Craig Smorynski. The incompleteness theorems. In Jon Barwise, editor, Handbook of Mathematical Logic, pages 821--865. North-Holland, 1977.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....all there is to finitistic mathematics (modulo the arithmetization of syntax) is defended by William Tait in [Tai81] 2. More precisely: If S is a theory that purports to formalize infinitistic mathematics, then the consistency of S is equivalent to the reflection principle for #1 sentences (see [Smo77]) 3. Parsons result appears in the last theorem of [Par70] In its proof, Parsons refers to the abstract [Par71] where it is stated that the theory I# 1 (actually, a seemingly stronger but equivalent theory) has a functional interpretation in T 0 , a fragment of Godel s T. The proof of this ....

Craig Smorynski. The incompleteness theorems. In Jon Barwise, editor, Handbook of Mathematical Logic, pages 821--865. North-Holland, 1977.

....that can be decided in polynomial time (some other complexity classes can similarly be characterized) It is well known ( Li97] that Q are conservative extensions of Q and PA. We use here the same symbol for fA in PR and in our metalanguage. The notation here follows that of [Sm77]. The class of b e ective relations is a proper subclass of the class of PR e ective relations. Our general de nition allows us, accordingly, to capture different notions of e ectiveness . None of them can exactly capture the intuitive notion of constructive e ectiveness (by a ....

C. Smorynski, The Incompleteness Theorems, in Handbook of Mathematical Logic (J. Barwise, Ed.), North-Holland, Amsterdam, 1977.

....1. Although Theorem 5.3.1.6 was hailed as a dramatic extension of Gdel s theorem , we should not forget that there is a big difference between the two results. Gdel s first incompleteness theorem is an explicit construction of an undecidable (hence true) # 1 formula: the fixed point lemma [91,827] associates with any formal system S in a primitive recursive way a formula # S which says of itself I am unprovable in S . But Theorem 5.3.1.6 provides no such explicit construction. First, its proof shows that the characteristic constant c(S) is not a recursive function of S. Second, suppose we ....

....contains too much information, which is something entirely different. This being said, it must be acknowledged that some true statements are undecidable in PA precisely because they contain too much information.The construction of such a statement utilizes the fixed point lemma: 5.3.2. 1 Lemma [91,827] Let # be an arithmetical formula in one free variable. Then, for infinitely many #, PA (# # #( # ) We use the fixed point lemma to define a sentence # which says intuitively I contain too much information for PA . Put k 0 : max k I(k) # c(PA) Choose (non effectively ) # such that # k 0 ....

C. Smorynski, The incompleteness theorems, in: J. Barwise (ed.), Handbook of mathematical logic, North-Holland (1977), 821-865.

.... By examining the mechanism of a proof of the completeness theorem, he also gave a model theoretic proof of the second incompleteness theorem of set theory ( Kr2] page 383, footnote) and the proof can be modified to that of arithmetic by means of the arithmetized completeness theorem (cf. [Sm1]) Recently, Kotlarski [Ko] showed new proofs of the incompleteness theorems by using the arithemtized completeness Typeset by A M S T E X 1 theorem. Jech [Je] gave a simple model theoretic proof of the second incompleteness theorem of set theory, and the proof also can be modified to that of ....

....theorems by using the arithemtized completeness Typeset by A M S T E X 1 theorem. Jech [Je] gave a simple model theoretic proof of the second incompleteness theorem of set theory, and the proof also can be modified to that of arithmetic by means of the arithmetized completeness theorem. See [Sm1], Ka] KT] for proofs of the arithmetized completeness theorem. The following corollary to the arithmetized completeness theorem is more useful than the theorem itself: Every model of PA Con(T ) has a definable end extension which is a model of T , where a definable end extension of a model M ....

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Smory'nski, C.A., The incompleteness theorems, Handbook of Math. Logic (J. Barwise, ed.), North-Holland, 1977, pp. 821--865.

....notation, assume that F v (B) fx; zg, F v (C) fx; y; zg, B is SPR w.r.t. fxg, and C is SPR w.r.t. fyg (and so A is SPR w.r.t. fx; yg) It is easy to prove (1) 14 For convenience we use for a function the same symbol in oe PA and in our metalanguage. 15 The notation here follows that of [Sm77]. 13 for A, given the induction hypotheses for B and C. We now show (2) By induction hypothesis there are p.r. functions f B and f C such that PA B x f B (z) PA C y f C (x; z) Let f A (z) max(fB (z) maxffC (i; z) j 0 i f B (z)g) Then f A is p.r. and A (x f A (z) ....

