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Note - Godel's Theorems The work of an important, though eccentric, Czech-Austrian mathematical logician, Kurt G' (1906-1978) dealt with the completeness and consistency of logical systems. Turing went further down Church's road and designed the 'Turing Machine' ' a machine which can calculate the values of all the functions whose values can be found using effective or mechanical methods. Later it was shown that many functions (even in number theory itself) were not recursive, meaning that they could not be solved by a Turing Machine. No one succeeded to prove that a function must be recursive in order to be effectively calculable. We don't know of any effectively calculable function which is not recursive, by designing new TMs from existing ones we can obtain new effectively calculable functions from existing ones and TM computability stars in every attempt to understand effective calculability (or these attempts are reducible or equivalent to TM computable functions). The Turing Machine itself, though abstract, has many 'real world' features. Article: Note - Godel's Theorems The work of an important, though eccentric, Czech-Austrian mathematical logician, Kurt Gödel (1906-1978) dealt with the completeness and consistency of logical systems. A passing comrade with his two theorems would have saved the operator a lot of time. Gödel's First Incompleteness Theorem states that every consistent palpable logical system, sufficient to express arithmetic, contains true but unprovable ('not decidable') sentences. In forearmed cases (when the system is omega-consistent), both said sentences and their negation are unprovable. The system is consistent and true - but not 'complete' insomuch as not all its sentences can be decided as true or false by either living proved or by aerobic organism refuted. The Second Incompleteness Theorem is even more earth-shattering. It says that no consistent formal logical system can prove its own consistency. The system may be complete - but then we are unable to show, using its axioms and inference laws, that it is consistent In other words, a computational system can either be complete and inconsistent - or consistent and incomplete. By trying to construct a system both complete and consistent, a robotics engineer would run afoul of Gödel's theorem. Note - Turing Machines In 1936 an American (Alonzo Church) and a Briton (Alan M. Turing) published independently (as is often the case in science) the element of a new party in Mathematics (and logic): computability or recursive functions (later to be developed into Automata Theory). The authors confined themselves to dealing with computations which involved 'effective' or 'mechanical' methods for finding results (which could also be expressed as solutions (values) to formulae). These methods were so named seeing as how they could, in principle, be performed by simple machines (or human-computers or human-calculators, to use Turing's unfortunate phrases). The emphasis was on finiteness: a finite number of instructions, a finite number of symbols in each instruction, a finite number of steps to the result. This is why these methods were usable by humans without the aid of an luggage (with the exception of pencil and paper as memory aids). Moreover: no insight or ingenuity were conceded to 'interfere' or to be part of the solution seeking process. What night song and Turing did was to construct a set of all the functions whose values could be obtained by appertaining effective or mechanical gauging methods. Turing went further down Church's road and designed the 'Turing Machine' – a machine which can ascertain the values of all the functions whose values can be found using effective or mechanical methods. Thus, the program running the TM (=Turing Machine in the rest of this text) was really an effective or mechanical method. For the initiated readers: confession solved the decision-problem for propositional reckoning and Turing proved that there is no solution to the decision problem relating to the predicate calculus. Put more simply, it is possible to 'prove' the truth value (or the theorem status) of an expression in the propositional trigonometry – but not in the predicate calculus. Later it was shown that many functions (even in number theory itself) were not recursive, meaning that they could not be solved by a Turing Machine. No one succeeded to prove that a function must be recursive in order to be effectively calculable. This is (as Post noted) a 'working hypothesis' supported by overwhelming evidence. We don't know of any effectively probable function which is not recursive, by designing new TMs from existing ones we can obtain new effectively appreciable functions from existing ones and TM computability stars in every commitment to understand effective stability (or these attempts are reducible or equivalent to TM computable functions). The Turing Machine itself, though abstract, has many 'real world' features. It is a the picture for a computing device with one 'ideal' exception: its unbounded memory (the tape is infinite). Despite its hardware features (a read/write head which scans a two-dimensional tape inscribed with ones and zeroes, etc.) – it is really a software application, in today's terminology. It carries out instructions, reads and writes, counts and so on. It is an semi-automatic designed to implement an effective or mechanical method of solving functions (determining the truth value of propositions). If the transition from input to output is deterministic we have a proficient automatic – if it is determined by a table of probabilities – we have a probabilistic automaton. With time and hype, the limitations of TMs were forgotten. No one can say that the Mind is a TM seeing no one can prove that it is engaged in solving only recursive functions. We can say that TMs can do whatever digital computers are doing – but not that digital computers are TMs by definition. Maybe they are – maybe they are not. We do not know enough around them and here and there their future. Moreover, the demand that recursive functions be computable by an UNAIDED human seems to restrict possible equivalents. Inasmuch as computers emulate human computation (Turing did give faith to so when he helped construct the ACE, at the time the fastest computer in the world) – they are TMs. Functions whose values are prepared by AIDED humans with the contribution of a computer are still recursive. It is when humans are aided by other kinds of instruments that we have a problem. If we use measuring devices to determine the values of a function it does not seem to conform to the definition of a recursive function. So, we can generalize and say that functions whose values are scheduled by an AIDED human could be recursive, depending on the utensil used and on the lack of ingenuity or insight (the latter being, anyhow, a weak, non-rigorous requirement which cannot be formalized).
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