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Church’s Thesis for Turing Machine

In 1936, A method named as lambda-calculus was created by Alonzo Church in which the Church numerals are well defined, i.e. the encoding of natural numbers. Also in 1936, Turing machines (earlier called theoretical model for machines) was created by Alan Turing, that is used for manipulating the symbols of string with the help of tape.

Church Turing Thesis :

Turing machine is defined as an abstract representation of a computing device such as hardware in computers. Alan Turing proposed Logical Computing Machines (LCMs), i.e. Turing’s expressions for Turing Machines. This was done to define algorithms properly. So, Church made a mechanical method named as ‘M’ for manipulation of strings by using logic and mathematics. This method M must pass the following statements:

  • Number of instructions in M must be finite.
  • Output should be produced after performing finite number of steps.
  • It should not be imaginary, i.e. can be made in real life.
  • It should not require any complex understanding.

Using these statements Church proposed a hypothesis called

Church’s Turing thesis

that can be stated as: “The assumption that the intuitive notion of computable functions can be identified with partial recursive functions.”

Or in simple words we can say that “Every computation that can be carried out in the real world can be effectively performed by a Turing Machine.”

In 1930, this statement was first formulated by Alonzo Church and is usually referred to as Church’s thesis, or the Church-Turing thesis. However, this hypothesis cannot be proved. The recursive functions can be computable after taking following assumptions:

  • Each and every function must be computable.
  • Let ‘F’ be the computable function and after performing some elementary operations to ‘F’, it will transform a new function ‘G’ then this function ‘G’ automatically becomes the computable function.
  • If any functions that follow above two assumptions must be states as computable function.

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Church Turing Thesis in Theory of Computation

Theory of computation.

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In this article, we have explain the meaning and importance of Church Turing Thesis in Theory of Computation along with its applications and limitations.

Table of contents:

  • Introduction to Turing Church Thesis

Applications of Church Turing Thesis

Limitations of church turing thesis.

Prerequisites: Algorithms, Turing Machine

Let us get started with Church Turing Thesis in Theory of Computation.

Definition of Church Turing Thesis

Church Turing Thesis states that:

A computation process that can be represented by an algorithm can be converted to a Turing Machine.

In simple words, any thing that can be done by an Algorithm can be done by a Turing Machine as well. So, all algorithms can be implemented in a Turing Machine.

Specific Computation Models are equivalent which means any one model can be coverted to another model. These Computation Models include:

  • One tape Turing Machine
  • K tape Turing Machine where K >= 1
  • Non Deterministic Turing Machine
  • Programs in Programming Languages such as Java, C++, Lisp and others.

So, a program in C++ can be converted to a K tape Turing Machine and vice versa.

The applications of Church Turing Thesis are as follows:

  • Church Turing Thesis is used to define an Algorithm (traditionally)
  • Used in solving 10th Problem by Hilbert.
  • Used in defining modern computing devices including Molecular and Quantum Computers.

10th Problem by Hilbert

It has been used to solve the 10th Problem by Hilbert in 8th August 1900 at the Second International Congress of Mathematicians in Paris. These problems were listed as critical problems that should be solved for progress in Mathematics.

The 10th Problem by Hilbert was:

Does there exists a process with finite number of steps that can determine if a given polynomial with integer coefficients has integral roots?

Another way to look at the problem is to find if there is an Algorithm to find if there exists an integral root for a given polynomial or not.

For example: This is a Polynomial:

35x 10 y 2 z 9 + 11x 6 z 7 + 103xyz + 17y 31 z 3 = 0.

Is there an algorithm that can find if there exist a solution in integers?

Note the solution is not needed. Only we need to find if such a solution exists or not.

In 1970, it was proved that no such algorithm exists. This was done by Matiyasevich.

Algorithm = Church Turing Thesis

To solve the 10th Hilbert Problem, one needs to understand what is meant by an algorithm. In fact, there have been different definitions and all have proved to be equivalent. Some definitions were:

  • 1936: Algorithm = Turing Machine
  • 1936: Algorithm = Lambda Calculus
  • 1970+: Algorithm = Implementation in Programming Languages like C and Lisp
  • Final: Algorithm = process converted to Turing Machine.

Finally, it was agreed that an Algorithm is based on Church Turing Thesis which said:

"Any computational process can be considered as an Algorithm if it can be converted to a Turing Machine." Note: This does not hold true as of now.

Modern Computing Devices

Traditional Computers which are in use today, are limited by Church Turing Thesis. This is because Church Turing Thesis defines an Algorithm which can be implemented in a real system.

Therefore, the Computing Device you are using is basically a Turing Machine.

The only difference is that Computing Devices are efficient while Turing Machine is inefficient. Theoretically, from a point of view of algorithms, there is no difference.

