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A simple proof of Parsons' theorem - Ferreira (1 citation) (Correct)
....sense. As noted, Mints argument uses the no counterexample interpretation which, being restricted here to one quantifier formulas, reminds one of Samuel Buss technique of witness functions [Bus85] For a witness function account of Parsons theorem, see [Bus98] Takeuti s proof appears in [Tak75]. 5. Sieg has an earlier, rather convoluted, proof of Parsons theorem in [Sie85] The proof technique used in [Sie91] was foreshadowed by an argument in [Fer90] 6. More precisely, we need a version of the Propriete A of first order validities (of the form ###) introduced by Jacques ....
Gaisi Takeuti. Proof Theory. North-Holland, 1975.
An Exercise in Programming in PA: Fast Growing Hierarchy Functions - Voda (Correct)
....= 0 0 and [0] from Prog x . In the inductive case we assume Prog x and 0 we have either a = 0 and so [0] from Prog x , or else a = 0; b where b 0 . We then get [b] from IH and [0; b] from 2. 5(2) The proof of (2) is based on a jump construction of Gentzen [Ge43] see also Takeuti [Tak75]) For a given m 1 we take any m or m formula [x] We set 0 [x] x] and n 1 [x] 8a( n [a] n [x; a] Clearly, n 1 [x] is n m 1 in I 1 and so we can use n as induction formulas in I n m because n induction is admissible in I n for all n. For any n 0 4 we prove rst: ....
G. Takeuti. Proof Theory. North-Holland, 1975.
Designing a Theorem Prover - Lawrence Paulson Computer (1990) (9 citations) (Correct)
....as far as possible. So #x.P # Q abbreviates #x. P # Q) and #y.P ##z.Q# R abbreviates #y. P # (#z. Q R) 1. 1 Representation of rules Given that Folderol will handle classical first order logic, what formal system is best suited for automation Gentzen s sequent calculus LK [Takeuti 1987] supports backwards proof in a natural way. A nave process of working upwards from the desired conclusion yields many proofs. Most rules act on one formula, and they come in pairs: one operating on the left and one on the right. We must inspect the rules carefully before choosing data structures. ....
....the proviso of the rule that created b. To perform the assignment a t, replace a by t throughout the proof. Replace each parameter depending on a by one depending on the meta variables in t. A lemma about LK justifies the substitution of terms for meta variables under suitable conditions [Takeuti 1987, Lemma 2.11] There is no logical need to distinguish meta variables from parameters. But the distinction helps us and Folderol remember which variables are candidates for substitution. 2 Basic data structures and operations Only now, having studied proof construction, can we choose a ....
G. Takeuti. Proof Theory. North Holland, 2nd edition.
Sketch-as-Proof - Preining (Correct)
....the labeled points the Points of the minimal projective plane. There are no more Points, Lines, especially no intersections as the holes in the gure should suggest. 3 The Calculus LPGK The calculus LPGK is based on Gentzen s LK, but extends it by certain means. The usual notations as found in [6] are used. ione and only onej can be replaced by ionej, because the fact that there is not more than one Point can be proven from axiom (PG1) More precise: The iideal Pointsj are the congruence classes of the lines with respect to the parallel relation and the iideal Linej is the class of ....
....notation in projective geometry. Finally I(P; g) will be written PIg. We also lose the subscript P and L in 8 P , since the right quantier is easy to deduce from the bound variable. The formulization of terms, atomic formulas and formulas is a standard technique and can be found in [6]. Capital letters are also used for formulas, but this shouldn t confuse the reader, since the context in each case is totally dioeerent. 3.2 The Rules and Initial Sequents of LPGK Denition 1. A logical initial sequent is a sequent of the form A A, where A is atomic. The mathematical ....
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Gaisi Takeuti. Proof Theory. North Holland, 1987.
Sketch-as-proof: A proof-theoretic analysis of axiomatic.. - Preining (Correct)
....introduction to model theory) On the other hand there is proof theory, which investigates the syntactical properties of sentences and proofs. Since the proof is the basic derivation concept maybe the best 10 of a mathematician, it is interesting to learn what the meaning of a proof is (see [15] and [4] for good introductions to proof theory) In a sense, mathematics is a collection of proofs. Therefore, in investigating imathematicsj, a fruitful method is to formalize proofs of mathematics and investigate the structure of these proofs. This is what proof theory is concerned with. So ....
....thesis the following logical symbols are used: for inotj for iandj for iorj oe for iimpliesj 8 for ifor allj 9 for ithere existsj and that parentheses are used freely for better readability. A detailed exposition of the formalization can be found in various books on proof theory (e.g. [15], 4] In the following, let Greek capital letters Gamma , Delta, Pi, Gamma 0 , denote nite (possibly empty) sequences of formulas separated by commas. Definition 3.1 For arbitrary Gamma and Delta, Gamma Delta is called a sequent. Gamma and Deltaare called the antecedent and ....
