A key method in the usual proofs of the first incompleteness theorem is the arithmetization of the formal language, or Gödel numbering: certain natural numbers. Gödel Number. DOWNLOAD Mathematica Notebook. Turing machines are defined by sets of rules that operate on four parameters: (state, tape cell color. Gödel’s numbering system is a way of representing any sentence of the formal language as a number. That means that every sentence of the formal language.
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Gödel numbering for sequences – Wikipedia
Of course, the embedding is surely possible set theoretically, but the emphasis is on the effectiveness of the functions manipulating such representations of sequences: For example, recursive function theory can be regarded as a formalization of the notion of an algorithmand if we regard it as a programming languagewe can mimic lists by encoding a sequence of natural numbers in a single natural number.
To achieve this, we can use various number theoretic ideas; using the fundamental theorem of arithmetic is a straightforward way, but there are also more economic approaches, such as using the pairing function combined with the Chinese remainder theorem in a sophisticated numberihg. For example, we can encode Markov algorithms or Turing machines  into natural numbers and thereby prove that the expressive power of recursive function theory is no less than that of the former machine-like formalizations of algorithms.
Any such representation of sequences should contain all the information as in the original sequence—most importantly, goel individual member must be retrievable. However, the length does not have to match directly; even if we want to handle sequences of different length, we can store length data as a surplus member,  or as the other member of an ordered pair by using a pairing function.
We expect that there is an effective way for this information retrieval process in form of an appropriate total recursive function.
Moreover, we can define some godeo them in a constructive way, so we can go well beyond mere proofs of existence. This is a primitive recursive function. Our specific solution will depend on a pairing function—there are several ways to implement the pairing function, so one method must be selected.
Now, we can abstract from the details of the implementation of the pairing function. We shall not discuss and formalize the axiom for excluding alien objects here, as it is not required to understand the method. We shall use another auxiliary function that will compute the remainder for natural numbers.
We can use a more concise form by an abuse of notation constituting a sort of pattern matching:. Let us achieve even more readability by more modularity and reuse as these notions are used in computer science : These have their own assumptions. Now we try to find out these assumptions, calibrating and tuning their strength carefully: Let m be chosen to satisfy.
It is needed to meet an assumption of the Chinese remainder theorem that of being pairwise coprime. In the literature, sometimes this requirement is replaced with a stronger one, e.
The second assumption does not concern the Chinese remainder theorem in any way. Since p is a prime element note that the irreducible element property is usedwe get. The assumption was chosen carefully to be as weak as possible, but strong enough to enable us to use it now.
We can prove by several means  known in propositional calculus that. Thus as equality axioms postulate identity to goddl a congruence relation unmbering. Thus, summarizing the above three statements, by transitivity of the equality. This establishes the contradiction we wanted to reach. Now by transitivity of equality we get. This can be seen now by transitivity of equalitylooking at nmbering above three equations.
Although proving this was most important for establishing an encoding scheme for sequences, we have to fill in some gaps yet. Our ultimate question is: The specification declares only an existential quantification, not yet a functional connection. Numebring want a constructive and algorithmic connection: This gap can be filled in in a straightforward way: In fact, the specification.
The importance of this notion is that it enables us to split off the sub class of total recursive functions from the super class of gdoel recursive functions.
In brief, the specification says that a function f  satisfying the specification. It can be proven using the notions of the previous section that g is total recursive.
If we use the above scheme for encoding sequences only in contexts where the length of the sequences is fixed, then no problem arises. In other words, we can use them in an analogous way as arrays are used in programming. But sometimes we need dynamically stretching sequences, or we need to deal with sequences whose length cannot be typed in a static way. In other words, we may encode sequences fodel an analogous way to lists in programming. To illustrate both cases: We can mimic dynamically stretching sequences by representing sequence concatenation or at least, augmenting a sequence with one more element with a totally recursive function.
Length can be stored simply as a surplus member: Also, the assumptions have to be modified accordingly. From Wikipedia, the free encyclopedia. This article may require cleanup to meet Wikipedia’s quality standards.
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