Looking at this today, it seems sort-of obvious. So he created a really clever mechanism for numerical encoding. The Principia logic is minimal and a bit cryptic, but it was built for a specific purpose: to have a minimal syntax, and a complete but minimal set of axioms. The whole idea of the Principia logic is to be purely syntactic.

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Turing machines are defined by sets of rules that operate on four parameters: state, tape cell color, operation, state. Let the states and tape cell colors be numbered and represented by quadruples of ordinal numbers. Then there exist algorithmic procedures that sequentially list all consistent sets of Turing machine rules.

A set of rules is called consistent if any two quadruples differ in the first or second element out of the four. Any such procedure gives both an algorithm for going from any integer to its corresponding Turing machine and an algorithm for getting the index of any consistent set of Turing machine rules. Assume that one such procedure is selected. The result of application of Turing machine with Godel number to is usually denoted. Hence, by Cantor's theorem , there exist functions which are not recursive.

Determining the convergence of is also recursively undecidable. For example, the statement that reads "there exists an such that is the immediate successor of " can be coded.

Portions of this entry contributed by Alex Sakharov author's link. Davis, M. Computability and Unsolvability. New York: Dover Hofstadter, D. New York: Vintage Books, p. Kleene, S. Mathematical Logic. New York: Dover, Rogers, H. Theory of Recursive Functions and Effective Computability.

Wolfram, S. A New Kind of Science. Champaign, IL: Wolfram Media, pp. Sakharov, Alex and Weisstein, Eric W. Explore thousands of free applications across science, mathematics, engineering, technology, business, art, finance, social sciences, and more. Walk through homework problems step-by-step from beginning to end.

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## Gödel numbering

Turing machines are defined by sets of rules that operate on four parameters: state, tape cell color, operation, state. Let the states and tape cell colors be numbered and represented by quadruples of ordinal numbers. Then there exist algorithmic procedures that sequentially list all consistent sets of Turing machine rules. A set of rules is called consistent if any two quadruples differ in the first or second element out of the four. Any such procedure gives both an algorithm for going from any integer to its corresponding Turing machine and an algorithm for getting the index of any consistent set of Turing machine rules. Assume that one such procedure is selected. The result of application of Turing machine with Godel number to is usually denoted.

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## Gödel number

These sequences of natural numbers can again be represented by single natural numbers, facilitating their manipulation in formal theories of arithmetic. The numbers involved might be very long indeed in terms of number of digits , but this is not a barrier; all that matters is that we can show such numbers can be constructed. Clearly there are many ways this can be done. He first assigned a unique natural number to each basic symbol in the formal language of arithmetic with which he was dealing. This allowed him to show a correspondence between statements about natural numbers and statements about the provability of theorems about natural numbers, the key observation of the proof.