There are 3 different approaches future computers may take:

  • Quantum Computer : Solve Computing Problems using atoms by quantum rules. This is an active area of research.
  • Molecular Computer : Solve Computing Problems using Molecules by taking advantage of Physical laws of Moleculars. This includes replicating the idea of DNA.
  • Super Recursive Algorithm : This domain has not been realized yet and exists in theory but this is the part where Church Turing Thesis fail. We have covered this in the next section on "Limitations of Church Turing Thesis".

Two different futuristic models of Computer which follows Church Turing Thesis:

  • Quantum Computers can be represented as Non Deterministic Turing Machine
  • Molecular Computers can be represented by Turing Machine with many tapes and heads

Therefore, Quantum and Molecular Computers are same fundamentally and they are only more efficient than Mechnical Computers.

Super Recursive Algorithms proved Church Turing Thesis wrong. The first Super Recursive algorithm was introduced in 1965 by Mark Gold and Hillary Putnam by using ideas of limit recursive and limit partial recursive functions. It was based on ideas from non standard analysis by Abraham Robinson in 1966 and Inductive Definition of sets by Spector in 1959. This resulted in Inductive Inference by Gasarch and Smith in 1997 and is used in Machine Learning.

Super Recursive Algorithms can solve problems that are unsolvable by Turing Machines. To account for this, a new idea was introduced: Inductive Turing Macine. These were not accepted as Algorithm for a long time as it was refuting Church Turing Thesis and Godel Incompleteness Theorem (as proved in 1987 by Burgin).

The idea of Inductive Turing Machine is as follows:

  • Turing Machine has a property that it stops after giving a result.
  • Most programs stop after giving a result and this seems to be reasonable as what a program should do once it has found the answer.
  • Operating Systems are also programs but it does not give a standard output. It gives some strings to the users during its use but it cannot be considered as a output. The functionality of an Operating System is considered to be the output. It does not stop like standard program. If it stops, it cannot give any output.
  • There can be programs which give a result at the moment which is good enough but
  • This idea of not stopping after giving a result is the basis.

Inductive Turing Machine is more powerful than Conventional Turing Machine. Inductive Turing Machine can solve the Halting Problem which is known to be unsolvable by Conventional Turing Maching.

There are different types of Inductive Turing Machine:

  • Inductive Turing Machine + Structured Memory
  • Inductive Turing Machine + Structured Rules (control device)
  • Inductive Turing Machine + Structured Head (Operating Device)

Today, Church Turing Thesis is not considered as an Universal Principle. Inductive Turing Machine is the most powerful super recursive algorithm.

This lead to the formulation of "Extended Church Turing Thesis".

There are three open questions:

  • How to realize Super Recursive algorithms in technological devices?
  • How modern computing devices are related to Super Recursive Algorithm?
  • What are the new possibilities with Super Recursive Algorithm?

Think about these research open ended problems in Theory of Computation.

With this article at OpenGenus, you must have the complete idea of Church Turing Thesis in Theory of Computation.

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church turing thesis in hindi

Church-Turing Thesis

The Church-Turing thesis (formerly commonly known simply as Church's thesis) says that any real-world computation can be translated into an equivalent computation involving a Turing machine . In Church's original formulation (Church 1935, 1936), the thesis says that real-world calculation can be done using the lambda calculus , which is equivalent to using general recursive functions .

The Church-Turing thesis encompasses more kinds of computations than those originally envisioned, such as those involving cellular automata , combinators , register machines , and substitution systems . It also applies to other kinds of computations found in theoretical computer science such as quantum computing and probabilistic computing.

There are conflicting points of view about the Church-Turing thesis. One says that it can be proven, and the other says that it serves as a definition for computation. There has never been a proof, but the evidence for its validity comes from the fact that every realistic model of computation, yet discovered, has been shown to be equivalent. If there were a device which could answer questions beyond those that a Turing machine can answer, then it would be called an oracle .

Some computational models are more efficient, in terms of computation time and memory, for different tasks. For example, it is suspected that quantum computers can perform many common tasks with lower time complexity , compared to modern computers, in the sense that for large enough versions of these problems, a quantum computer would solve the problem faster than an ordinary computer. In contrast, there exist questions, such as the halting problem , which an ordinary computer cannot answer, and according to the Church-Turing thesis, no other computational device can answer such a question.

The Church-Turing thesis has been extended to a proposition about the processes in the natural world by Stephen Wolfram in his principle of computational equivalence (Wolfram 2002), which also claims that there are only a small number of intermediate levels of computing power before a system is universal and that most natural systems are universal.

