Evidence record

Can everything be computed?

Every sentence on the exploration belongs to at least one claim below. Each says what kind of statement it is, what it rests on, what the evidence does not establish, and how far it was checked. Nothing here is a score, and none of it is a verdict on a belief or a theory.

Last checked 24 September 2026.

A judgement, not a fact: 10
Source checked: 26
Presented as open: 4
Partly checked: 1

If something here is wrong, misquoted, out of date or unfair, write to support@unknownoceans.com and give the claim number, which does not change.

What the question takes for granted

A question with a mathematical answer at its core: some precisely stated problems have no algorithm, and that is a theorem. Around it sit two questions that are not settled, whether physical systems can always be simulated and whether minds are machines of Turing’s kind.

“Computed” means what a Turing machine can do
Established
For step-by-step procedures carried out by rule, this is the Church-Turing thesis, which is accepted across the field but is not a theorem and cannot be proved. It does not extend by itself to every physical machine: that is a separate thesis, and it is open (claim 35).
Physical processes can always be simulated by such a machine
Open
Stated precisely, this is the Deutsch-Wolfram thesis, and the reference literature treats it as open. The page reports the known counterexamples as constructed or artificial, which is how their authors describe them.
Minds are, or are not, machines of Turing’s kind
Open
The page does not settle it. It reports that the argument from Gödel’s theorem to an answer is widely held to fail, which is a narrower thing.

How the question is labelled

Claim: The questions no machine answers

Kind: This page’s reading. A judgement the page makes from the evidence. Others read it differently.
Review: A judgement, not a fact. Reviewed for whether it is fairly put and follows from the checked claims. No source can confirm it.

The page’s title, drawn from Turing’s result and Rice’s theorem below. “No machine” means no machine of Turing’s kind, which by the Church-Turing thesis is any step-by-step procedure.

Claim 1 · checked 24 September 2026 · AI-assisted audit

Claim: Computing

Kind: Framing. Signposting. It asserts nothing a source could check.

The question

Claim: No.

Kind: This page’s reading. A judgement the page makes from the evidence. Others read it differently.
Review: A judgement, not a fact. Reviewed for whether it is fairly put and follows from the checked claims. No source can confirm it.

The page’s answer to its question, resting on the proved existence of problems no Turing machine decides (claim 11).

What this does not establish

It answers for computation by rule. Whether every physical process can be simulated is a different question, and the page leaves it open.

Claim 3 · checked 24 September 2026 · AI-assisted audit

Claim: That was proved in 1936, before any modern computer existed, and the proof covers every possible program rather than any particular machine.

Kind: Historical claim. A statement about the past that historical evidence can test.
Review: Source checked. The cited passage was read, and it says this.

Supports
On Computable Numbers, with an Application to the Entscheidungsproblemp. 230, head of the paper
Cited · Link answered, 19 October 2026 · Read in full, 24 September 2026 · Primary text
The source reads: Received 28 May, 1936.–Read 12 November, 1936.
Supports
Alan Turingsection 3
Cited · Link answered, 19 October 2026 · Read in full, 24 September 2026 · Reference work
The source reads: the reader must always remember that in 1936 there were no such computers; indeed the modern computer arose out of the formulation of ‘behaving mechanically’ that Turing found in this work
Supports
Turing Machinessection 1.1
Cited · Link answered, 19 October 2026 · Read in full, 24 September 2026 · Reference work
The source reads: If then we accept the Turing machine model as a reasonable model of the modern computer, then any result which shows that a function is not Turing computable is very strong, since it would imply that no computer that we could ever build could carry out the computation.

What this does not establish

“Every possible program” means every program for a Turing machine. That this covers every step-by-step procedure is the Church-Turing thesis, accepted but not provable. The paper was printed in the 1937 volume.

Corrected

24 September 2026. Previously: “That was proved in 1936, before any computer existed, and the proof is about all possible machines rather than any particular one.” Calculating machines, and human “computers”, existed in 1936; the source says there were no computers of the modern kind. And the proof is about every Turing machine, not every possible machine: the encyclopedia is explicit that Turing proved nothing about machines of some other sort.

Claim 4 · checked 24 September 2026 · AI-assisted audit

Claim: There are precisely stated questions that no algorithm can answer, and the proof does not depend on the technology.

Kind: What the field holds. The settled view of a field, as a review, a consensus statement or a reference work gives it. Not one study.
Review: Source checked. The cited passage was read, and it says this.

Supports
Turing Machinessection 1.1
Cited · Link answered, 19 October 2026 · Read in full, 24 September 2026 · Reference work
The source reads: It ensures that no computable function will fail to be Turing-computable solely because there is insufficient time or memory to complete the computation.
Supports
The Church-Turing Thesissection 5.3.2
Cited · Link answered, 19 October 2026 · Read in full, 24 September 2026 · Reference work
The source reads: What is proved is that no Turing machine can always determine, when provided with the description of any Turing machine together with its initial state, whether or not that machine will reach its halt state.

What this does not establish

“Algorithm” here means a step-by-step procedure. That every such procedure is a Turing machine is the Church-Turing thesis, which is not a theorem.

