Consistency, completeness of undecidable preposition of Principia Mathematica. Tanmay Jaipurkar

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1 Consistency, completeness of undecidable preposition of Principia Mathematica Tanmay Jaipurkar October 21, 2013

2 Abstract The fallowing paper discusses the inconsistency and undecidable preposition of Principia Mathematica as discussed by Kurt Godel in On formally undecidable propositions of Principia Mathematica and related systems I. The paper also talks briefly about the Liar s Paradox, Russel s Paradox and Diagonal Proof. Efforts are made to simplify the proof in of undecidable preposition giving a rough sketch of proof. Keywords- Consistency, Completeness, Principia Mathematica, Paradox, Logic.

3 Contents 1 Introduction Consistency Definition What if a system is inconsistent? Completeness Definition Relation with Godel s Paper Rough Sketch Of Proof Formal Proof Liar s Paradox Russel s Paradox Diagonal Proof Conclusion Bibliography

4 1 Introduction The current paper presupposes some knowledge of the state of meta-mathematics and logic.here we have discussed the need of considering new logic system and problem associated with prepositional logic.readers will get good insights and motivation for the development and need of epistemic logic and Kripke model of logic.the author also tries to give some basic idea of some famous Paradox. 2 Consistency 2.1 Definition In logic, a consistent theory is one that does not contain a contradiction. The lack of contradiction can be defined in either semantic or syntactic terms. The semantic definition states that a theory is consistent if and only if it has a model, i.e. there exists an interpretation under which all formulas in the theory are true. This is the sense used in traditional Aristotelian logic, although in contemporary mathematical logic the term satisfying is used instead. The syntactic definition states that a theory is consistent if and only if there is no formula P such that both P and its negation are provable from the axioms of the theory under its associated deductive system. A set of formula F is said to be consistent if and only if for no formula P of F, both P and the negation of P are theorems of F. A deductive theory is called CONSISTENT or NON-CONTRADICTORY if no two asserted statements of this theory contradict each other, or in other words, if of any two contradictory sentences... at least one cannot be proved. 2.2 What if a system is inconsistent? Assume a statement P such that P and P are correct. Consider statement of the form PQ. Now P is correct therefore, PQ is correct. Also P is correct therefore Q has to be true because PQ is correct. 1. P 2. P 3. Show Q 4. PQ 1,Addition 5. Q 2,4 Modus Tollendo Ponens Therefore we have proved every statement in the system because Q can be any statement. Hence if a system is inconsistent we can actually prove anything. 2

5 3 Completeness 3.1 Definition In theories of arithmetic, such as Peano arithmetic, there is an intricate relationship between the consistency of the theory and its completeness. A theory is complete if, for every formula f in its language, at least one of f or f is a logical consequence of the theory. In logic,a formal system is semantically complete when all its tautologies* are theorems. *TAUTOLOGY - In logic, a tautology is a formula which is true in every possible interpretation. A formal system S is syntactically complete or deductively complete or maximally complete or simply complete if and only if for each formula f of the language of the system either f or f is a theorem of S. This is also called negation completeness. In another sense, a formal system is syntactically complete if and only if no improvable axiom can be added to it as an axiom without introducing an inconsistency. Truth-functional propositional logic and firstorder predicate logic are semantically complete, but not syntactically complete (for example, the propositional logic statement consisting of a single variable a is not a theorem, and neither is its negation, but these are not tautologies). Gdel s incompleteness theorem shows that any recursive system that is sufficiently powerful, such as Peano arithmetic, cannot be both consistent and complete.a theory is called COMPLETE, if of any two contradictory sentences formulated exclusively in terms of the theory under consideration (and the theories preceding it) at least one sentence can be proved in this theory. Of a sentence which has the property that its negation can be proved in a given theory, it is usually said that it can be DISPROVED in that theory... a theory is complete... if every sentence formulated in the terms of this theory can be proved or disproved in it. 3.2 Relation with Godel s Paper Gdel research paper under our study deals with his first incompleteness theorem. The first incompleteness theorem (which our study concerns with) states that no consistent system of axioms whose theorems can be listed by an effective procedure (essentially, a computer program) is capable of proving all truths about the relations of the natural numbers (arithmetic). For any such system, there will always be statements about the natural numbers that are true, but that are improvable within the system. The Principia Mathematica(which Gdel discusses about) is so far developed that one can reduce these proof methods to a few axioms and deduction rules. Therefore, the conclusion that these deduction rules are sufficient to decide all mathematical questions expressible in those systems doesn t seem deniable. But Gdel shows that this is not true, while stating the theorem of incompleteness. This is not only for the PM, but it is true for a very wide class of formal systems, which in particular includes all those that you get by adding a finite number of axioms to the above mentioned systems, provided the additional axioms don t make false theorems provable. 3