This entry contributed by Todd Rowland

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Rowland, Todd . "Church-Turing Thesis." From MathWorld --A Wolfram Web Resource, created by Eric W. Weisstein . https://mathworld.wolfram.com/Church-TuringThesis.html

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Truth, Existence and Explanation pp 225–248 Cite as

Church-Turing Thesis, in Practice

  • Luca San Mauro 6  
  • First Online: 25 October 2018

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Part of the Boston Studies in the Philosophy and History of Science book series (BSPS,volume 334)

We aim at providing a philosophical analysis of the notion of “proof by Church’s Thesis”, which is – in a nutshell – the conceptual device that permits to rely on informal methods when working in Computability Theory. This notion allows, in most cases, to not specify the background model of computation in which a given algorithm – or a construction – is framed. In pursuing such analysis, we carefully reconstruct the development of this notion (from Post to Rogers, to the present days), and we focus on some classical constructions of the field, such as the construction of a simple set. Then, we make use of this focus in order to support the following encompassing claim (which opposes to a somehow commonly received view): the informal side of Computability, consisting of the large class of methods typically employed in the proofs of the field, is not fully reducible to its formal counterpart.

  • Church-Turing Thesis
  • Informal Methods
  • Informal Algorithm
  • Acceptable Number

These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

The author was partially supported by the Austrian Science Fund FWF through project P 27527.

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A classical introduction to CTT can be found in Kleene ( 1952 ). See also Church ( 1936 ), Turing ( 1936 , 1948 ), Post ( 1936 ), and Gödel ( 1946 ). Soare ( 1987b ) contains, in its first part, an accurate reconstruction of the role of Turing’s work in the acceptance of the thesis. For recent philosophical work concerning CTT, the reader is referred, for instance, to Olszewski et al. ( 2006 ).

The standard interpretation is that CTT is indeed a thesis , or, in Post’s words, “a working hyphotesis” (Post 1936 ). That is to say, something that cannot be subject of a mathematical proof. Yet, it has been argued that CTT has not necessarily an hypothetical status, but rather that it can be susceptible of a rigorous mathematical proof, or even that such a proof is already contained in Turing ( 1936 ) (for this line of thought see, e.g., Mendelson ( 1990 ), Gandy ( 1988 ), Sieg ( 1994 ), and the discussion contained in Shapiro ( 2006 )). Responses to this latter position can be found in Folina ( 1998 ) and Black ( 2000 ).

See Welch ( 2007 ) for a rich survey on models of transfinite computation. On the other hand, Davis ( 2006 ) denies any theoretical significance to “hypercomputationalism” as such.

For instance, the following is the first proof-theoretic reference to CTT in Rogers ( 1967 ):

There are exactly ℵ 0 partial recursive functions, and there are exactly ℵ 0 recursive functions.

All constants functions are recursive, by Church’s Thesis. Hence there are at least ℵ 0 recursive functions. The Gödel numbering shows that there are at most ℵ 0 partial recursive functions.

To be fair, Post mainly speaks of computably enumerable sets, there introduced for the first time. But since, by definition, a set is computably enumerable if it is the range of a computable function, then one can trivially translate Post’s formulations in instances of our prototype.

It is worth noticing that the Leibnizian ideal is by no means archeological. Quite to the contrary. Hacking reports Voevodsky’s opinion that “in a few years, journal will accept only articles accompanied by their machine-verifiable equivalents”. More generally – and less radically – research on proof-assistants can be (partially) motivated as a way of improving automatic verification of proofs.

For an accurate reconstruction of Post’s thought see De Mol ( 2006 )

From now on, in describing the practical side of CTT, we will mostly refer to textbooks. This is a natural choice. Since, as already said, there are no philosophical studies concerning the practice of Computability, the most immediate source of observations regarding how such practice has to be intended comes from the kind of expository remarks that abound in books such as Rogers’.

The interested reader can consult Mancosu ( 2008 ) for an anthology of papers in Philosophy of Mathematical Practice.

\(\mathbb {I}\) does clearly correspond to a pre-theoretic object whose formalization would be far from trivial. For instance, there could be a worry concerning a sort of Berry-like paradox, inasmuch we admit a too relaxed notion on what counts as an informal description for an algorithm. Nonetheless, we can suppose to deal with sufficiently clear descriptions. This is because, although border-cases cannot arguably be expunged, we are more interested, as we will see, in a somewhat global tendency.

All of this is of course related to the philosophical problem of determining if one can possibly formulate a definition for algorithms that would be correct in the sense of Shore: “Find, and argue conclusively for, a formal definition of algorithm and the appropriate analog of the Church-Turing thesis. Here we want to capture the intuitive notion that, for example, two particular programs in perhaps different languages express the same algorithm, while other ones that compute the same function represent different algorithms for the function. Thus we want a definition that will up to some precise equivalence relation capture the notion that two algorithms are the same as opposed to just computing the same function”(Buss et al. 2001 ). See also Dean ( 2007 ) for a rich discussion on whether algorithms can be fairly regarded as abstract mathematical objects.