Later work looked for

Both entries revised 2023 and 2025; Crossref, 2018 on, "Church-Turing thesis": textbook and survey treatments, nothing that changes the result.

Claim 5 · checked 24 September 2026 · AI-assisted audit

Claim: What remains open is which physical processes can be simulated, and whether the universe itself is computable.

Kind: Open question. Something the page says is not known.
Review: Presented as open. The page says this is unresolved, and the review found that it is.

Supports
The Church-Turing Thesissection 6.4.1
Cited · Link answered, 19 October 2026 · Read in full, 24 September 2026 · Reference work
The source reads: Every finite physical system can be simulated to any specified degree of accuracy by a universal Turing machine. ... Is the Deutsch-Wolfram thesis true? This is an open question

Claim 6 · checked 24 September 2026 · AI-assisted audit

Claim: The limits of computing are usually imagined as speed and memory: problems that are too big for now.

Kind: This page’s reading. A judgement the page makes from the evidence. Others read it differently.
Review: A judgement, not a fact. Reviewed for whether it is fairly put and follows from the checked claims. No source can confirm it.

The page’s opening, setting up the contrast with a limit that no amount of speed or memory removes (claim 24).

Claim 7 · checked 24 September 2026 · AI-assisted audit

Claim: There is a harder limit underneath, and it was established before the first modern computer was built.

Kind: Historical claim. A statement about the past that historical evidence can test.
Review: Source checked. The cited passage was read, and it says this.

Supports
Alan Turingsection 3
Cited · Link answered, 19 October 2026 · Read in full, 24 September 2026 · Reference work
The source reads: the reader must always remember that in 1936 there were no such computers; indeed the modern computer arose out of the formulation of ‘behaving mechanically’ that Turing found in this work

Corrected

24 September 2026. Previously: “There is a harder limit underneath, and it was established before the first electronic computer was built.” Aligned with the subtitle and with what the source says, which is about computers of the modern, stored-program kind.

Claim 8 · checked 24 September 2026 · AI-assisted audit

The proof

Claim: The proof

Kind: Framing. Signposting. It asserts nothing a source could check.

The halting problem

Claim: The halting problem

Kind: Framing. Signposting. It asserts nothing a source could check.

Claim: In 1936 Alan Turing defined a precise notion of a machine following instructions, and then showed that no such machine can decide, for every machine it is shown, whether that machine goes on printing forever, or whether it ever prints a given symbol.

Kind: Historical claim. A statement about the past that historical evidence can test.
Review: Source checked. The cited passage was read, and it says this.

Supports
On Computable Numbers, with an Application to the Entscheidungsproblem§ 8, pp. 246-247
Cited · Link answered, 19 October 2026 · Read in full, 24 September 2026 · Primary text
The source reads: we have no general process for doing this in a finite number of steps. In fact, by applying the diagonal process argument correctly, we can show that there cannot be any such general process.
“This” is deciding whether a number describes a circle-free machine, one that goes on printing figures forever.
Supports
On Computable Numbers, with an Application to the Entscheidungsproblem§ 8, p. 248
Cited · Link answered, 19 October 2026 · Read in full, 24 September 2026 · Primary text
The source reads: We can show further that there can be no machine ... the S.D of an arbitrary machine ... ever prints a given symbol (0 say).
Supports
Turing Machinessection 2.4.2
Cited · Link answered, 19 October 2026 · Read in full, 24 September 2026 · Reference work
The source reads: Based on the uncomputability of CIRC?, Turing then shows that also PRINT? is not computable.

Corrected

24 September 2026. Previously: “In 1936 Alan Turing defined a precise notion of a machine following instructions, and then showed that no such machine can decide, for every program and input, whether that program eventually stops.” Turing’s paper proves the undecidability of circle-freeness and of printing a given symbol. The halting problem in its familiar form is not in it; claim 12 says where it comes from.

Claim 11 · checked 24 September 2026 · AI-assisted audit

Claim: The halting problem as now taught, whether a program eventually stops, is a close relative of these, first set out by Martin Davis in the 1950s.

Kind: Historical claim. A statement about the past that historical evidence can test.
Review: Source checked. The cited passage was read, and it says this.

Supports
The Church-Turing Thesissection 5.3.2
Cited · Link answered, 19 October 2026 · Read in full, 24 September 2026 · Reference work
The source reads: although the Halting Problem is very commonly attributed to Turing, as Langton does here, Turing did not in fact formulate it. The Halting Problem originated with Davis in the early 1950s (Davis 1958: 70).
Supports
Turing Machinessection 2.4.3; section 5.1, on Davis
Cited · Link answered, 19 October 2026 · Read in full, 24 September 2026 · Reference work
The source reads: Note that Turing’s problem PRINT? is very close to HALT? ... people like Martin Davis

Corrected

24 September 2026. Previously: “” Added so the passage can keep its familiar heading without attributing to Turing a formulation that is not in his paper.

Claim 12 · checked 24 September 2026 · AI-assisted audit

Claim: The argument is a construction: assume such a decider exists, use it to build a machine that works through every machine in turn, and show that when it reaches its own description the decider can give neither verdict. The assumption collapses.

Kind: Historical claim. A statement about the past that historical evidence can test.
Review: Source checked. The cited passage was read, and it says this.

