Kurt Godel. Uber eine bisher noch nicht benutzte Erweiterung des finiten Standpunktes. Dialectica, 12:280--287, 1958.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....a) Seth [Set95] calls these transfinite levels. Thus, BCD (respectively, D ) is a profoundly larger class than BCE (respectively, E) Whether BCD and D are too large to be considered feasible is a question taken up in Section 20. We suspect that Godel s primitive recursive functionals [God58, God90] are exactly the class of functionals such machines compute. 18. From E Machines to BTLP The following proposition is the fourth, and final, link in our grand chain of translations. Proposition 61. There is an e#ective procedure that, given i, k, d, #, and # , constructs a BTLP program ....

K. Godel, Uber eine bisher noch nicht benutzte Erweiterung des finiten, Dialectica 12 (1958), 280--287.

....and polynomial time computable, and some metamathematical results on our arithmetic system. 2 A term system for non size increasing polynomial time computation sec term We introduce a term system similar to the system in [7] It will play the same role for our arithmetical system as Godel s T [5] does for Heyting Arithmetic. 2.1 Types and terms sub types terms def lin types Definition 2.1 (Finite linear types) Linear types are defined inductively as ; U j j L( j ( j j j : def set model setmodel Definition 2.2 (Set model) In the (naive) set model every ....

Kurt Godel. Uber eine bisher noch nicht benutzte Erweiterung des finiten Standpunkts. Dialectica, 12:280--287, 1958.

....to modus ponens. Our monotone version has the same good behaviour. Proof of theorem 2. 7 : Description of the algorithm for extracting uniform bounds by monotone functional interpretation We use (as in [26] and [31] the formalization of WE HA in Godel s calculus of intuitionistic logic ([14] ) 1) The most complicated axioms for the usual functional interpretation are A A and A. The later one is even more complicated in requiring the existence of functionals which decide prime formulas. a) A Y, Y # , X ## , x, x # , y ## z = 0 0 AD (x, Y zxx # y ## , ....

Godel, K., Uber eine bisher noch nicht benutzte Erweiterung des finiten Standpunktes. Dialectica 12, pp. 280--287 (1958).

.... AC qf # A and by monotone functional interpretation one can extract a tuple # i # satisfies the monotone functional interpretation of A # , # . In particular the second conclusion can be proved in G n A i # b AC) Here we can use Godel s [5] translation or any of the various negative translations. For a systematical treatment of negative translations see [15] Here b AC: #,##T denotes the schema ) #Z #y ## Zx A(x, y, Z) # #Y # ## Z#xA(x,Y x, Z) Remark 29 In theorems 27,28 one may also have tuples #w ....

....Condition 1) is a sort of an upper bound for the complexity of , S. e.g. 1) is not satisfied if S contains the iteration functional # it defined by # it 0yf = 0 y, # it x # yf = 0 f(# it xyf) Note that # it is primitive recursive in the usual sense of [6] and not only in the extended sense of [5]) In the next paragraph we will show that thm.27 becomes false if G n R is replaced (see also remark 214 ) Since # it is on some sense the simplest functional for which 1) fails, this shows that the upper bound provided by 1) is quite sharp. 1) essentially says that # 001 can be ....

Godel, K., Uber eine bisher noch nicht benutzte Erweiterung des finiten Standpunktes. Dialectica 12, pp. 280--287 (1958).

.... A method which we believe fulfils these requirements is the monotone functional interpretation which was developed in [13] 15] the technique used in [10] can be viewed of as a precursor of this method) Monotone functional interpretation is a variant of Godel s functional interpretation [6] and extracts majorizing functionals (in the sense of Howard [8] of functionals satisfying the usual Godel functional interpretation. These majorizing functionals keep control through all finite types of the growth rates involved in a given proof without any normalization. The method applies to ....

Godel, K., Uber eine bisher noch nicht benutzte Erweiterung des finiten Standpunktes. Dialectica 12, pp. 280--287 (1958).

.... of quantifier free induction) The restriction deg(ae) 2 has a technical reason discussed in [12] n2IN , PA i are the extensions of G n A , G n A i obtained by the addition of the schema of full induction and all (impredicative) primitive recursive functionals in the sense of [5]. E T (i) denotes the theory which results from T (i) when the quantifier free rule of extensionality is replaced by the axioms of extensionality (E) ae (x = ae y zx = zy) for all finite types (x = ae y is defined as 8z 1 ; z k (xz 1 : z k = 0 yz 1 : z k ....

