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Another reason why the undecidability of the Halting problem is interesting is because it provides an alternative proof of Gödel's incompleteness theorem. This is because if there were a complete and consistent axiomatization of all true statements about natural numbers, then we would be able to construct an algorithm that decides wheter such a statement is true or not. (This will be discussed in more detail later on.) But we saw above that that is impossible. |
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Another reason why the undecidability of the Halting problem is interesting is because it provides an alternative proof of Gödel's incompleteness theorem (more precisely, a weaker form of the First Incompleteness Theorem). This is because if there were a complete and consistent axiomatization of all true statements about natural numbers, then we would be able to construct an algorithm that decides wheter such a statement is true or not. (This will be discussed in more detail later on.) But we saw above that that is impossible. |
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The concepts raised by the Gödel's incompleteness theorem are very similar to those raised by the halting problem, and the proofs are quite similar. In fact, the incompleteness theorem follows quite directly from the undecidability of the Halting problem. This can be demonstrated as follows. Assume that we have a consistent and complete axiomatization of all true first-order logic statements about natural numbers. Then we can build an algorithm that enumerates all these statements. This means that there is an algorithm N(n) that given a natural number n computes a true first-order logic statement about natural numbers such that for all the true statements there is at least one n such that N(n) is that statement. Now suppose we want to decide if the algorithm with representation a halts on input i. We know that this statement can be expressed with a first-order logic statement, say H(a, i). Since the axiomatization is complete it follows that either there is an n such that N(n) = H(a, i) or there is an n' such that N(n' ) = ¬ H(a, i). So if we iterate over all n until we either find H(a, i) or its negation, we will always halt. This means that this gives us an algorithm to decide the Halting problem. Since we know that there cannot be such an algorithm it follows that the assumption that there is a consistent and complete axiomatization of all true first-order logic statements about natural numbers, must be false. |
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The concepts raised by the Gödel's incompleteness theorems are very similar to those raised by the halting problem, and the proofs are quite similar. In fact, a weaker form of the First Incompleteness Theorem is an easy consequence of the undecidability of the Halting Problem. This weaker form differs from the standard statement of the incompleteness theorem by asserting that a complete, consistent and sound axiomatization of all statements about natural numbers is unacheivable. The "sound" part is the weakening: it means that we require the axiomatic system in question to prove only ''true' statements about natural numbers (it's very important to observe that the statement of the standard form of Godel's First Incompleteness Theorem is completely unconcerned with the question of truth, but only concerns the issue of provability). The weaker form of the theorem can be proved from the undecidability of the Halting Problem as follows. Assume that we have a consistent and complete axiomatization of all true first-order logic statements about natural numbers. Then we can build an algorithm that enumerates all these statements. This means that there is an algorithm N(n) that given a natural number n computes a true first-order logic statement about natural numbers such that for all the true statements there is at least one n such that N(n) is that statement. Now suppose we want to decide if the algorithm with representation a halts on input i. We know that this statement can be expressed with a first-order logic statement, say H(a, i). Since the axiomatization is complete it follows that either there is an n such that N(n) = H(a, i) or there is an n' such that N(n' ) = ¬ H(a, i). So if we iterate over all n until we either find H(a, i) or its negation, we will always halt. This means that this gives us an algorithm to decide the Halting problem. Since we know that there cannot be such an algorithm it follows that the assumption that there is a consistent and complete axiomatization of all true first-order logic statements about natural numbers, must be false. |
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