The version usually taught, a program that asks about itself and does the opposite, is a later popular proof of the halting problem; Turing chose a different construction on purpose.

Supports
On Computable Numbers, with an Application to the Entscheidungsproblem§ 8, p. 247
Cited · Link answered, 19 October 2026 · Read in full, 24 September 2026 · Primary text
The source reads: In the N-th section the machine ... tests the number N. ... It must test whether K is satisfactory ... Thus both verdicts are impossible and we conclude that there can be no machine
Supports
Turing Machinessection 2.4.2
Cited · Link answered, 19 October 2026 · Read in full, 24 September 2026 · Reference work
The source reads: becomes a circular machine when it is provided with its own description number, hence the assumption of a machine which is capable of solving CIRC? must be false

Corrected

24 September 2026. Previously: “The argument is a construction: assume such a decider exists, build a program that asks the decider about itself and then does the opposite, and the assumption collapses.” Turing names the do-the-opposite proof and sets it aside for one that builds a machine running through every description number and stalls on its own.

Claim 13 · checked 24 September 2026 · AI-assisted audit

Claim: It rules out an entire class of possible machines rather than reporting a failure to find one.

Kind: This page’s reading. A judgement the page makes from the evidence. Others read it differently.
Review: A judgement, not a fact. Reviewed for whether it is fairly put and follows from the checked claims. No source can confirm it.

Follows from the form of Turing’s conclusion, an impossibility proof rather than a failed search.

Context
On Computable Numbers, with an Application to the Entscheidungsproblem§ 8, p. 247
Cited · Link answered, 19 October 2026 · Read in full, 24 September 2026 · Primary text
The source reads: we conclude that there can be no machine

Claim 14 · checked 24 September 2026 · AI-assisted audit

It is not an isolated curiosity

Claim: It is not an isolated curiosity

Kind: This page’s reading. A judgement the page makes from the evidence. Others read it differently.
Review: A judgement, not a fact. Reviewed for whether it is fairly put and follows from the checked claims. No source can confirm it.

A heading, borne out by Rice’s theorem in claim 17.

Claim 15 · checked 24 September 2026 · AI-assisted audit

Claim: Many ordinary-sounding questions turn out to be undecidable too, and one theorem says how many.

Kind: What the field holds. The settled view of a field, as a review, a consensus statement or a reference work gives it. Not one study.
Review: Source checked. The cited passage was read, and it says this.

Supports
Recursive Functions, section 3.4: Rice’s Theoremsection 3.2, after Theorem 3.4
Cited · Link answered, 19 October 2026 · Read in full, 24 September 2026 · Reference work
The source reads: provides a means of showing that many decision problems of practical import are undecidable

Later work looked for

A theorem, not a finding; the entry was revised in 2024.

Corrected

24 September 2026. Previously: “Many ordinary-sounding questions turn out to be equivalent to it, and one theorem says how many.” Rice’s theorem shows these questions undecidable, not equivalent to the halting problem; some are strictly harder.

Claim 16 · checked 24 September 2026 · AI-assisted audit

Claim: Rice proved in 1953 that every non-trivial question about what a program computes, as opposed to how it is written, is undecidable.

Kind: Historical claim. A statement about the past that historical evidence can test.
Review: Source checked. The cited passage was read, and it says this.

Supports
Recursive Functions, section 3.4: Rice’s Theoremsection 3.2, Theorem 3.4
Cited · Link answered, 19 October 2026 · Read in full, 24 September 2026 · Reference work
The source reads: Theorem 3.4 (Rice 1953): If ... is a non-trivial index set, then ... is undecidable.
Supports
Recursive Functions, section 3.4: Rice’s Theoremsection 3.2, before Theorem 3.4
Cited · Link answered, 19 October 2026 · Read in full, 24 September 2026 · Reference work
The source reads: Sets with this property are known as index sets as they collect together the indices of all partial computable functions which share a common “semantic” property–i.e., one which is completely determined by their graphs

Claim 17 · checked 24 September 2026 · AI-assisted audit

Claim: Whether a program correctly computes a given function is such a question.

Kind: What the field holds. The settled view of a field, as a review, a consensus statement or a reference work gives it. Not one study.
Review: Source checked. The cited passage was read, and it says this.

Supports
Recursive Functions, section 3.4: Rice’s Theoremsection 3.2, after Theorem 3.4
Cited · Link answered, 19 October 2026 · Read in full, 24 September 2026 · Reference work
The source reads: of determining whether a program always returns an output or whether it correctly computes a given function (e.g., addition or multiplication)

Later work looked for

A theorem, not a finding; the entry was revised in 2024.

Corrected

24 September 2026. Previously: “Whether two programs compute the same function is such a question.” Rice’s theorem is about a property of one program. Equivalence of two follows from it but is not what the source states; the source’s own example is used instead.

Claim 18 · checked 24 September 2026 · AI-assisted audit

Claim: So is whether a piece of code ever produces a given output.

Kind: What the field holds. The settled view of a field, as a review, a consensus statement or a reference work gives it. Not one study.
Review: Source checked. The cited passage was read, and it says this.