C. Smorynski, The Incompleteness Theorems, in Handbook of Mathematical Logic (J. Barwise, Ed.), North-Holland, Amsterdam, 1977.

.... of F is there a theorem of F to the e ect that S 0 is a non theorem of F (if F is consistent) There are more problems with this EWD, e.g. it is claimed that c0 (I have a proof that I have a proof that A (holds) is equivalent to c1 (I have a proof that A (holds) but this is not the case (see [26]) The proof of their equivalence (hinted at, but not given) uses: if and have the same truth value, then is provable i is. But truth and provability are di erent, e.g. replace by true and by CON F (the sentence asserting the consistency of F , the suciently strong formal theory ....

C. Smorynski. The incompleteness theorems. In Jon Barwise, editor, Handbook of Mathematical Logic. North-Holland, 1977.

....f0; 1g , can be defined by an algorithm that raises no provability problems. 3 All of our results can be easily translated to similar theorems for any recursive formal system that extends PA (in particular, to ZFC set theory) 4 Here we differ from the common definition of Pi 1 formulas [Sm77] that allows only primitive recursive bounds. It is not hard to realize that the relevant proof theoretic results (mainly Lemma 5 below) can be strengthened to apply for our definition. This is shown explicitly in [BD91] 6 PA 1 enjoys a property that guarantees that non provability phenomena is ....

Smorynski C., "The Incompleteness theorem", Handbook of mathematical logic, J.Barwise ed., North-Holland, New-York (1977), 821-865.

....situation when a veri er uses the power of all of mathematics, not only the elementary methods formalizable in PA. Here is the sketch of the standard metamathematical argument which under certain assumptions about ZF concludes that in fact PA Provable(F ) assume that ZF is consistent (cf. [15], 7] 16] since Provable(F ) is an arithmetical 1 statement, this yields that Provable(F ) is true and, by the 1 completeness of PA, PA Provable(F ) On the one hand, we have succeeded in establishing that PA Provable(F ) On the other hand, at the metalevel of this argument we ....

....proof is represented implicitly by the existential quanti er, which does not provide any speci cation of this proof. The implicit provability predicate has been studied extensively since its invention by G odel in 1930. The milestone results here are the second G odel incompleteness theorem (cf. [15], 7] which states that If V is consistent, then 6 Consis(V) and the L ob theorem which says that V implies V : By the well known Hilbert Bernays lemma (cf. 15] 7] V implies V : This lemma can be considered as a justi cation of the formalization rule = for V , ....

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Smorynski, C.: The Incompleteness Theorems. In: Barwise, J (ed.): Handbook of Mathematical Logic, Vol. 4. North-Holland, Amsterdam (1977) 821-865

.... from codes of proofs to codes of proofs corresponding to Delta and : Delta stands for a operation on proof sequences which realizes the modus ponens rule in arithmetic, and is the proof checker operation, appearing in the proof of the second Godel Incompleteness theorem (cf. [17], 5] However, the choice operation is already incompatible with the deterministic character of Proof (x; y) where a proof x proves only one formula y. Indeed, if s : F and t : G, then both (s t) F and (s t) G, i.e. s t proves at least two different formulas F and G. The usual ....

C. Smorynski, "The incompleteness theorems", in Handbook of mathematical logic, Amsterdam; North Holland, 1977, pp. 821-865.

....as well as 9 x: F (x) are provable [God86] In the original statement the system S is the logic of Principia Mathematica [WR25] with the axiom of choice (for all types) and the natural numbers as individuals. The theorem, however, can be proved for any formal theory containing arithmetic [Smo77]. We shall prove in GETFOL the following formula 1 : OCONS oe 9 w ( PROVABLE(w) PROVABLE( w) 1) where OCONS means that maths a fixed but unspecified theory containing arithmetic is consistent, w is a variable ranging over formulas, PROVABLE(w) means that the formula w is provable ....