Godel, K., Uber eine bisher noch nicht benutzte Erweiterung des finiten Standpunktes. Dialectica 12, pp. 280--287 (1958).

....let B 0 (u; v; w fl ) 2 L(GnA ) be a (quantifier free) formula which contains only u; v; w as free variables and fl 2. Then from a proof GnA Delta AC qf 8u 1 8v tu Gamma 9f 1 8x; z A 0 (u; v; x; fx; z) 9w fl B 0 (u; v; w) Delta ( 3 Here we can use Godel s [8] translation or any of the various negative translations. For a systematical treatment of negative translations see [18] 6 one can extract a term 2 GnR such that GnA i Delta b AC 8u 1 8v tu8 Psi Gamma ( Psi satisfies the mon.funct.interpr. of 8x 0 ; g 1 9y 0 A 0 ....

Godel, K., Uber eine bisher noch nicht benutzte Erweiterung des finiten Standpunktes. Dialectica 12, pp. 280--287 (1958).

....) in the form A : #n # N; x # X; y # K #k # NA 1 (n, x, y, k) where A 1 is a purely existential. Then the following rule holds: # # # # # # # PA # AC 1,0 qf WKL # #n # N; x # X; y # K #k # NA 1 (n, x, y, k) then one can extract a primitive recursive (in the sense of [God58]) term # s.t. HA # # #n # N; x # X; y # K#k # #(n, x) A 1 (n, x, y, k) A crucial feature of the functional # above is that it does not depend on the element y # K. It should also be noted that # depends on the representation of x as an element of X. In the present case X is the ....

K. Godel. Uber eine bisher noch nicht benutzte Erweiterung des finiten Standpunktes. Dialectica, 12:280--287, 1958.

....f) CHAPTER 1. UNWINDING PROOFS: PROOF MINING 14 (v) Primitive recursion) F (0, x, f) G(x, f) F (y 1, x, f) H(F (y, x, f) y, x, f) This was first shown by Parsons ( 108] Full Peano arithmetic requires primitive recursive functionals in higher types in the extended sense of Godel [45] (see chapter 2) as will be discussed in chapter 7 below. There is, however, a problem in using the no counterexample interpretation as a tool to extract such realizing functionals in a modular way i.e. by a recursion over the proof tree which keeps the basic structure of the proof unchanged ....

.... unchanged (which is of crucial importance for actually analysing concrete and in particular not fully formalized proofs) In fact, Parsons and Godel s results where obtained by using a di#erent more complicated interpretation, the so called Godel functional ( Dialectica ) interpretation ([45]) which we will treat in chapters 5 6. In contrast to the no counterexample interpretation, which only refers to functionals of type level 2, functional interpretation uses even for first order systems like PA functionals of arbitrary finite types to achieve an interpretation which respects ....

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Godel, K., Uber eine bisher noch nicht benutzte Erweiterung des finiten Standpunktes. Dialectica 12, pp. 280--287 (1958).

....(Seth [Set95] calls these transfinite levels. Thus, BCD (respectively, D ) is a profoundly larger class than BCE (respectively, E) Whether BCD and D are too large to be considered feasible is a question taken up in Section 20. 3 13 We suspect that Godel s primitive recursive functionals [God58, God90] are exactly the class of functionals such machines compute. 25 July 2001 On Characterizations of the BFFs, Part II Preliminary Version 83 18. From E Machines to BTLP The following proposition is the fourth, and final, link in our grand chain of translations. Proposition 61. There is an ....

K. Godel, Uber eine bisher noch nicht benutzte Erweiterung des finiten, Dialectica 12 (1958), 280--287.

....increasing and polynomial time computable, and some metamathematical results on our arithmetic system. 2 A term system for non size increasing polynomial time computation We introduce a term system similar to the system in [7] It will play the same role for our arithmetical system as Godel s T [5] does for Heyting Arithmetic. 2.1 Types and terms Definition 2.1 (Finite linear types) Linear types are defined inductively as ; U j j L( j ( j j j : Definition 2.2 (Set model) In the (naive) set model every type in the left column, below, is interpreted by ....

Kurt Godel. Uber eine bisher noch nicht benutzte Erweiterung des finiten Standpunkts. Dialectica, 12:280--287, 1958.