Supports
Turing Machinessection 2.4.2
Cited · Link answered, 19 October 2026 · Read in full, 24 September 2026 · Reference work
The source reads: Based on the uncomputability of CIRC?, Turing then shows that also PRINT? is not computable.
Supports
On Computable Numbers, with an Application to the Entscheidungsproblem§ 8, p. 248
Cited · Link answered, 19 October 2026 · Read in full, 24 September 2026 · Primary text
The source reads: We can show further that there can be no machine ... the S.D of an arbitrary machine ... ever prints a given symbol (0 say).

Later work looked for

A theorem, not a finding; the entry was revised in 2025.

Claim 19 · checked 24 September 2026 · AI-assisted audit

Claim: The same shape appears in mathematics as Gödel’s incompleteness theorems.

Kind: This page’s reading. A judgement the page makes from the evidence. Others read it differently.
Review: A judgement, not a fact. Reviewed for whether it is fairly put and follows from the checked claims. No source can confirm it.

Both results are diagonal constructions about what a rule-bound system can settle about itself.

Context
On Computable Numbers, with an Application to the Entscheidungsproblemintroduction, p. 230
Cited · Link answered, 19 October 2026 · Read in full, 24 September 2026 · Primary text
The source reads: conclusions are reached which are superficially similar to those of
Turing’s own comparison with Gödel. Gödel’s theorems (1931) came first.

Claim 20 · checked 24 September 2026 · AI-assisted audit

Claim: Any consistent formal system strong enough to describe arithmetic contains true statements it cannot prove.

Kind: What the field holds. The settled view of a field, as a review, a consensus statement or a reference work gives it. Not one study.
Review: Source checked. The cited passage was read, and it says this.

Supports
Gödel’s Incompleteness Theoremssection 1, statement of the first theorem
Cited · Link answered, 19 October 2026 · Read in full, 24 September 2026 · Reference work
The source reads: Any consistent formal system ... within which a certain amount of elementary arithmetic can be carried out is incomplete; i.e., there are statements of the language of ... which can neither be proved nor disproved in
Supports
Gödel’s Incompleteness Theoremssection 2.5
Cited · Link answered, 19 October 2026 · Read in full, 24 September 2026 · Reference work
The source reads: is true if and only if it is not provable in ... In sum, if ... is consistent, then ... is not provable in
The Gödel sentence of a consistent system is unprovable in it, and true.
Qualifies
Gödel’s Incompleteness Theoremssection 1
Cited · Link answered, 19 October 2026 · Read in full, 24 September 2026 · Reference work
The source reads: A common misunderstanding is to interpret Gödel’s first theorem as showing that there are truths that cannot be proved.

What this does not establish

Unprovable in that system, not unprovable anywhere: a stronger system can prove the statement, and has its own unprovable one.

Later work looked for

A theorem, not a finding; the entry was revised in October 2025.

Claim 21 · checked 24 September 2026 · AI-assisted audit

Claim: What the limit is and is not

Kind: Framing. Signposting. It asserts nothing a source could check.

Claim: A faster computer could solve these.

Kind: Framing. Signposting. It asserts nothing a source could check.

Claim: The proof already allows unlimited time and memory, so no faster machine of the same kind escapes it.

Kind: What the field holds. The settled view of a field, as a review, a consensus statement or a reference work gives it. Not one study.
Review: Source checked. The cited passage was read, and it says this.

Supports
Turing Machinessection 1.1
Cited · Link answered, 19 October 2026 · Read in full, 24 September 2026 · Reference work
The source reads: regardless of the amount of time it takes. One can think of this as assuming the availability of potentially infinite time to complete the computation.
Supports
Turing Machinessection 1.1
Cited · Link answered, 19 October 2026 · Read in full, 24 September 2026 · Reference work
The source reads: It ensures that no computable function will fail to be Turing-computable solely because there is insufficient time or memory to complete the computation.

Later work looked for

A consequence of the definition; the entry was revised in 2025.

Corrected

24 September 2026. Previously: “The proof covers any machine following any finite set of rules, at any speed. Speed is irrelevant to it.” The proof covers Turing machines, not any machine, and speed is not wholly irrelevant: the Church-Turing entry describes accelerating machines, notional ones, that compute the halting function by speeding up without limit.

Claim 24 · checked 24 September 2026 · AI-assisted audit

Claim: Only a notional machine that completes infinitely many steps in a finite time would.

Kind: What a model says. What a theory or model predicts or describes, which is not the same as what has been observed.
Review: Source checked. The cited passage was read, and it says this.

Supports
The Church-Turing Thesissection 5.4.1
Cited · Link answered, 19 October 2026 · Read in full, 24 September 2026 · Reference work
The source reads: an ATM is able to perform infinitely many operations in two moments of operating time. This enables ATMs to generate functions that cannot be computed by any standard Turing machine ... One example of such a function is the halting function
Context
The Church-Turing Thesissection 6.4.2
Cited · Link answered, 19 October 2026 · Read in full, 24 September 2026 · Reference work
The source reads: emphasize that the Németi computer is “not in conflict with presently accepted scientific principles”
The relativistic proposal the alternatives mention.

What this does not establish

A thought experiment. Nothing on the page, or in the source, says such a machine could be built.

