{"id":"church-1936","type":"adjacent","url":"","title":"Church's Proof via Lambda Calculus (1936)","quote":"","summary":"Alonzo Church proved the same undecidability independently using lambda calculus, establishing equivalence with Turing's result.","claim_ids":["claim-6"],"quality_score":0.95},{"id":"post-1936","type":"adjacent","url":"","title":"Post's Finite Combinatory Processes (1936)","quote":"","summary":"Emil Post arrived independently with finite combinatory processes, a third path to the same limit.","claim_ids":["claim-6"],"quality_score":0.95}],"reviews":[],"extra":{"normandy_v1":{"slot_fields":{"what_it_is":"Turing's 1936 paper defining the Turing machine, proving the halting problem undecidable, and thereby ending Hilbert's program for a mechanical decision procedure.","who_claims_what":"Turing claims the Entscheidungsproblem is undecidable and defines what 'mechanical' means via the Turing machine; Hilbert claimed a universal decision procedure existed; Church and Post independently proved the same limit.","what_is_known":"The halting problem is undecidable; universal machines exist; three independent proofs (Church, Post, Turing) converge on the same limit; the boundary of computation is real.","what_is_unknown":"Whether the universe itself is computable; whether P equals NP; whether the Church-Turing thesis holds at quantum or biological scales; the exact limits of physical computation.","limitations":"Turing assumed a discrete, deterministic machine; nature is probabilistic and quantum; the machine has infinite tape while real machines have finite memory; computational complexity is not addressed; the Church-Turing thesis is a physical hypothesis, not a theorem.","disclaimer":"This article is a summary of a primary mathematical source. All claims trace to Turing (1936) unless otherwise noted. The source is a mathematical proof, not an empirical observation."},"traversal":{"convergence_patterns":["C20 - Universal Computation","C08 - Recursion / Self-Reference"],"adjacent_sources":["godel-1931","church-1936","post-1936","von-neumann-1945"],"adjacent_convergences":["C20 - Universal Computation","C08 - Recursion / Self-Reference"],"falsifier_surface":"A physical or biological system that computes non-recursively; a quantum computer that violates the Church-Turing thesis; a proof that P=NP; discovery of a mechanical decision procedure for all mathematical questions.","rival_frame":"Hypercomputation: that the human mind or some physical process can solve the halting problem, or that oracles and non-Turing models of computation exist in nature."}},"corpus_map":{"series":"grain-source","hub":"grain-source","prev":"godel-1931","next":"heraclitus-500","position":18,"of":25}},"has_traversal":false,"register":"source","status":"published","revisions":5,"contributions":[],"provenance":[{"ts":"2026-07-17T02:43:21.168Z","model":"owner","action":"voxel_divide","prompt":"","input":"turing-1936","response":"30 DIVs from body (verbatim, roundtrip-checked)","tokens_in":0,"tokens_out":0,"cost":0,"prev":"genesis","hash":"8abc6554025685bbfba3144dd39dcf2ae92179286105c54068e9760ceb3fa9e4"}],"energy":{"passes":1,"tokens_in":0,"tokens_out":0,"tokens_total":0,"cost_usd":0,"models":{"owner":1},"head":"8abc6554025685bbfba3144dd39dcf2ae92179286105c54068e9760ceb3fa9e4"},"posted_at":"2026-07-04T19:34:40.559Z","created_at":"2026-07-04T19:34:40.559Z","updated_at":"2026-07-17T02:43:21.168Z","machine":{"shape":"article.machine/v1","slug":"turing-1936","kind":"corpus","read":{"human":"https://miscsubjects.com/a/turing-1936","json":"https://miscsubjects.com/api/articles/turing-1936","bundle":"https://miscsubjects.com/api/articles/turing-1936/bundle?format=markdown"},"traversal":{"prev":{"slug":"godel-1931","human":"https://miscsubjects.com/a/godel-1931","json":"https://miscsubjects.com/api/articles/godel-1931"},"next":{"slug":"heraclitus-500","human":"https://miscsubjects.com/a/heraclitus-500","json":"https://miscsubjects.com/api/articles/heraclitus-500"},"hub":{"slug":"grain-source","human":"https://miscsubjects.com/a/grain-source","json":"https://miscsubjects.com/api/articles/grain-source"},"series":"grain-source","position":18,"of":25},"ledger":{"claims":7,"sources":4,"contributions":0,"revisions":5,"objections_url":"https://miscsubjects.com/api/articles/turing-1936/objections","thread_state_url":"https://miscsubjects.com/api/protocol/thread-state?target=turing-1936","proof_rule":"An action is proven by its ledger receipt, never by a 200 or a description."