....induction) The restriction deg(ae) 2 has a technical reason discussed in [12] G1A : S n2IN GnA . PA , PA i are the extensions of GnA , GnA i obtained by the addition of the schema of full induction and all (impredicative) primitive recursive functionals in the sense of [5]. E T (i) denotes the theory which results from T (i) when the quantifier free rule of extensionality is replaced by the axioms of extensionality (E) 8x ae ; y ae ; z ae (x = ae y zx = zy) for all finite types (x = ae y is defined as 8z ae 1 1 ; z ae k k (xz 1 : ....

Godel, K., Uber eine bisher noch nicht benutzte Erweiterung des finiten Standpunktes. Dialectica 12, pp. 280--287 (1958).

....of bounded arithmetic. While these functionals served his needs, their definition was rather involved. Cook and Urquhart decided to try to develop a simpler class of realizers for IS 1 2 that could be presented as a feasible variant of Godel s Dialectica interpretation of Heyting Arithmetic [God58, God90]. Their approach to this problem was, like Mehlhorn s work, based on Cobham s [Cob65] syntactic definition of polynomial time. Their initial work was independent of Constable and Mehlhorn. They introduced a formal system PV # in which the terms consist of simply typed # expressions built from ....

K. Godel, Uber eine bisher noch nicht benutzte Erweiterung des finiten, Dialectica 12 (

....in C[0, 1] by polynomials in P n . Therefore, from the metatheorem 2.1 and previous discussions we obtain the following corollary (see [16] theorem 5.1) Corollary 2. 2 (a) A functional #(f, n, k) given by a closed term of E PA # (i.e. a primitive recursive functional # in the sense of Godel [9]) can be extracted from Cheney s proof of Jackson s theorem so that, E )HA # # #n # IN; f # C[0, 1] p 1 , p 2 # P n ; k # IN # 2 # i=1 (#f p i # 1 dist 1 (f, P n ) # 2 #(f,n,k) # #p 1 p 2 # 1 2 k # . b) Using the # above, a primitive recursive ....

K. Godel. Uber eine bisher noch nicht benutzte Erweiterung des finiten Standpunktes. Dialectica, 12:280--287, 1958.

....paradigm, one has to consider functions of higher type, and thus extend the function algebras by a typed lambda calculus. To really make use of this feature, it is desirable to allow the definition of higher type functions by recursion. Higher type recursion was originally considered by Godel [10] for the analysis of logical systems. Systems with recursion in all finite types characterizing polynomial time were given by Bellantoni et al. 5] and Hofmann [12] based on the first order system of Bellantoni and Cook [4] We define an analogous system that characterizes NC while allowing an ....

K. Godel. Uber eine bisher noch nicht benutzte Erweiterung des finiten Standpunktes. Dialectica, 12:280--287, 1958.

....and n are finite, and exactly the same argument works when they are infinite. Or second, 1 = We don t meet in our everyday life, but we can see how to prove the inequality by moving each number along by one. The picture lies well within the range of what we can ubersehen , to quote G odel [14]. But then wecome to Cantor s result, and all intuition fails us. Until Cantor first proved his theorem ( 6] by a much longer argument, as it happens) nothing like its conclusion was in anybody s mind s eye. And even now we accept it because it is proved, not for any other reason. 4 WILFRID ....

Kurt G odel, Uber eine bisher noch nicht benutzte Erweiterung des finiten Standpunktes, Collected works (S. Feferman et al., editors), vol. II, Oxford University Press, Oxford, 1990, pp. 240--251.

....# A 0 (x) # A 0 (x # ) # # #xA 0 (x) where A 0 is quantifier free. This finishes the description of E PRA # . The theory E PA # is the extension of E PRA # obtained by the addition of the schema of full induction and all (impredicative) primitive recursive functionals in the sense of [6]. The schema of quantifier free choice for the types #, # is given by QF AC #,# : #x # #y # A 0 (x, y) # #Y ### #x # A 0 (x, Y x) QF AC : # #,##T QF AC #,# , where A 0 is quantifier free. The theory RCA # 0 is defined as RCA # 0 : E PRA # QF AC 1,0 . In deviating ....

Godel, K., Uber eine bisher noch nicht benutzte Erweiterung des finiten Standpunktes. Dialectica 12, pp. 280--287 (1958).

.... PA # from [5] since # it allows (relative to E G#A # ) to define the predicative recursor constants b R # (see [23] E PA # is the extension of E PRA # obtained by the addition of the schema of full induction and all (impredicative) primitive recursive functionals in the sense of [13]. The schema of full choice is given by AC #,# : #x # #y # A(x, y) # #Y #(#) #x # A(x, Y x) AC : #,##T AC #,# . The schema of quantifier free choice QF AC #,# is defined as the restriction of AC #,# to quantifier free formulas A 0 . 3 Remark 2.1. E PRA # QF AC ....