Claim: Quantum computers get around it.

Kind: Framing. Signposting. It asserts nothing a source could check.

Claim: They change which problems are efficiently solvable, not which are solvable at all. The undecidable stays undecidable.

Kind: What a model says. What a theory or model predicts or describes, which is not the same as what has been observed.
Review: Partly checked. Only part of the source could be read. What was not read is listed.

Supports
Quantum theory, the Church–Turing principle and the universal quantum computerabstract
Cited · Link answered, 19 October 2026 · Abstract only, 24 September 2026 · Theory · Model
The source reads: These do not include the computation of non-recursive functions, but they do include ‘quantum parallelism’, a method by which certain probabilistic tasks can be performed faster by a universal quantum computer than by any classical restriction of it.

What this does not establish

Read in the abstract only. It is shown for Deutsch’s model of a universal quantum computer.

Claim 27 · checked 24 September 2026 · AI-assisted audit

Claim: It means most useful questions are unanswerable.

Kind: Framing. Signposting. It asserts nothing a source could check.

Claim: It means no single procedure answers every instance.

Kind: What the field holds. The settled view of a field, as a review, a consensus statement or a reference work gives it. Not one study.
Review: Source checked. The cited passage was read, and it says this.

Supports
Gödel’s Incompleteness Theoremssection 2.5, on the two senses of “undecidable”
Cited · Link answered, 19 October 2026 · Read in full, 24 September 2026 · Reference work
The source reads: a theory may be undecidable (see below) in the sense that there does not exist a decision method for determining of an arbitrary given sentence of the language whether or not it is derivable in the theory

Later work looked for

A definition, not a finding; the entry was revised in October 2025.

Corrected

24 September 2026. Previously: “It means no single procedure answers all instances. Particular programs are analysed successfully every day.” No source read carries a claim about daily practice; the replacement says what the sources say, that undecidability is about an infinite family and that particular cases are settled.

Claim 29 · checked 24 September 2026 · AI-assisted audit

Claim: Undecidability concerns an infinite family of cases, and particular cases can still be settled.

Kind: What the field holds. The settled view of a field, as a review, a consensus statement or a reference work gives it. Not one study.
Review: Source checked. The cited passage was read, and it says this.

Supports
Gödel’s Incompleteness Theoremssection 2.5, on the two senses of “undecidable”
Cited · Link answered, 19 October 2026 · Read in full, 24 September 2026 · Reference work
The source reads: so this latter sense of “undecidable” concerns, so to speak, an infinite class of statements
Supports
Undecidability of the spectral gapDiscussion; and introduction
Cited · Link answered, 19 October 2026 · Read in full, 24 September 2026 · Theory · Argument
The source reads: Whilst algorithmic undecidability always concerns infinite families of systems ... Many seminal results in condensed matter theory prove that specific systems are gapped or gapless.

What this does not establish

Not every particular case: Cubitt and colleagues show some individual systems whose answer can be neither proved nor disproved.

Later work looked for

A definition, not a finding; nothing to supersede it.

Claim 30 · checked 24 September 2026 · AI-assisted audit

What it reaches

Claim: What it reaches

Kind: Framing. Signposting. It asserts nothing a source could check.

Claim: Proved in 1936, about machines that did not yet exist, and still true of every one built since.

Kind: This page’s reading. A judgement the page makes from the evidence. Others read it differently.
Review: A judgement, not a fact. Reviewed for whether it is fairly put and follows from the checked claims. No source can confirm it.

Follows from claim 4 on the date, and from treating every computer built as a machine of Turing’s kind with finite memory, which the source calls a reasonable model.

Context
Turing Machinessection 1.1
Cited · Link answered, 19 October 2026 · Read in full, 24 September 2026 · Reference work
The source reads: If then we accept the Turing machine model as a reasonable model of the modern computer, then any result which shows that a function is not Turing computable is very strong, since it would imply that no computer that we could ever build could carry out the computation.

Claim 32 · checked 24 September 2026 · AI-assisted audit

Whether physics is computable

Claim: Whether physics is computable

Kind: Framing. Signposting. It asserts nothing a source could check.

Claim: A separate question is whether the behaviour of physical systems can always be simulated by a machine of this kind.

Kind: Framing. Signposting. It asserts nothing a source could check.

Claim: The claim that every finite physical system can be simulated to any desired accuracy has been called the Deutsch-Wolfram thesis, and whether it is true is open.

Kind: Open question. Something the page says is not known.
Review: Presented as open. The page says this is unresolved, and the review found that it is.

Supports
The Church-Turing Thesissection 6.4.1
Cited · Link answered, 19 October 2026 · Read in full, 24 September 2026 · Reference work
The source reads: Every finite physical system can be simulated to any specified degree of accuracy by a universal Turing machine. ... Is the Deutsch-Wolfram thesis true? This is an open question

Corrected

24 September 2026. Previously: “Most physical law is expressed in equations that can be approximated numerically to any desired accuracy.” The old sentence stated as settled what the reference literature calls an open question, and passed over the best-known counterexample.