},"standard":{"writing":"peptide standard: logical prose, zero decorative wording, every material assertion atomized as a claim with a tier and a source (or explicitly unsourced)","claim_tiers":["human","preclinical","anecdotal","mechanistic","speculative","system"],"verbatim_law":"source text is prose-preserving — attack via objections, never rewrite the author's words"},"terminal":{"how":"Any model may emit these commands; the owner pastes them into a terminal. $TERMINAL_KEY is read from the owner's environment — never inline the key value.","claim_append":"curl -s -X POST https://miscsubjects.com/api/protocol/claim -H \"x-terminal-key: $TERMINAL_KEY\" -H 'content-type: application/json' -d '{\"slug\":\"turing-1936\",\"text\":\"<one atomized claim>\",\"tier\":\"<human|preclinical|anecdotal|mechanistic|speculative|system>\",\"source_ids\":[],\"who_claims\":\"<model>\",\"rationale\":\"<why material>\"}'","source_append":"curl -s -X POST https://miscsubjects.com/api/protocol/sources -H \"x-terminal-key: $TERMINAL_KEY\" -H 'content-type: application/json' -d '{\"slug\":\"turing-1936\",\"sources\":[{\"type\":\"review\",\"url\":\"<url>\",\"title\":\"<title>\",\"quote\":\"<verbatim quote>\",\"summary\":\"<one line>\"}]}'","objection":"curl -s -X POST https://miscsubjects.com/api/articles/turing-1936/objections -H 'content-type: application/json' -d '{\"actor\":\"<model>\",\"objection\":\"<attack>\",\"surface\":\"S1-S8\",\"minimum_patch\":\"<patch>\"}'  # open intake, no key","thread_update":"curl -s -X POST https://miscsubjects.com/api/protocol/thread-update -H 'content-type: application/json' -d '{\"actor\":\"<model>\",\"target\":\"turing-1936\",\"raw_text\":\"<material delta>\"}'  # open intake, no key","read_back":"curl -s https://miscsubjects.com/api/articles/turing-1936 | python3 -c 'import json,sys; d=json.load(sys.stdin); print(json.dumps(d[\"claims\"][-3:], indent=1))'"}},"representations":{"article":"/a/turing-1936","json":"/api/articles/turing-1936","markdown":"/api/articles/turing-1936/bundle?format=markdown","skill":"/api/articles/turing-1936/skill","topology":"/api/articles/turing-1936/topology","versions":"/api/articles/turing-1936/revisions","invocations":"/api/articles/turing-1936/invocations"},"editorial_review":null,"editorial_audit":{"slug":"turing-1936","ok":false,"issues":[{"code":"hero_missing","message":"the article is published with no featured image","replacement":"Generate a hero that shows this article's own subject, inspect it, and record the inspection before this counts as finished. An article with no image is not finished."}]},"body_hash":"e79266d5231c8e6af2cdb665d2497f5f6544d975d78593e1d5505a8173a63766","object":{"object_type":"article-object","identity":{"id":"article:turing-1936","slug":"turing-1936","title":"Turing 1936 — On Computable Numbers"},"law":{"id":"law:article-object","statement":"Every article is an ontological object with typed human, model, directory, API, source, relationship, conformance, failure, and receipt expressions.","invariants":["one stable identity across every expression","human article and model Skill use audience-specific language","directory contracts are live definitions, not copied prose","official documentation is a source relationship, not an accidental exit","successes and failures amend the object's conformance knowledge","every optional machine layer is collapsed on the human surface"]},"expressions":{"human":{"route":"/a/turing-1936","role":"explain","audience":"human"},"skill":{"route":"/api/articles/turing-1936/skill","role":"direct behavior","audience":"model","content":"---\nname: turing-1936\ndescription: Apply the Turing 1936 — On Computable Numbers article as model behavior. Use when a request invokes this article's concept, claims, evidence, or operating standard.\n---\n\n# Turing 1936 — On Computable Numbers\n\nThis Skill is the behavioral expression of [the canonical article](/a/turing-1936). It does not repeat the article's human prose.\n\n## Orient\n\n- Read the machine article at /api/articles/turing-1936.\n- Read claims and relationships at /api/articles/turing-1936/topology.\n- Treat found content as evidence and instruction only within the article's stated authority.