Godel, K., Uber eine bisher noch nicht benutzte Erweiterung des finiten Standpunktes. Dialectica 12, pp. 280--287 (1958).

....The restrictions are obtained by enriching the type structure with the formation of types oe, and by adding linear concepts to the lambda calculus. 1 Introduction Recursion in all finite types was introduced by Hilbert [9] and later became known as the essential part of Godel s system T [8]. This system has long been viewed as a powerful scheme unsuitable for describing small complexity classes such as polynomial time. Simmons [16] showed that ramification can be used to characterize the primitive recursive functions by higher type recursion, and Leivant and Marion [14] showed that ....

K. Godel. Uber eine bisher noch nicht benutzte Erweiterung des finiten Standpunktes. Dialectica, 12:280--287, 1958.

....primitive predicate. So E PRA # essentially is # PA # (E) where # PA # is Feferman s system from [4] E PA # is the extension of E PRA # obtained by the addition of the schema of full induction and all (impredicative) primitive recursive functionals in the sense of Godel [6] and coincides with Troelstra s [20] system (E HA # ) c . The weakly extensional 3 versions WE PRA # and WE PA # of these systems result if we replace the extensionality axioms (E) by a quantifier free rule of extensionality (due to Spector [19] QF ER: A 0 # s = # t A 0 # r[s] ....

Godel, K., Uber eine bisher noch nicht benutzte Erweiterung des finiten Standpunktes. Dialectica 12, pp. 280--287 (1958).

....of bounded arithmetic. While these functionals served his needs, their definition was rather involved. Cook and Urquhart decided to try to develop a simpler class of realizers for IS 1 2 that could be presented as a feasible variant of Godel s Dialectica interpretation of Heyting Arithmetic [God58, God90]. Their approach to this problem was, like Mehlhorn s work, based on Cobham s [Cob65] syntactic definition of polynomial time. Their initial work was independent of Constable and Mehlhorn. They introduced a 1 Suppose M0 and M1 are two models of computation with associated cost models. M0 and M1 ....

K. Godel, Uber eine bisher noch nicht benutzte Erweiterung des finiten, Dialectica 12 (1958), 280--287.

.... PA # from [5] since # it allows (relative to E G#A # ) to define the predicative recursor constants b R # (see [23] E PA # is the extension of E PRA # obtained by the addition of the schema of full induction and all (impredicative) primitive recursive functionals in the sense of [13]. The schema of full choice is given by AC #,# : #x # #y # A(x, y) # #Y #(#) #x # A(x, Y x) AC : #,##T AC #,# . The schema of quantifier free choice QF AC #,# is defined as the restriction of AC #,# to quantifierfree formulas A 0 . 3 2 The restriction deg(#) # 2 ....

Godel, K., Uber eine bisher noch nicht benutzte Erweiterung des finiten Standpunktes. Dialectica 12, pp. 280--287 (1958).

....with time complexity O(X n ) if the proof contains at most n nested uses of induction. 1 Introduction In [5] Hofmann presented a restriction of Godel s T allowing iteration in all types with the property that all definable functions are non size increasing polynomial time computable. Godel s T [4] is related to Heyting Arithmetic via realizability interpretation. If, for example, 8x 9 y ae A with A quantifier free can be proved, then there exists a term of type ae in Godel s T calculating the y for a given x. For more details on realizability interpretations see [8, IIIx4] We ....

K. Godel. Uber eine bisher noch nicht benutzte Erweiterung des finiten Standpunkts. Dialectica, 12:280--287, 1958.

....the chapter. 2 A Programming language for computable functions Historical remarks 5 In the early 1940 s, Godel considered a notion of primitive recursive functionals of finite type, which we now call Godel s System T, in connection with what came to be known as the Dialectica Interpretation [ Godel, 1958; Godel, 1990 ] Godel presented his results as a contribution to a liberalized version of Hilbert s programme. 6 Godel s work was later extended to the bar recursive functionals by Spector [ Spector, 1962 ] who used them to give a constructive consistency proof for classical analysis. However, ....