Claim 35 · checked 24 September 2026 · AI-assisted audit

Claim: In 1981 Pour-El and Richards showed that the ordinary wave equation, started from computable conditions, can behave in a way no such machine computes;

Kind: Historical claim. A statement about the past that historical evidence can test.
Review: Source checked. The cited passage was read, and it says this.

Supports
The Church-Turing Thesissection 6.4.1
Cited · Link answered, 19 October 2026 · Read in full, 24 September 2026 · Reference work
The source reads: Pour-El and Richards showed in their 1981 article that a system evolving from computable initial conditions in accordance with the familiar three-dimensional wave equation is capable of exhibiting behavior that falsifies the Deutsch-Wolfram thesis.

What this does not establish

Checked against the encyclopedia’s account; the 1981 paper itself was not read.

Claim 36 · checked 24 September 2026 · AI-assisted audit

Claim: whether those starting conditions can occur in nature is also open.

Kind: Open question. Something the page says is not known.
Review: Presented as open. The page says this is unresolved, and the review found that it is.

Supports
The Church-Turing Thesissection 6.4.1
Cited · Link answered, 19 October 2026 · Read in full, 24 September 2026 · Reference work
The source reads: However, now as then, it is an open question whether these initial conditions are physically possible.

Claim 37 · checked 24 September 2026 · AI-assisted audit

Claim: There are results showing that particular questions about models of quantum many-body systems, such as whether one has an energy gap, are undecidable.

Kind: What the field holds. The settled view of a field, as a review, a consensus statement or a reference work gives it. Not one study.
Review: Source checked. The cited passage was read, and it says this.

Supports
The Church-Turing Thesissection 6.4.3
Cited · Link answered, 19 October 2026 · Read in full, 24 September 2026 · Reference work
The source reads: This was a curtain-raiser to a series of dramatic results about the uncomputability of quantum phase transitions, by Cubitt and his group
Supports
Undecidability of the spectral gapabstract
Cited · Link answered, 19 October 2026 · Read in full, 24 September 2026 · Theory · Argument
The source reads: We construct families of quantum spin systems on a 2D lattice with translationally-invariant, nearest-neighbour interactions for which the spectral gap problem is undecidable.

Later work looked for

Crossref, 2016 on, "undecidability spectral gap": Bausch, Cubitt, Lucia and Perez-Garcia (Physical Review X, 2020) extend it to one dimension, and Castilla-Castellano and Lucia (Physical Review Research, 2026) to rotation-symmetric models; both widen the result and neither concerns a real material.

Corrected

24 September 2026. Previously: “There are results showing that particular questions about quantum many-body systems, such as whether a material has an energy gap, are undecidable.” The results are about constructed models, not about any material.

Claim 38 · checked 24 September 2026 · AI-assisted audit

Claim: Whether any real material is affected is open: the models are, in their authors’ words, highly artificial, and whether the result extends to more natural ones is not known.

Kind: Open question. Something the page says is not known.
Review: Presented as open. The page says this is unresolved, and the review found that it is.

Supports
Undecidability of the spectral gapDiscussion
Cited · Link answered, 19 October 2026 · Read in full, 24 September 2026 · Theory · Argument
The source reads: the models we construct in the proof are highly artificial. Whether the results can be extended to more natural models is open.
Supports
The Church-Turing Thesissection 6.4.3
Cited · Link answered, 19 October 2026 · Read in full, 24 September 2026 · Reference work
The source reads: it is an open–and fascinating–question whether there are realistic physical systems that fail to satisfy the Total thesis

Corrected

24 September 2026. Previously: “Whether that has consequences for anything measurable is disputed, because the constructions involve infinite idealised systems.” No source read reports a dispute; the authors and the encyclopedia both call it open. And the authors give the artificiality of the models as the limitation, while arguing that the result does reach finite systems, where the gap can switch on at a size no algorithm predicts.

Claim 39 · checked 24 September 2026 · AI-assisted audit

One case where it bites

Claim: One case where it bites

Kind: Framing. Signposting. It asserts nothing a source could check.

Claim: In 2015 a paper proved that whether a model of a quantum material has an energy gap, a property that shapes much of how a material behaves, is undecidable in general.

Kind: Historical claim. A statement about the past that historical evidence can test.
Review: Source checked. The cited passage was read, and it says this.

Supports
Undecidability of the spectral gapabstract and first paragraph
Cited · Link answered, 19 October 2026 · Read in full, 24 September 2026 · Theory · Argument
The source reads: Here we prove that this is an undecidable problem. ... determining much of its low-energy physics
Supports
The Church-Turing Thesissection 6.4.3
Cited · Link answered, 19 October 2026 · Read in full, 24 September 2026 · Reference work
The source reads: an important determinant of the properties of a substance

What this does not establish

Published in Nature in December 2015 (Crossref record); the arXiv text read here is the authors’ version.

Corrected

24 September 2026. Previously: “In 2015 a paper proved that whether a quantum material has an energy gap, a basic property that determines whether it conducts, is undecidable in general. There is no algorithm that answers it for every material.” The paper proves a result about models, and describes the gap as governing low-energy physics, not conduction.

Claim 41 · checked 24 September 2026 · AI-assisted audit

Claim: There is no algorithm that answers it for every such model.