Indeed, Blass et al. ( 2009 ) argue that “one cannot give a precise equivalence relation capturing the intuitive notion of ‘the same algorithm.”’

The reader is referred to Rogers ( 1967 ) for the proof of this fact, and to Odifreddi ( 1989 ) for additional results concerning numberings.

Some equivalence between classical models of computation can be found in Odifreddi ( 1989 ).

For a classical defense of structuralism in philosophy of mathematics, the reader is referred to Resnik ( 1997 ).

Two noteworthy exception being Carter ( 2008 ) and McLarty ( 2008 ).

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Church, A. 1936. An unsolvable problem of elementary number theory. American Journal of Mathematics 58(2): 345–363.

Davis, M. 2006. Why there is no such discipline as hypercomputation. Applied Mathematics and Computation 178(1): 4–7.

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Acknowledgements

A preliminary version of this paper appeared as a chapter of my PhD thesis. I would like to thank my supervisors, Gabriele Lolli and Andrea Sorbi, for their guidance and support. I have presented this work at several conferences. In particular, I am grateful to the participants of APMP 2014, in Paris, and of FilMat 2016, in Chieti, for their comments. Finally, Richard Epstein’s remarks were fundamental in rethinking the organization of the present material.

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Mauro, L.S. (2018). Church-Turing Thesis, in Practice. In: Piazza, M., Pulcini, G. (eds) Truth, Existence and Explanation. Boston Studies in the Philosophy and History of Science, vol 334. Springer, Cham. https://doi.org/10.1007/978-3-319-93342-9_13

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What is The Church-Turing Thesis in TOC?

The Church-Turing thesis says that every solvable decision problem can be transformed into an equivalent Turing machine problem.

It can be explained in two ways, as given below −

The Church-Turing thesis for decision problems.

The extended Church-Turing thesis for decision problems.

Let us understand these two ways.

The Church-Turing thesis for decision problems

There is some effective procedure to solve any decision problem if and only if there is a Turing machine which halts for all input strings and solves the problem.

The extended Church-Turing thesis for decision problems

A decision problem Q is said to be partially solvable if and only if there is a Turing machine which accepts precisely the elements of Q whose answer is yes.

A proof by the Church-Turing thesis is a shortcut often taken in establishing the existence of a decision algorithm.

For any decision problem, rather than constructing a Turing machine solution, let us describe an effective procedure which solves the problem.

The Church-Turing thesis explains that a decision problem Q has a solution if and only if there is a Turing machine that determines the answer for every q ϵ Q. If no such Turing machine exists, the problem is said to be undecidable.

Bhanu Priya

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Supplement to The Church-Turing Thesis

The rise and fall of the entscheidungsproblem, a. stating the entscheidungsproblem, b.1 a “philosophers’ stone”, b.2 the consistency of mathematics, c. partial solutions, d. negative vibes, e. the church-turing result, f. aftermath.

Turing gave two formulations of what he called “the Hilbert Entscheidungsproblem ” (1936 [2004: 84, 87]):

[Is there a] general process [and a few paragraphs later, he emphasizes “general (mechanical) process”] for determining whether a given formula \(A\) of the functional calculus K is provable
[Is there a] machine [Turing machine] which, supplied with any one \(A\) of these formulae, will eventually say whether \(A\) is provable.

Given Turing’s thesis, the two formulations are equivalent.

Church stated the Entscheidungsproblem more generally:

By the Entscheidungsproblem of a system of symbolic logic is here understood the problem to find an effective method by which, given any expression Q in the notation of the system, it can be determined whether or not Q is provable in the system. (Church 1936b: 41)

While Turing and Church both formulated the Entscheidungsproblem in terms of determining whether or not a formula is provable , Hilbert and Ackermann had formulated it in terms of universal validity (“ allgemeingültigkeit ”):

The Entscheidungsproblem is solved once we know a procedure that allows us to decide, by means of finitely many operations, whether a given logical expression is universally valid or, alternatively, satisfiable. (Hilbert & Ackermann 1928: 73)