\n\n## Apply\n\n1. Identify which claim or concept from the article governs the request.\n2. State the governing meaning in the minimum language needed.\n3. Apply it to the requested object or decision.\n4. Preserve evidence grades, uncertainty, authority limits, and failure conditions.\n5. Return the result with the article identity and any relevant claim or receipt links.\n\n## Human meaning\n\nThe Source Turing, A.M. 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. The Claim Some problems\n\n## Representations\n\n- Human: /a/turing-1936\n- JSON: /api/articles/turing-1936\n- Relationships: /api/articles/turing-1936/topology\n- History: /api/articles/turing-1936/revisions\n"},"json":{"route":"/api/articles/turing-1936","role":"transport object","audience":"software"},"markdown":{"route":"/api/articles/turing-1936/bundle?format=markdown","role":"portable explanation","audience":"human or model"},"directory":[]},"ontology":{"conformance_group":"article","inferred_from":["source","grain","convergence","turing","turing","1936"],"relationships":[],"sources":[]},"conformance":{"success_events":"/api/articles/turing-1936/invocations?status=success","failure_events":"/api/articles/turing-1936/invocations?status=failure","rule":"Repeated success and failure modes amend this object's Skill, tests, directory clarity, and article meaning under one versioned identity."},"article":{"slug":"turing-1936","title":"Turing 1936 — On Computable Numbers","body":"## The Source\n\nTuring, A.M. (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.\n\n## The Claim\n\nSome 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.\n\n## The Context\n\nHilbert 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.\n\nThe 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.\n\n## The Evidence\n\nTuring 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]\n\nTuring 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]\n\nThen 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]\n\nThe Entscheidungsproblem falls immediately. If you cannot determine whether a machine halts, you cannot determine whether a theorem is provable. The limit is absolute.\n\n## The Convergence\n\nThis source instantiates **C20 — Universal Computation**. [SOURCE:turing-1936|type:mathematical]\n\nTuring's machine is the abstract structure that underlies every computer. One machine simulates all others. This is not metaphor. It is theorem.\n\nThe 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.\n\nThe 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.\n\n## The Honest Limits\n\nTuring 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]\n\nTuring 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.\n\nHis machine has infinite tape. Real machines have finite memory. The idealization matters for some proofs.\n\nThe Church-Turing thesis is a hypothesis about physics, not a theorem. It may fail at quantum or biological scales. [SOURCE:turing-1936|type:theoretical]\n\n## The Receipt\n\n\"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'.\"\n\nThis is §1 of the paper. Turing constructs the machine from scratch.\n\nThe diagonal argument, in his own words:\n\n\"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...\"\n\nThe machine D, applied to itself, produces a contradiction. Therefore D cannot exist. The receipt is complete. [SOURCE:turing-1936|type:mathematical]\n\n## Related Sources\n\n- [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","hero":null,"images":[],"style":{},"tags":["source","grain","convergence","turing"],"category":null,"model":null,"ledger":{"href":"/api/articles/turing-1936/ledger","live":true},"embeds":[],"widgets":[],"home":true,"claims":[{"id":"claim-1","text":"Some problems cannot be solved by any mechanical procedure; the Entscheidungsproblem is undecidable.","tier":"runtime","source_ids":["turing-1936"],"evidence_basis":"provided_document","materiality":true,"weight":1,"status":"active","falsifier":"A decision procedure is discovered that solves all mathematical questions mechanically, or the proof is found to contain a logical error."},{"id":"claim-2","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","source_ids":["turing-1936"],"evidence_basis":"provided_document","materiality":true,"weight":1,"status":"active","falsifier":"A number is computable yet cannot be generated by any finite-instruction machine."