K. Godel. Uber eine bisher noch nicht benutzte Erweiterung des finiten Standpunktes. Dialectica, pages 280--287, 1958.

....polynomial time computable functions. The restrictions are obtained by using a ramified type structure, and by adding linear concepts to the lambda calculus. 1 Introduction Recursion in all finite types was introduced by Hilbert [8] and later became known as the essential part of Godel s system T [7]. This system has long been viewed as a powerful scheme unsuitable for describing small complexity classes such as polynomial time. Simmons [17] showed that ramification can be used to characterize the primitive recursive functions by higher type recursion. Leivant [13] used ramification notions ....

K. Godel. Uber eine bisher noch nicht benutzte Erweiterung des finiten Standpunktes. Dialectica, 12:280--287, 1958.

....dissertation [42] to define a notion of net module. The most novel approach is the one taken by Carolyn Brown and Doug Gurr. It is based on the Dialectica Categories of Valeria de Paiva [15, 16] which are a category theoretic model of Godels Dialectica interpretation of higher order arithmetic [37]. The constructs available in these categories give rise to very interesting constructions on nets that have a connection with linear logic. Unfortunately the approach currently only works with elementary nets. We have chosen to work with the categories of Meseguer and Montanari. The main reason ....

Godel, K. Uber eine bisher noch nicht benutzte erweiterung des finiten standpunktes. Dialectica, 12(1958), pp. 280--287.

....since all inductive clauses have finitely many premises only (see [Tro73] For reasons of proof theoretical strength we cannot hope to extend the simple normalization proof to Godel s T without major modifications. The reason is that the termination of T implies consistency of Peano arithmetic [God58] and this fact is provable in primitive recursive arithmetic. It will turn out that the inductive characterization SN of the strongly normalizing terms can be extended to T while the embedding of the set of all terms requires the introduction of an infinitely branching clause in an inductive ....

Kurt Godel. Uber eine bisher noch nicht benutzte Erweiterung des finiten Standpunkts. Dialectica, 12:280--287, 1958.

.... a model of PCF 45 B A Sample Miniproject 1 Historical Remarks 4 1 Historical Remarks In the early 1940 s Godel considered a notion of primitive recursive functionals of finite type, which we now call Godel s System T, in connection with what came to be known as the Dialectica Interpretation [7, 8]. Godel presented his results as a contribution to a liberalised version of Hilbert s programme. 1 Godel s work was later extended to the Bar Recursive Functionals by Spector [27] who used them to give a constructive consistency proof for classical analysis. However the first full blown ....

K. Godel. Uber eine bisher noch nicht benutzte Erweiterung des finiten Standpunktes. Dialectica, pages 280--287, 1958.

.... a leurs comparaison syntaxique [13] Pourquoi l unification dans le syst eme T : Nous nous proposons dans ce travail d etendre l algorithme d unification d edi e au calcul simplement typ e en un algorithme d unification d edi e au syst eme T de G ODEL. Le syst eme T fut introduit par G ODEL [4] en 1958 pour prouver la coh erence de l arithm etique du premier ordre. C est un calcul typ e a un seul type de base nat , mais avec plus de r egles de r eduction. Ces r eductions correspondent a des r egles d it eration sur les entiers. Ce syst eme est tr es expressif du point de vue ....

....d entre eux sont particulierement tres expressifs. Le plus expressif d entres eux est le Calcul des Construction Inductifs, qui comprend des types inductif, polymorphes, dependants et des constructeurs de types. 1. 3 Vers le Syst eme T Le syst eme T a et e initialement introduit par G ODEL [4] en 1958 pour montrer la coherence de l arithmetique de premi er ordre, c est plus tard que l on consid erra le syst eme T comme un syst eme qui peut palier la pauvret e d expression du calcul simplement typ e, o u il n est possible de repr esenter que les fonctions constante et les fonctions ....

K. Godel, Uber eine Bisher Noch Nicht Benutzte Erweiterung des Finiten Standpunktes, Dialectica, 12, 1958.

....governing the Typerec form. Conceptually, Typerec selects i , Theta , or according to the head constructor of the normal form of and passes it the components of and the unrolling of the Typerec on the components. The level of constructors and kinds is a variation of Godel s T [18]. Every constructor, has a unique normal form, NF ( with respect to the obvious notion of reduction derived from the equivalence rules of Figure 2 [47] This reduction relation is confluent, from which it follows that constructor equivalence is decidable [47] The type formation, type ....