A universally valid formula of the functional calculus (e.g., \((x)(Fx \lor \neg Fx)\) or, in modern notation, \(\forall x(Fx \lor \neg Fx)\)) must contain neither free variables nor individual constants, and is such that a true assertion results no matter which replacements (of suitable adicity) are made for the formula’s predicate symbols, and no matter which objects are allocated to its variables (Hilbert & Ackermann 1928: 72–73). In the case of a satisfiable formula (e.g., \((Ex)Fx\) or, in modern notation, \(\exists xFx\)) there must be at least one way of replacing its predicate symbols and of allocating objects to its variables so that a true assertion results. Universal validity and satisfiability are related as follows: \(A\) is universally valid if and only if \(\neg A\) is not satisfiable (1928: 73). In the above quotation, therefore, Hilbert and Ackermann are presenting two different but equivalent forms of the Entscheidungsproblem , one employing the concept of universal validity (or simply validity, as we would say today) and the other the closely related concept of satisfiability. The hunt was on for what they called “a determinate, finite procedure” (1928: 17) for deciding, of any given formula \(A\) of the functional calculus, whether or not \(A\) is universally valid, or, equivalently, for deciding whether or not \(\neg A\) is satisfiable.

Turing’s and Church’s provability formulation of the Entscheidungsproblem and the Hilbert-Ackermann formulation in terms of validity are in fact logically equivalent, as Church noted in 1936 (1936b: 41). This equivalence is a consequence of Gödel’s proof that (where \(A\) is any formula of the functional calculus) if \(A\) is universally valid then \(A\) is provable in the calculus. (The proof was presented originally in Gödel’s 1929 doctoral dissertation and then published as Gödel 1930.) Nevertheless, the provability version of the Entscheidungsproblem is arguably superior, since it asks a question about a specific axiom system, as do the allied problems of consistency and completeness. In modern treatments, the problems of consistency, completeness and decidability for an axiom system lie at the heart of proof theory, the area of modern logic founded by Hilbert.

In 1928, Hilbert and Ackermann were certainly aware of the provability formulation (1928: 86, and see below) but they gave the validity formulation the starring role. It was von Neumann who emphasized the provability formulation, calling the process of proof the “real heart of mathematics” (von Neumann 1927: 10).

B. Why the problem mattered

Why was it that the Entscheidungsproblem was regarded as “the main problem of mathematical logic” and “the most general problem of mathematics”? There were fundamentally two reasons.

In his turn-of-the-century Paris lecture, Hilbert explained the idea of axiomatizing a subject-matter and using provability from the axioms as the touchstone of truth:

When we are engaged in investigating the foundations of a science, we must set up a system of axioms which contains an exact and complete description of the relations subsisting between the elementary ideas of that science…. [N]o statement within the realm of the science whose foundation we are testing is held to be correct unless it can be derived from those axioms by means of a finite number of logical steps. (Hilbert 1900: 264 [trans. 1902: 447])

He also famously said in his lecture that there is “no ignorabimus ” in mathematics—there is nothing that we shall not know (Hilbert 1900: 262). It was a phrase he would often use to express his “conviction of the solvability of every mathematical problem” (Hilbert 1900: 262 [trans. 1902: 445]). Thirty years later he reiterated the same standpoint, in a valedictory lecture in Königsberg:

[I]n my opinion, there is no such thing as an unsolvable problem. On the contrary, instead of foolish ignorabimus, our slogan is: We must know, We will know. (Hilbert 1930b: 963)

It was Hilbert’s great invention, proof theory, that would, he thought, supply the basis for answering all “meaningful questions”:

[O]n the basis of the proof theory I have outlined, every fundamental question can be answered in a way that is unambiguous and mathematically precise. (Hilbert 1930a: 8, 9)

However, proving a statement in an axiom system can be a tricky matter. Suppose the mathematician fails to discover a derivation of the statement from the axioms, what follows then? The failure could be because the statement is indeed not provable—but, on the other hand, there might be a way to prove it that escaped the mathematician’s attention. Success in finding a proof brings certainty, whereas failure leaves us uncertain. That is why a decision method is so desirable. The method enables the mathematician to determine whether or not the statement is provable. Thinking up proofs is a serendipitous activity that involves, as Behmann put it, “inventive insight”, “creative thought”, and searching “in every direction of the compass” (Behmann 1921: 2–3 [trans. 2015: 174]). Whereas a decision method is a purely mechanical process that is guaranteed to produce a definite answer.

Herbrand wrote of the “present great ambition, the solution of the Entscheidungsproblem ”, saying this “would provide a general method in mathematics” and enable us to “to decide with certainty whether a proposition is true in a given theory” (Herbrand 1930b: 254–255). In his 1921 lecture, Behmann had referred to this desired general method as a “philosophers’ stone” (“ Stein der Weisen ”), by means of which mathematicians “would be able to solve every problem posed ”—or even “delegate the work of proving mathematical theorems to mathematical laborers” (Behmann 1921: 3–4 [trans. 2015: 175], emphasis added). Turing’s mentor Newman, familiar of course with the work of the Hilbert group, also referred to the solution of the Entscheidungsproblem as a philosophers’ stone. Hilbert and Ackermann themselves said, in 1928:

Solving this general Entscheidungsproblem [for the second-order functional calculus] would … enable us to decide on the provability or unprovability of any mathematical proposition , at least in principle. (Hilbert & Ackermann 1928: 86, emphasis added)

Schönfinkel went even further. He was another member of the Göttingen group who praised “Leibniz’s bold conjectures” (Schönfinkel 192?: 2). (For an excellent biographical article on Schönfinkel, see Wolfram 2021.) In an early draft of what would become Bernays and Schönfinkel (1928), Schönfinkel wrote of “the great Entscheidungsproblem of mathematical logic” which, thanks to Hilbert, mathematicians now had “the courage and the boldness to dare to touch”. He described the Entscheidungsproblem as “the problem of ‘solving all problems’”—not just all mathematical problems but all problems. Because once there is a method for solving all mathematical problems:

thereafter everything else is indeed lightweight, once this “Gordian knot” is undone (since “the world is written in mathematical characters”). (Schönfinkel 192?: 1–2)

The second reason the Entscheidungsproblem was regarded as so important was its connection with the quest for proofs of the consistency of mathematical systems. A system is said to be inconsistent if there is some statement \(A\) such that both \(A\) and \(\neg A\) are provable from the axioms. The system is consistent if (and only if) there is no statement for which this sorry situation is the case.

By the early twentieth century, mathematics—until then a “paragon of reliability and truth”, Hilbert said—was in trouble (Hilbert 1926: 170, trans. in van Heijenoort 1967: 375). Its reputation had been “lost due to the paradoxes of set theory” (Hilbert 1922: 160). Hilbert explained:

[C]ontradictions appeared, sporadically at first, then ever more severely and ominously…. In particular, a contradiction discovered by Zermelo and Russell [“Russell’s paradox”] had, when it became known, a downright catastrophic effect in the world of mathematics. (Hilbert 1926: 169 [trans. 1967: 375])

Herbrand takes up the story:

One must take great care … Set theory has produced famous examples [of inconsistencies] … But there is nothing to show us that similar issues do not arise for the theories that seem to us the most finished, like arithmetic. Could it be that arithmetic is inconsistent? (Herbrand 1930b: 250–251)

“[P]roofs of consistency would be very useful to dispel lingering doubts”, wrote Herbrand (1930b: 251). Hilbert put it even more forcefully:

[T]he situation in which we presently find ourselves with respect to the paradoxes is in the long run intolerable. (Hilbert 1926: 170 [trans. 1967: 375])

Proving “the consistency of the arithmetic axioms” is “urgent”, he said—“a burning question” (Hilbert 1926: 179 [trans. 1967: 383]; Hilbert 1930a: 3). Hilbert’s overarching concern was to “re-establish mathematics’ old reputation for incontestable truth” (Hilbert 1922: 160), and he was “convinced that it must be possible to find a direct proof of the consistency of the arithmetical axioms” (Hilbert 1900: 265).

A solution to the Entscheidungsproblem would furnish a route to establishing consistency: “By means of the decision method, issues of consistency would also be able to be resolved” (Hilbert & Ackermann 1928: 76). This is because the decision method could be used to establish whether or not \(A\) and \(\neg A\) are both provable—so gaining a definite answer to the question “Is the system consistent?”. Hilbert believed that, following the discovery of the decision method, arithmetic and analysis would be proved consistent, thereby “establish[ing] that mathematical propositions are indeed incontestable and absolute truths” (Hilbert 1922: 162). Mathematics’ reputation would be regained.

By the time Turing and Church engaged with the Entscheidungsproblem , a number of decision methods were known for parts of the functional calculus.

Besides the previously mentioned Hilbert-Bernays decision method for the propositional calculus (and Peirce’s much less well-known method), and also the Löwenheim method for the monadic fragment of the functional calculus (later improved by Behmann in his 1922 paper, where it was additionally proved that the monadic fragment of the second-order functional calculus is decidable), there were various decision methods that succeeded providing the functional calculus was restricted in one way or another. Bernays and Schönfinkel (1928) showed there is a decision method when formulae containing at most two individual variables are considered. At Cambridge, Ramsey devised a method that worked provided existential quantifiers were omitted from the calculus, and any universal quantifiers were stacked one after another at the very beginning of a formula (with no negation sign preceding any of them, and with their scope extending to the end of the formula). Such formulae are interesting, Ramsey pointed out, since they represent “general laws” (Ramsey 1930: 272; Langford 1926b: 115–116). Ackermann, Bernays, Schönfinkel, Gödel, Kálmar, and Herbrand devised methods for other fragments of the calculus, in which only certain patterns of quantifiers were permitted. Gödel’s 1933 paper on the Entscheidungsproblem gave a summary of some of the developments.