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Gaisi Takeuti. Proof Theory. North Holland, 1987. 66 BIBLIOGRAPHY 67
On Herbrand's Theorem - Buss (1995) (6 citations) (Correct)
....incompleteness. All proofs in this paper are presented in terms of the sequent calculus; however for space reasons, background material and definitions for the sequent calculus are not included in this paper. A reader unfamiliar with the sequent calculus should either skip all proofs or refer to [1,17] for definitions. We are grateful to R. Parikh and W. Goldfarb for comments on an earlier draft of this paper. 2 The weak form of Herbrand s theorem Herbrand s theorem is one of the fundamental theorems of mathematical logic and allows a certain type of reduction of first order logic to ....
G. TAKEUTI, Proof Theory, North-Holland, Amsterdam, 2nd ed., 1987.
On Gödel's Theorems on Lengths of Proofs I: Number of Lines and.. - Buss (Correct)
....for all formulas of Z i , and (4) if i 0, comprehension axioms (#X) #x) x X for all formulas # and variables x and X of orders j 1 and j i, respectively. For the purposes of this paper, Z i must be formalized in a Hilbert style system, not in the Gentzen style G LC of Takeuti [16]. If the We are using and as function symbols but our methods work equally well if they are three place relation symbols. The di#culty with G LC is that it allows the use of substitution of formulas for variables. This is logically equivalent to the use of the comprehension axiom; ....
....By Kreisel s conjecture we mean the statement that, for all formulas A(x) if there is a fixed k such that for all n, Z i A(n) then Z i # #xA(x) The reason Kreisel s conjecture is usually stated with the term S (0) instead of n . See Parikh [11] or Kreisel s footnote on page 400 of [16] for more information on Kreisel s conjecture. 14 that Kreisel s conjecture implies the last question has answer No is that otherwise Z i would prove its own consistency. On the other hand, we have the following example of a theory that does not satisfy Kreisel s conjecture and does not have ....
G. Takeuti, Proof Theory, North-Holland, 2nd ed., 1987.
On Partial and Paraconsistent Logics - Muskens (1999) (Correct)
....will also be two notions of positive and negative occurrences of formulas. It will turn out that the theorem holds for both these notions. The proof of the theorem was inspired by the proof in Kleene [20] for the classical case, but also bears resemblance to the Maehara method discussed in Takeuti [27]. Proofs of interpolation for have been known before (see [21] but since it is unclear how interpolation for could be obtained from these, we give a direct proof here. 23 We colour signed sentences in proofs in order to be able to keep track of them. A coloured signed sentence is a ....
G. Takeuti. Proof Theory. North-Holland, Amsterdam, 1975.
Provably Total Functions in - Bounded Arithmetic Theories Self-citation (Takeuti) (Correct)
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G. Takeuti, Proof Theory, North-Holland, Amsterdam, 2nd ed., 1987.
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K. Schutte. Proof Theory. North-Holland, 1962.
Decision Problems for Second-Order Linear Logic - Lincoln, Scedrov, Shankar (1995) (6 citations) (Correct)
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The Deduction Rule and Linear and Near-linear Proof Simulations - Bonet, Buss (1993) (Correct)
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G. Takeuti, Proof Theory, North-Holland, Amsterdam, 2nd ed., 1987. 34
On Herbrand's Theorem - Samuel Buss University (1995) (6 citations) (Correct)
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G. Takeuti, Proof Theory, North-Holland, Amsterdam, 2nd ed., 1987.
On the Deduction Rule and - The Number Of (Correct)
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G. Takeuti, Proof Theory, North-Holland, Amsterdam, 2nd ed., 1987.
An Equivalence between Second Order Bounded Domain Bounded.. - Razborov (1993) (7 citations) (Correct)
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G. Takeuti. Proof Theory. North-Holland, 1975. Rus. per.: G. Takeuti, Teori dokazatel~stv, M., Mir, 1978.
Intuitionistic Autoepistemic Logic - Aiello, Amati, Pirri (1997) (1 citation) (Correct)
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Bounded Arithmetic and Constant Depth Frege Proofs - Buss (2004) (Correct)
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G. Takeuti, Proof Theory, North-Holland, Amsterdam, 2nd ed., 1987.
Explicit Provability And Constructive Semantics - Artemov (2001) (1 citation) (Correct)
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G. Takeuti, Proof theory, North-Holland, 1975.
On Herbrand's Theorem - Samuel Buss University (1995) (6 citations) (Correct)
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G. Takeuti, Proof Theory, North-Holland, Amsterdam, 2nd ed., 1987.
Some Remarks on Lengths of Propositional Proofs - Buss (2002) (3 citations) (Correct)
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G. Takeuti, Proof Theory, North-Holland, Amsterdam, 2nd ed., 1987.
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G. Takeuti. Proof Theory. North-Holland, 2nd edition, 1987.
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G. Takeuti. Proof Theory. North-Holland, Amsterdam, 2nd edition, 1987.
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