C. Smorynski. The Incompleteness Theorems. In Jon Barwise, editor, Handbook of Mathematical Logic, pages 821--865. North Holland Publishing Company, 1977.

....of proofs which correspond to Theta and . So, Theta corresponds to a concatenation of proof sequences which realizes the modus ponens rule in arithmetic, and is represented by a special case of a Godel function appearing in the proof of Sigma 1 completeness of arithmetic (cf. [11]) The usual proof predicate has a natural nondeterministic version PROOF (x; y) called standard nondeterministic proof predicate x is a code of a derivation containing a formula with a code y . PROOF already has all three operations of the LP language: ffl u Omega v is the code of the ....

C. Smorynski, "The incompleteness theorems", in Handbook of mathematical logic, Amsterdam; North Holland, 1977, pp. 821-865.

....107) The fact that IF logic defines truth in N should not be seen as an extraordinary thing, as similar results are known in the literature. For example, Myhill [Myhill, 1950] shows that an existentially quantified fragment of ordinary first order logic defines its own truth predicate. Smorynski [Smorynski, 1977] systematises similar results for the existential fragments of the arithmetical hierarchy, and these have been considered in many other occasions in the literature as well. The result in PMR states the same thing for the first existential level of the hyper arithmetical hierarchy. The point of the ....

Smorynski, C.: The incompleteness theorems, in Barwise, J., (ed.), Handbook of Mathematical Logic, Amsterdam: North-Holland, pp. 821-- 865.

....1998, p. 107) The fact that IF logic defines truth in N should not be seen as an extraordinary thing, as similar results are known in the literature. Historically, Myhill (1950) shows that an existentially quantified fragment of ordinary first order logic defines its own truth predicate. Smorynski (1977) systematises similar results for the existential fragments of the arithmetical hierarchy, and related cases have been considered in many other occasions in the literature as well. The definability result states the same thing for the first existential level of the hyper arithmetical hierarchy. ....

Smorynski, C.: (1977) The incompleteness theorems, in Barwise, J., (ed.), Handbook of Mathematical Logic, Amsterdam: North-Holland, pp. 821--865.

....language. We should add that these scopes seem to be quite efficiently implementable (cf. the dictionaries of Barklund [2] and the substitutions of Sato [26] 7 Related Work In constructing this language we have tried to remain closer to work on reflection and encodings in mathematical logic [27] than most other approaches. The language was obviously inspired by Reflective Prolog [18] differing from it in mainly three aspects: ffl Alloy has names for variables, so nonground expressions can be represented directly. This should make the language suitable for writing program ....

Smorynski, C., The Incompleteness Theorems, in: J. Barwise (ed.), Handbook of Mathematical Logic, North-Holland, Amsterdam, 1977.

....such a representation an amalgamation of the metalanguage and the object language, but this term has since come to denote a stronger concept as discussed in Sect. 2.3. 4 We could weaken this restriction to require only that the representing term is closed, as is common in mathematical logic [80], but the terminology in the field has long distinguished mainly between ground and nonground representations. consequence of this restriction for metaprogramming was observed by Bowen Kowalski [9] However, as Konolige points out [50] being able to show that a sentence is not a consequence ....

....sentences can be constructed [9] This is not harmful in itself, but a theory with such a language can express paradoxes. The concept of self reference has been studied quite extensively, e.g. by Perlis [70, 71] There is an overview of self reference and incompleteness in logic by Smorynski [80]. 2.3 Amalgamation: Theories that Represent Themselves Given a self representable language LA , we may go one step further and develop a theory TA that represents itself, i.e. having the property that Demo TA (P 0 ) is a theorem of TA if and only if P is a theorem of TA , where P 0 ....

Smorynski, C., The Incompleteness Theorems, in: J. Barwise (ed.), Handbook of Mathematical Logic, North-Holland, Amsterdam, 1977.

.... the concepts of provability and truth [24] both provided with definitions which can be technically handled within a syntax and a semantics of theories; the definition of consistency of a theory, and the possibility of proving this consistency only by referring to more powerful theories [16, 28, 29, 14, 5]: the consistency of a theory including Arithmetic cannot be proved using constructive methods. In addition, the realization of two facts: the metatheory on a theory T can be formalized using the language of T; it can be formally discussed what parts of the metatheory of T can be proved in T. ffl ....