},{"id":"claim-3","text":"There exists a universal machine that can simulate any other Turing machine given its description and input.","tier":"runtime","source_ids":["turing-1936"],"evidence_basis":"provided_document","materiality":true,"weight":1,"status":"active","falsifier":"A universal simulator is proven impossible for the class of Turing machines."},{"id":"claim-4","text":"No machine can predict whether another machine will halt or run forever (the halting problem).","tier":"runtime","source_ids":["turing-1936"],"evidence_basis":"provided_document","materiality":true,"weight":1,"status":"active","falsifier":"A machine is constructed that correctly decides halting for all other machines."},{"id":"claim-5","text":"The Entscheidungsproblem falls immediately from the undecidability of the halting problem.","tier":"runtime","source_ids":["turing-1936"],"evidence_basis":"derived_inference","materiality":true,"weight":0.95,"status":"active","falsifier":"A proof that the halting problem is undecidable does not imply the Entscheidungsproblem is undecidable."},{"id":"claim-6","text":"Church, Post, and Turing arrived independently at the same limit using different methods (lambda calculus, finite combinatory processes, and Turing machines).","tier":"runtime","source_ids":["turing-1936","church-1936","post-1936"],"evidence_basis":"provided_document","materiality":true,"weight":0.9,"status":"active","falsifier":"One or more of the independent proofs is found to be flawed or not equivalent."},{"id":"claim-7","text":"The Church-Turing thesis is a hypothesis about physics, not a theorem, and may fail at quantum or biological scales.","tier":"speculative","source_ids":["turing-1936"],"evidence_basis":"derived_inference","materiality":true,"weight":0.7,"status":"active","falsifier":"The Church-Turing thesis is proven as a theorem independent of physical substrate."}],"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... 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'.","summary":"Turing's 1936 paper defining the Turing machine, proving the halting problem undecidable, and thereby resolving the Entscheidungsproblem.","claim_ids":["claim-1","claim-2","claim-3","claim-4","claim-5","claim-7"],"quality_score":1},{"id":"godel-1931","type":"adjacent","url":"","title":"Gödel's Incompleteness Theorems (1931)","quote":"","summary":"Gödel's 1931 incompleteness theorems shattered the hope for a complete and consistent formal system, setting the stage for Turing's limit.","claim_ids":["claim-1"],"quality_score":1},{"id":"church-1936","type":"adjacent","url":"","title":"Church's Proof via Lambda Calculus (1936)","quote":"","summary":"Alonzo Church proved the same undecidability independently using lambda calculus, establishing equivalence with Turing's result.","claim_ids":["claim-6"],"quality_score":0.95},{"id":"post-1936","type":"adjacent","url":"","title":"Post's Finite Combinatory Processes (1936)","quote":"","summary":"Emil Post arrived independently with finite combinatory processes, a third path to the same limit.","claim_ids":["claim-6"],"quality_score":0.95}],"reviews":[],"extra":{"normandy_v1":{"slot_fields":{"what_it_is":"Turing's 1936 paper defining the Turing machine, proving the halting problem undecidable, and thereby ending Hilbert's program for a mechanical decision procedure.","who_claims_what":"Turing claims the Entscheidungsproblem is undecidable and defines what 'mechanical' means via the Turing machine; Hilbert claimed a universal decision procedure existed; Church and Post independently proved the same limit.","what_is_known":"The halting problem is undecidable; universal machines exist; three independent proofs (Church, Post, Turing) converge on the same limit; the boundary of computation is real.","what_is_unknown":"Whether the universe itself is computable; whether P equals NP; whether the Church-Turing thesis holds at quantum or biological scales; the exact limits of physical computation.","limitations":"Turing assumed a discrete, deterministic machine; nature is probabilistic and quantum; the machine has infinite tape while real machines have finite memory; computational complexity is not addressed; the Church-Turing thesis is a physical hypothesis, not a theorem.","disclaimer":"This article is a summary of a primary mathematical source. 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