Kind: What the field holds. The settled view of a field, as a review, a consensus statement or a reference work gives it. Not one study.
Review: Source checked. The cited passage was read, and it says this.

Supports
Undecidability of the spectral gapabstract
Cited · Link answered, 19 October 2026 · Read in full, 24 September 2026 · Theory · Argument
The source reads: Our result implies that there exists no algorithm to determine whether an arbitrary model is gapped or gapless.
Supports
The Church-Turing Thesissection 6.4.3
Cited · Link answered, 19 October 2026 · Read in full, 24 September 2026 · Reference work
The source reads: The uncomputability results of Cubitt et al. stem from their discovery that the halting problem can be encoded in the spectral gap problem.

Later work looked for

Crossref, 2016 on, "undecidability spectral gap": the full version (Forum of Mathematics, Pi, 2022) and extensions in 2020 and 2026 confirm it.

Claim 42 · checked 24 September 2026 · AI-assisted audit

Claim: The construction embeds a halting problem into a lattice of interacting particles, and it applies to an idealised infinite system rather than to any object on a bench.

Kind: Historical claim. A statement about the past that historical evidence can test.
Review: Source checked. The cited passage was read, and it says this.

Supports
Undecidability of the spectral gapabstract; Discussion
Cited · Link answered, 19 October 2026 · Read in full, 24 September 2026 · Theory · Argument
The source reads: construct a Hamiltonian whose ground state encodes the evolution of a quantum phase-estimation algorithm followed by a universal Turing Machine. The spectral gap depends on the outcome of the corresponding Halting Problem. ... we usually probe the idealised infinite thermodynamic limit

What this does not establish

The authors also draw a consequence for finite systems of the same models: the gap can appear or vanish at a size that cannot be computed.

Claim 43 · checked 24 September 2026 · AI-assisted audit

Claim: What it establishes is that undecidability is not confined to questions about programs.

Kind: This page’s reading. A judgement the page makes from the evidence. Others read it differently.
Review: A judgement, not a fact. Reviewed for whether it is fairly put and follows from the checked claims. No source can confirm it.

Follows from claims 41 and 42: the undecidable question is one about a physical model.

Claim 44 · checked 24 September 2026 · AI-assisted audit

Claim: It can be a property of a physical model, which is a stranger result than it first sounds.

Kind: This page’s reading. A judgement the page makes from the evidence. Others read it differently.
Review: A judgement, not a fact. Reviewed for whether it is fairly put and follows from the checked claims. No source can confirm it.

The page’s judgement on claim 41.

Claim 45 · checked 24 September 2026 · AI-assisted audit

What minds do

Claim: What minds do

Kind: Framing. Signposting. It asserts nothing a source could check.

Claim: It has been argued that human mathematicians see the truth of statements a formal system cannot prove, and therefore are not machines of this kind.

Kind: An argument. An argument a named thinker makes. That it is made is checked; whether it succeeds is argued.
Review: Source checked. The cited passage was read, and it says this.

Supports
Gödel’s Incompleteness Theoremssection 6.3
Cited · Link answered, 19 October 2026 · Read in full, 24 September 2026 · Reference work
The source reads: There have been repeated attempts to apply Gödel’s theorems to demonstrate that the powers of the human mind outrun any mechanism or formal system.

Claim 47 · checked 24 September 2026 · AI-assisted audit

Claim: Lucas put it in 1961: given any consistent machine capable of simple arithmetic, there is a formula it cannot produce as true but which we can see to be true.

Kind: An argument. An argument a named thinker makes. That it is made is checked; whether it succeeds is argued.
Review: Source checked. The cited passage was read, and it says this.

Supports
Minds, Machines and Gödelp. 113
Cited · Link answered, 19 October 2026 · Read in full, 24 September 2026 · Primary text
The source reads: given any machine which is consistent and capable of doing simple arithmetic, there is a formula which it is incapable of producing as being true ... but which we can see to be true

Claim 48 · checked 24 September 2026 · AI-assisted audit

Claim: Penrose later made very similar claims.

Kind: What scholars argue. A report of what specialists argue, or where they disagree.
Review: Source checked. The cited passage was read, and it says this.

Supports
Gödel’s Incompleteness Theoremssection 6.3
Cited · Link answered, 19 October 2026 · Read in full, 24 September 2026 · Reference work
The source reads: More recently, very similar claims have been put forward by Roger Penrose (1989, 1994).

Corrected

24 September 2026. Previously: “Penrose later argued something close to it at length.” Brought to what the source says.

Claim 49 · checked 24 September 2026 · AI-assisted audit

Claim: The specialists have largely rejected it.

Kind: What the field holds. The settled view of a field, as a review, a consensus statement or a reference work gives it. Not one study.
Review: Source checked. The cited passage was read, and it says this.

Supports
Gödel’s Incompleteness Theoremssection 6.3
Cited · Link answered, 19 October 2026 · Read in full, 24 September 2026 · Reference work
The source reads: These Gödelian anti-mechanist arguments are, however, problematic, and there is wide consensus that they fail.

Later work looked for

Crossref, 2016 on, "Lucas Penrose Gödel argument": a 2024 textbook chapter and several papers continue the debate; the entry, revised in October 2025, still reports wide consensus that the arguments fail.