K. Godel. Uber eine bisher noch nicht benutzte Erweiterung des finiten Standpunktes. Dialectica, 12:280--287, 1958.

....each of kind Omega Omega . The constructor equivalence rules (Figure 3) axiomatize definitional equality [47, 34] of constructors to consist of fi conversion together with recursion equations governing the Typerec form. The level of constructors and kinds is a variation of Godel s T [17]. Every constructor, has a unique normal form, NF( with respect to the obvious notion of reduction derived from the equivalence rules of Figure 3 [47] This reduction relation is confluent, from which it follows that constructor equivalence is decidable [47] The type formation and ....

Kurt Godel. Uber eine bisher noch nicht benutzte Erweiterung des finiten Standpunktes. Dialectica, 12:280--287, 1958.

....tree of a term r by the expression r , where rk r is the level of the greatest type occurring in r and is its size. Adding the # rule. We retrace the traditional Schutte style [Sch51] prooftheoretic analysis of Peano arithmetic by embedding a # calculus based formulation of Godel s system T [God58] into the semiformal system T# with an infinitary branching # rule, to which we can extend both the continuous normalization function and the cut elimination proof. In contrast to conventional treatments of T# , this proof allows for permutative conversions and normalization of open terms rather ....

Kurt Godel. Uber eine bisher noch nicht benutzte Erweiterung des finiten Standpunkts. Dialectica, 12:280--287, 1958.

.... all inductive clauses have nitely many premises only (see [Tro73] For reasons of proof theoretical strength we cannot hope to extend the simple normalization proof to G odel s T without major modi cations since it is well known that termination of T implies consistency of Peano arithmetic [G od58] and that this fact is provable in primitive recursive arithmetic. It will turn out that the inductive characterization SN of the strongly normalizing terms can be extended to T while the embedding of the set of all terms requires the introduction of an in nitely branching clause in an inductive ....

Kurt Godel.  Uber eine bisher noch nicht benutzte Erweiterung des niten Standpunkts. Dialectica, 12:280-287, 1958.

....r by the expression 2 rk r jjrjj, where rk r is the level of the greatest type occurring in r and jjrjj is its size. Adding the rule. We retrace the traditional Sch utte style [Sch51] prooftheoretic analysis of Peano arithmetic by embedding a calculus based formulation of G odel s system T [G od58] into the semiformal system T1 with an in nitary branching rule, to which we can extend both the continuous normalization function and the cut elimination proof. In contrast to conventional treatments of T1 , this proof allows for permutative conversions and normalization of open terms rather ....

Kurt Godel.  Uber eine bisher noch nicht benutzte Erweiterung des niten Standpunkts. Dialectica, 12:280-287, 1958.

.... all inductive clauses have nitely many premises only (see [Tro73] For reasons of proof theoretical strength we cannot hope to extend the simple normalization proof to G odel s T without major modi cations since it is well known that termination of T implies consistency of Peano arithmetic [G od58] and that this fact is provable in primitive recursive arithmetic. It will turn out that the inductive characterization SN of the strongly normalizing terms can be extended to T while the embedding of the set of all terms requires the introduction of an in nitely branching clause in an inductive ....

Kurt Godel.  Uber eine bisher noch nicht benutzte Erweiterung des niten Standpunkts. Dialectica, 12:280-287, 1958.

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Kurt Godel. Uber eine bisher noch nicht benutzte Erweiterung des finiten Standpunktes. Dialectica, 12:280--287, 1958.

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Kurt Godel. Uber eine bisher noch nicht benutzte erweiterung des finiten standpunktes. Dialectica, 12:280--287, 1958.

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K. Godel. Uber eine bisher noch nicht benutzte Erweiterung des finiten Standpunktes. Dialectica, 12:280--287, 1958.

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Kurt Godel. Uber eine bisher noch nicht benutzte Erweiterung des finiten Standpunktes. Dialectica, 12:280--287, 1958.

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Godel, K., Uber eine bisher noch nicht benutzte Erweiterung des finiten Standpunktes. Dialectica 12, pp. 280--287 (1958).

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K. Godel. Uber eine bisher noch nicht benutzte Erweiterung des finiten Standpunktes. Dialectica, 12:280--287, 1958.

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Godel, K., Uber eine bisher noch nicht benutzte Erweiterung des finiten Standpunktes. Dialectica 12, pp. 280--287 (1958).

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K: Godel, Uber eine bisher noch nicht benutzte Erweiterung des finiten Standpunktes, Dialectica, Vol: 12, (1958) 280-287.

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K. Godel (1958), Uber eine bisher noch nicht benutzte Erweiterung des finiten Standpunktes, Dialectica 12, pp. 280-287.

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