All these contributions involve proofs that the models are computationally equivalent to the Turing machine; such models are said to be . Because all these different attempts at formalizing the concept of "effective calculability/computability" have yielded equivalent results, it is now generally assumed that the Church–Turing thesis is correct. In fact, Gödel (1936) proposed something stronger than this; he observed that there was something "absolute" about the concept of "reckonable in S1":
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Informally the Church–Turing thesis states that if an (a procedure that terminates) exists then there is an equivalent , -definable function, or , for that algorithm. A more simplified but understandable expression of it is that "everything computable is computable by a Turing machine." Though not formally proven, today the thesis has near-universal acceptance.
Church-Turing thesis - Psychology Wiki
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On 13 March 2000, issued a set of to celebrate the greatest achievements of the twentieth century, one of which carries a recognisable portrait of Turing against a background of repeated 0s and 1s, and is captioned: "1937: Alan Turing's theory of digital computing".
theory - What is Turing Complete? - Stack Overflow
The Church-Turing thesis is credible because every singe model of computation that anyone has come up with so far has been proven to be equivalent to turing machines (well, or strictly weaker, but those aren't interesting here). Those models include
Turing Machine, Church-Turing Thesis, Turing Test ..
It would be quite implausible to suppose that the inchoate pre-theoretic cluster of ideas at the first level pins down anything very definite. No, the Church–Turing Thesis sensibly understood, in keeping with the intentions of the early founding fathers, is a view about the relations between concepts at the second and third level. The Thesis kicks in after some proto-theoretic work has been done. The claim is that the functions that fall under the proto-theoretic idea of an effectively computable function are just those that fall under the concept of a recursive function and under the concept of a Turing-computable function. NB: the Thesis is a claim about the extension of the concept of an effectively computable function.
ELI5: Church-Turing thesis and tests ..
The difference between the Church-Turing thesis and real theorems is that it seems impossible to formalize the Church-Turing thesis. Any such formalization would need to formalize what an arbitrary computable function is, which requires a model of computation to begin with. You can think of the Church-Turing thesis as a kind of meta-theorem which states
Talk:Church-Turing thesis - Esolang
The Church-Turing thesis asserts that the informal notion of a function that can be calculated by an (effective) algorithm is precisely the same as the formal notion of a recursive function. Since the prior notion is informal, one cannot give a formal proof of this equivalence. But one can present informal arguments supporting the thesis. For example, every known attempt at formally modeling this informal notion of computability has led to precisely the same class of recursive functions, whether it be via lambda-calculus,Post systems, Markov algorithms, combinatory logic, etc. This remarkable confluence lends strong support for the importance of this class of functions.