Claim 50 · checked 24 September 2026 · AI-assisted audit

Claim: The standard reply, which goes back to Putnam, is that the theorem is conditional: the truth of a system’s Gödel sentence depends on that system being consistent, so the argument needs a mind that can always tell whether an arbitrary formal theory is consistent, and that is highly implausible.

Kind: An argument. An argument a named thinker makes. That it is made is checked; whether it succeeds is argued.
Review: Source checked. The cited passage was read, and it says this.

Supports
Gödel’s Incompleteness Theoremssection 6.3
Cited · Link answered, 19 October 2026 · Read in full, 24 September 2026 · Reference work
The source reads: this objection goes back to Putnam 1960 ... Gödel’s theorem has in reality a conditional form ... The anti-mechanist’s argument thus also requires that the human mind can always see whether or not a given formalized theory is consistent. However, this is highly implausible

Corrected

24 September 2026. Previously: “The standard reply, which goes back to Putnam, is that the theorem is conditional: the truth of a system’s Gödel sentence depends on that system being consistent, so the argument needs a mind that can always tell whether an arbitrary formal theory is consistent, and nobody thinks it can.” “Nobody thinks it can” is not so: Lucas and Penrose replied. The source’s own words are used, and claim 52 records the replies.

Claim 51 · checked 24 September 2026 · AI-assisted audit

Claim: Lucas and Penrose have both replied.

Kind: What scholars argue. A report of what specialists argue, or where they disagree.
Review: Source checked. The cited passage was read, and it says this.

Supports
Gödel’s Incompleteness Theoremssection 6.3
Cited · Link answered, 19 October 2026 · Read in full, 24 September 2026 · Reference work
The source reads: Lucas, Penrose and others have attempted to reply to such criticism (see, e.g., Lucas 1996; Penrose 1995, 1997).

Corrected

24 September 2026. Previously: “” A real disagreement is reported as one, with who holds the other side.

Claim 52 · checked 24 September 2026 · AI-assisted audit

Claim: That is not the same as the question being settled about minds. It is the argument from this theorem being settled.

Kind: This page’s reading. A judgement the page makes from the evidence. Others read it differently.
Review: A judgement, not a fact. Reviewed for whether it is fairly put and follows from the checked claims. No source can confirm it.

Follows from claims 50 to 52: what the consensus rejects is an inference from Gödel’s theorem, not a view about minds.

Claim 53 · checked 24 September 2026 · AI-assisted audit

What was read

On Computable Numbers, with an Application to the EntscheidungsproblemTuring, Proceedings of the London Mathematical Society · Proceedings of the London Mathematical Society s2-42 (1937), pp. 230-265, scan of the printed paper with its OCR text (copy at cs.virginia.edu); §§ 1, 8 and the introduction read closelyRead in full, 24 September 2026 · Copyright; quoted brieflyThe OCR has errors in symbols and machine names; excerpts avoid those runs or skip them with " ... ". Crossref dates the volume 1937; the paper was received and read in 1936.
Recursive Functions, section 3.4: Rice’s TheoremStanford Encyclopedia of Philosophy · Stanford Encyclopedia of Philosophy, revision of 1 March 2024Read in full, 24 September 2026 · Copyright; quoted briefly
Gödel’s Incompleteness TheoremsStanford Encyclopedia of Philosophy · Stanford Encyclopedia of Philosophy, revision of 8 October 2025Read in full, 24 September 2026 · Copyright; quoted brieflyLaTeX symbols in the saved text are skipped with " ... " in excerpts.
Minds, Machines and GödelLucas, Philosophy 36 · Lucas’s own online text of the paper (users.ox.ac.uk/~jrlucas/Godel/mmg.html), read through the Internet Archive; it marks the page breaks of Philosophy 36 in round bracketsRead in full, 24 September 2026 · Copyright; quoted brieflyThe Cambridge version was not reachable. Crossref’s record carries the same opening paragraph.
The Church-Turing ThesisStanford Encyclopedia of Philosophy · Stanford Encyclopedia of Philosophy, revision of 18 December 2023Read in full, 24 September 2026 · Copyright; quoted briefly
Turing MachinesStanford Encyclopedia of Philosophy · Stanford Encyclopedia of Philosophy, revision of 21 May 2025Read in full, 24 September 2026 · Copyright; quoted briefly
Alan TuringStanford Encyclopedia of Philosophy · Stanford Encyclopedia of Philosophy, revision of 30 September 2013Read in full, 24 September 2026 · Copyright; quoted briefly
Quantum theory, the Church–Turing principle and the universal quantum computerDeutsch, Proceedings of the Royal Society of London A · Abstract in the Crossref record for 10.1098/rspa.1985.0070Abstract only, 24 September 2026 · Copyright; quoted brieflyThe Royal Society refused this environment and no open copy was found.
Undecidability of the spectral gapCubitt, Perez-Garcia and Wolf, Nature · arXiv:1502.04135v3, the authors’ “short version”, which arXiv links to the Nature DOI; Nature’s typeset pages not readRead in full, 24 September 2026 · Copyright; quoted brieflyLigatures in the extracted text, and one word split across a line break, were written out plainly in the saved copy.