6 4 Rough Sketch Of Proof An undecidable theorem of the system PM, i.e. a theorem A for which neither A nor A is provable (i.e. to prove the incompleteness of the system assuming its consistency). Method devised by Gdel (Arithmetization of syntax) The main problem in fleshing out the proof of the theorems mentioned before is that it seems at first that to construct a statement p that is equivalent to p cannot be proved, p would have to somehow contain a reference to p, which could easily give rise to an infinite regress. Gdel s ingenious trick is to show that statements can be matched with numbers (often called the arithmetization of syntax) in such a way that proving a statement can be replaced with testing whether a number has a given property. This allows a self-referential formula to be constructed in a way that avoids any infinite regress of definitions. In simple terms, a method was devised so that every formula or statement that can be formulated in the system gets a unique number, called its Gdel number, in such a way that it is possible to mechanically convert back and forth between formulas and Gdel numbers. 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 such numbers can be constructed. A simple example is the way in which English is stored as a sequence of numbers in computers using ASCII or Unicode: The word HELLO is represented by using decimal ASCII, i.e. the number Gdel has viewed the formulae of a formal system (PM here) syntactically as finite sequences of the basic symbols (variables, logical constants, and parentheses or separators). Similarly, proofs have been considered formally as finite sequences of formulae (with specific definable properties). And as signs taken for basic symbols don t carry much significance, natural numbers were chosen for them (namely the Gdel numbers as mentioned before). Hence, a formula is a finite sequence of natural numbers, and a proof schema turned out to be a finite sequence of finite sequences of natural numbers. In particular, one can show that the concepts formula, proof schema, provable formula are all expressible within the system PM, i.e. one can, for example, come up with a formula F(v) of PM that has one free variable v (whose type is sequence of numbers) such that the semantic interpretation of F(v) is: v is a provable formula. A formula of PM with exactly one free variable of type natural numbers has been called a class-sign. The class-signs are assumed to be somehow numbered, calling the nth one Rn, and note that both the concept class-sign and the ordering relation R are definable within the system PM. Let be an arbitrary class-sign; With (n) we denote the formula that we get when one substitutes n for the free variable of. Now we will define a class K of natural numbers (NATURAL NUMBERSAS EVERY- THING IN GODELS SYSTEM FINALLY BOILS DOWN TO NATURAL NUMBERS) as follows: K = nn provable(rn(n)) (1) (where provable(x) means x is a provable formula). With other words, K is the set of numbers n where the formula Rn(n) that we get when we insert n into its own formula Rn is improvable. 4

7 5 Formal Proof Given a set of axioms (as contained in the PM), Let K be the set of numbers which encode(according to Gdel numbering) sentences which are provable from the given axioms. Thus for any sentence s (i.e. for its corresponding Gdel number)1111 S(n) states that nk. (1) s is in K iff s is not provable. Since the set of axioms (or concepts as mentioned in the paper) is expressible, so is the set of proofs which use these axioms and so is K(this is what was said earlier), the set of Gdel numbers of not provable theorems. There is a class-sign S such that the formula As a class-sign, S is identical with a specific Rq, i.e. we have S <=> Rqf oraspecif icnumberq. (2) Claim : The theorem Rq(q) is undecidable within PM. Rq(q) = S(q) = qk, provable(rq(q)), in other words, Rq(q) states I am improvable. Assuming the theorem Rq(q) were provable, then it would also be true, i.e. provable(rq(q)) would be true in contradiction to the assumption. If on the other hand Rq(q) were provable, then we would have qk, i.e. provable(rq(q)). That means that both Rq(q) and Rq(q) would be provable, which again is impossible. 6 Liar s Paradox The liar paradox is the sentence This sentence is false. An analysis of the liar sentence shows that it cannot be true (for then, as it asserts, it is false), nor can it be false (for then, it is true). A Gdel sentence G for a theory T makes a similar assertion to the liar sentence, but with truth replaced by provability: G says G is not provable in the theory T. The analysis of the truth and provability of G is a formalized version of the analysis of the truth of the liar sentence. If This sentence is false is true, then the sentence is false, which would in turn mean that it is actually true, but this would mean that it is false, and so on ad infinitum. Similarly, if This sentence is false is false, then the sentence is true, which would in turn mean that it is actually false, but this would mean that it is true, and so on ad infinitum. 7 Russel s Paradox (informal discussion) Let us call a set abnormal if it is a member of itself, and normal otherwise. For example, take the set of all squares. That set is not itself a square, and therefore is not a member of the set of all squares. So it is normal. On the other hand, if we take the complementary set that contains all non-squares, that set is itself not a square and so should be one of its own members. It is abnormal. Now we consider the set of all normal sets, R. Attempting to determine whether R is normal 5

8 or abnormal is impossible: If R were a normal set, it would be contained in the set of normal sets (itself), and therefore be abnormal; and if it were abnormal, it would not be contained in the set of all normal sets (itself), and therefore be normal. This leads to the conclusion that R is neither normal nor abnormal: Russell s paradox 8 Diagonal Proof Cantor s diagonal argument, also called the diagonalisation argument, the diagonal slash argument or the diagonal method, was published in 1891 by Georg Cantor as a mathematical proof that there are infinite sets which cannot be put into one-to-one correspondence with the infinite set of natural numbers. Such sets are now known as uncountable sets, and the size of infinite sets is now treated by the theory of cardinal numbers which Cantor began. The diagonal argument was not Cantor s first proof of the uncountability of the real numbers; it was actually published much later than his first proof, which appeared in However, it demonstrates a powerful and general technique that has since been used in a wide range of proofs, also known as diagonal arguments by analogy with the argument used in this proof. The most famous examples are perhaps Russell s paradox, the first of Gdel s incompleteness theorems. s1 = (0, 0, 0, 0, 0, 0, 0,...) s2 = (1, 1, 1, 1, 1, 1, 1,...) s3 = (0, 1, 0, 1, 0, 1, 0,...) s4 = (1, 0, 1, 0, 1, 0, 1,...) s5 = (1, 1, 0, 1, 0, 1, 1,...) s6 = (0, 0, 1, 1, 0, 1, 1,...)... s0 = (1, 0, 1, 1, 1, 0, 1,...) 9 Conclusion In simple words, Gdel pointed out that the following statement is a part of the system: a statement P which states there is no proof of P. (as shown by class K by Gdel) If P is TRUE, there is no proof of it. If P is FALSE, there is a proof that P is true, which is a CONTRADICTION. Therefore it cannot be determined within the system whether P is true or not. 10 Bibliography [1] - Kurt Godel On formally undecidable propositions of Principia Mathematica and related systems I 6

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