Despite all this attention to the decision problem, no method had been found for the full functional calculus. But according to Hilbert it was just a matter of time. He and Ackermann emphasized that “finding a general decision process is an as yet unsolved and difficult problem” (1928: 77). They exuded optimism, writing buoyantly of moving “closer to the solution of the Entscheidungsproblem ” (1928: 81).

Even in the 1920s, however, negative opinion on the solvability of the Entscheidungsproblem was building. Behmann had pointed out in his 1921 lecture that, once the “philosophers’ stone” was found, “the entirety of mathematics would indeed be transformed into one enormous triviality ” (Behmann 1921: 5 [trans. 2015: 175]). Some found this consequence highly unpalatable. In 1927, Hilbert’s student Weyl insisted that “such a philosopher’s stone has not been discovered and never will be discovered”, and a good job too, Weyl thought, since if it were, “Mathematics would thereby be trivialized” (Weyl 1927: 20 [trans. Weyl 1949: 24]). Then a year later, in 1928, Hardy announced to the Cambridge mathematicians, assembled at his Rouse Ball Lecture, that he expected no “system of rules which enable us to say whether any given formula [is] demonstrable or not” (Hardy 1929: 16). “[T]his is very fortunate”, he continued, since if there were such a system,

we should have a mechanical set of rules for the solution of all mathematical problems, and our activities as mathematicians would come to an end. (ibid.)

Hardy explained that his description of Hilbert’s ideas was “based upon that of v. Neumann, a pupil of Hilbert’s”, saying that he found von Neumann’s exposition “sharper and more sympathetic than Hilbert’s own” (Hardy 1929: 13–14). Hardy was referring to von Neumann’s “Zur Hilbertschen Beweistheorie” ( On Hilbert’s Proof Theory ), published in 1927 but completed by the middle of 1925. Von Neumann said:

So it seems that there is no way to find the general decision criterion for whether a given normal formula [i.e., a well-formed formula with no free variables] is provable. (von Neumann 1927: 11, emphasis added)
The day that undecidability lets up, mathematics in its current sense would cease to exist; into its place would step a perfectly mechanical rule, by means of which anyone could decide, of any given proposition, whether this can be proved or not. (von Neumann 1927: 12)

But how to prove that there is no general decision criterion? Von Neumann confessed he did not know:

At present, of course, we cannot demonstrate this. Moreover, no clue whatsoever exists how such an undecidability proof would have to be conducted. (von Neumann 1927: 11)

Nor did he find a proof. With a hint of resignation he said in 1931 that the Entscheidungsproblem was “far too difficult” (1931: 121). Four years later, in 1935, he visited Cambridge for a term (arriving in April, a few weeks after Newman had lectured on the Entscheidungsproblem ) and he struck up an acquaintance with a young mathematician who admired his work (Copeland & Fan 2023). The young man was of course Alan Turing, who within a few months would devise his groundbreaking proof that—just as von Neumann had hypothesized—“It is generally undecidable whether a given normal formula is provable or not” (von Neumann 1927: 12). Was there discussion, during the spring of 1935, between von Neumann and Turing (the latter already primed by Newman’s lecture) about the Entscheidungsproblem —which was, after all, the main problem of mathematical logic? Did von Neumann perhaps play a role in Turing’s decision to take on this fundamental problem and prove it unsolvable? These are tantalizing questions, and we may never know for sure.

Gödel’s famous incompleteness theorems of 1931 placed unexpected new obstacles in the way of Hilbert’s desired consistency proof for arithmetic (Gödel 1931). Suspicion also began to build that Gödel’s incompleteness results might further imply the unsolvability of the Entscheidungsproblem . Herbrand said cautiously:

[A]lthough at present the possibility of a solution to the Entscheidungsproblem seems unlikely, its impossibility has not yet been proved. (Herbrand 1931a: 56)

Carnap, who took a close interest in Hilbert’s Göttingen group and had worked with Behmann (Schiemer, Zach, & Reck 2017) wrote the following about the Hilbertian idea of a “definite criterion of validity” or “decision procedure for mathematics”, using which “we could so to speak calculate the truth or falsity of every given proposition”:

[B]ased on Gödel’s latest results, the search for a definite criterion of validity for the entire system of mathematics appears hopeless. (Carnap 1935: 163–4)

But, as Herbrand noted, nobody had proved the Entscheidungsproblem to be unsolvable. That was where Turing and Church entered the story. Newman later summed up matters as they had appeared at the time, before Church and Turing published their transformational papers in 1936:

A first blow was dealt [to the “Hilbert decision-programme”] by Gödel’s incompleteness theorem (1931), which made it clear that truth or falsehood of \(A\) could not be equated to provability of \(A\) or not-\(A\) in any finitely based logic, chosen once for all; but there still remained in principle the possibility of finding a mechanical process for deciding whether \(A\), or \(\neg A\), or neither, was formally provable in a given system. (Newman 1955: 256)

Turing summed up his show-stopping result in his usual terse way: he had shown, he said, that “the Hilbert Entscheidungsproblem can have no solution” (Turing 1936 [2004: 84]). Church, also not one to waste words, compressed his proof into a paper barely two pages long, and concluded pithily:

The general case of the Entscheidungsproblem of the engere Funktionenkalkül is unsolvable. (Church 1936b: 41)

In establishing this, Turing and Church showed moreover that there can be no decision method for the second-order functional calculus (where quantification is permitted over not just individuals but also over properties and relations), since the second-order calculus contains the first-order calculus as a fragment. The same applies to every other mathematical system containing the first-order calculus, such as arithmetic.

However, it is one of the great ironies of the history of logic that, all along, Gödel’s first incompleteness theorem of 1931 did in fact suffice to establish the undecidability of the functional calculus—although this was certainly not apparent at the time. More than three decades passed after the publication of Gödel’s paper before this corollary of his theorem was noted, by Davis (Davis 1965: 109; Kleene 1986: 136).

Naturally, this unnoticed implication of Gödel’s theorem does not diminish Turing’s and Church’s great achievement, which at the time broke new ground. However, in recent times their work is sometimes regarded as amounting to merely a smallish development of Gödel’s previously published results, on which Church’s and Turing’s work was supposedly based. This is particularly true of Turing. Turing, it is said, “merely reformulated Gödel’s work in an elegant way” (Schmidhuber 2012: 1638) and “recast” Gödel’s findings “in the guise of the Halting Problem” (Dawson 2006: 133). (For further discussion of these and similar views, see Copeland & Fan 2022.) In this connection, it is worth remembering the words of Kleene, who worked closely with Church and played a leading part in the development of computability theory in the 1930s. Kleene noted that

One sometimes encounters statements asserting that Gödel’s work laid the foundation for Church’s and Turing’s results

and commented:

Whether or not one judges that Church would have proceeded from his thesis to these [undecidability] results without his having been exposed to Gödel numbering, it seems clear that Turing in [“On Computable Numbers”] had his own train of thought, quite unalloyed by any input from Gödel. One is impressed by this in reading Turing [“On Computable Numbers”] in detail. (Kleene 1987: 491–2)

What became of the Entscheidungsproblem after the Church-Turing negative result?

In letters written in the wake of the result, Turing and Newman discussed the idea Newman had presented in his 1935 lecture, a machine that is able to decide mathematical problems. Turing wrote “I think you take a much more radically Hilbertian attitude about mathematics than I do”, responding to Newman’s statement that

If all this whole formal outfit is not about finding proofs which can be checked on a machine it’s difficult to know what it is about. (Turing c.1940 [2004: 215])

Turing noted that he saw no essential difference between a proof-checking machine and a proof-finding machine. He challenged Newman:

When you say “on a machine” do you have in mind that there is (or should be or could be, but has not been actually described anywhere) some fixed machine on which proofs are to be checked, and that the formal outfit is, as it were about this machine?

Turing called this an “extreme Hilbertian” position, and said:

If you take this attitude … there is little more to be said: we simply have to get used to the technique of this machine and resign ourselves to the fact that there are some problems to which we can never get the answer.

Turing rejected this attitude to mathematics, because, he said, “there is a fairly definite idea of a true formula which is quite different from the idea of a provable one”—mathematicians’ judgements about whether formulae are true can outrun the pronouncements of the Hilbertian machine. Turing continued in his letter:

If you think of various machines I don’t see your difficulty. One imagines different machines allowing different sets of proofs, and by choosing a suitable machine one can approximate “truth” by “provability” better than with a less suitable machine … (Turing c.1940 [2004: 215])

In Turing’s opinion, that was how the debate on the Entscheidungsproblem had panned out. The Hilbertians had wanted a single decision method for the whole of mathematics—a single computing machine or algorithm—whereas Turing showed there can be no such thing; but he emphasized that there are, nevertheless, different decision methods (machines) for different areas of mathematics (see further Copeland & Shagrir 2013). In place of the one great unsolvable decision problem, there are many lesser, but often solvable, decision problems.

In the long aftermath of the Church-Turing result, as those rough pioneering days gave way to modern computer science, Turing’s opinion became the mainstream view: Today, computer science makes use of a multiplicity of algorithms for deciding different parts of the functional calculus and other mathematical systems.

Copyright © 2023 by B. Jack Copeland < jack . copeland @ canterbury . ac . nz >

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