C. Smorynski. The incompleteness theorems. In Barwise, editor, Handbook of Mathematical Logic. North Holland, Amsterdam, 1977.

....principle states that 1 p 2 j 3 q , 1 Pi 2 j 1 Pi 3 and can be seen as a correctness statement for coincidence of internal theories. Both these principles are valid for every theory system. The traditional local reflection principle for a single theory T in mathematical logic [30] reads Pr T (pOEq) OE and states the correspondence between a provability statement and what is to be proved, namely that if the provability predicate holds for an encoding of a formula OE, then OE holds as well. We call our statements reflection principles by analogy, as they state ....

Smorynski, C., The Incompleteness Theorems, in: J. Barwise (ed.), Handbook of Mathematical Logic, North-Holland, Amsterdam, 1977.

....see (Dauben 1979) 49. For an historical account of the controversy about the Axiom of Choice, see (Moore 1982) for a mathematical discussion of the significance of this axiom, see (Jech 1973) 50. For Godel s results, see (Godel 1931) For a discussion of the incompleteness theorems, see (Smorynski 1977); the results are proved in many standard texts, such as (Enderton 1972) 51. Cohen s original paper was (Cohen 1963 64) see also (Cohen 1966) There are now many methods in the standard texts of proving that CH is independent of ZFC, and other independence results; the methods are surveyed in ....

Smorynski, C. (1977). The incompleteness theorems. In Handbook of mathematical logic (ed. J. Barwise), pp. 821--65. North-Holland, Amsterdam.

....axioms for computational strategies, and for the internalization of those strategies in a reflective logic are also given. 1 Introduction Reflection is a fundamental idea. In logic it has been vigorously pursued by many researchers since the fundamental work of Godel and Tarski (see the surveys [52, 53]) In computer science it has been present from the beginning in the form of universal Turing machines. Many researchers have recognized its great importance and usefulness in programming languages [54, 51, 60, 56, 23, 19, 30] in theorem proving [62, 7, 48, 20, 2, 29, 14, 16] in concurrent and ....

C. Smorynski. The incompleteness theorems. In J. Barwise, editor, Handbook of Mathematical Logic, pages 821--865. North-Holland, 1977.

....reflection principles restricted to classes of formulas Sigma n and Pi n are equivalent to sentences (namely, the uniform reflection for the corresponding universal formula) therefore they will be treated as such. We shall use some well known results on reflection principles. Lemma 1 (see [13]) 1) Sigma 1 completeness of B: P r B d e, for every Sigma 1 sentence ; 2) Con T j Pi 1 Rfn T j Pi 1 RFN T ; 3) 1 Con T j Sigma 1 Rfn T . Let us recall that Godel s first incompleteness theorem asserts that for every consistent T , there is an independent sentence. The ....

.... T ( x)e we shall use a weaker assumption by taking only one special case of Sigma n RFN T , namely P r T d: P r T d ( x)e ( x) x)e: The formula within the outer d e reduces (using propositional calculus) so that we get Pr T dP r T d ( x)e ( x)e: By Lob s theorem (cf. [13]) it implies P r T d ( x)e. Now we can apply our assumption Sigma n RFN T and conclude (x) as required. 2 Let us analyze now the intuitive argument that we can add the consistency Con T Con T when we already know Con T . The usual argument goes roughly as follows: Suppose :Con T Con T , i.e. ....

C. Smorynski, The incompleteness theorems, in Handbook of Mathematical Logic, J. Barwise ed., North-Holland 1977, 821-865

....axioms leads directly to the question as to what additional power (if any) or expressivity (if any) is to be gained by removing the axioms and going to a more fully higher order language. Because of the desire to formalize mathematics, and because of the incompleteness theorems of Goedel [48], higher order types were explored to attain the expressivity necessary to define mathematics in a uniform context in a way that avoided the incompleteness problems. Hierarchies of types allowed the consistency and meta logical properties of each level to be expressed at a higher level. The ....

Smorynski, C. The incompleteness theorems. In Mathematical Logic, J. Barwise, Ed. North Holland, 1977, pp. 821--866.

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C. Smorynski, "The incompleteness theorems", in Handbook of mathematical logic, Amsterdam; North Holland, 1977, pp. 821-865.

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