Kurt Godel. Zur intuitionistischen Arithmetik und Zahlentheorie. Ergebnisse eines mathematischen Kolloquiums, 4:34--38, 1933. Reprinted in [31], pages 286--295. Home/Search Document Not in Database Summary Related Articles Check |
This paper is cited in the following contexts:
Analyzing Proofs in Analysis - Kohlenbach (Correct)
....AC qf of quantifier free choice for the types #, # is defined as AC qf : A 0 (x, y) A 0 (x, Y x) A 0 quantifier free) We now carry out our monotone functional interpretation for WE HA . By doing first elimination of extensionality ( 26] and then negative translation ([13] ) this interpretation also applies to classical systems e.g. E PA qf. We stress that our interpretation also works for various subsystems of WE HA , e.g. the system from [11] with quantifier free induction and elementary recursor constants only and also to much weaker systems ....
Godel, K., Zur intuitionistischen Arithmetik und Zahlentheorie. Ergebnisse eines Mathematischen Kolloquiums, vol. 4 pp. 34--38 (1933).
On the Arithmetical Content of Restricted Forms of.. - Kohlenbach (1997) (Correct)
....of Pi k CA(f) i.e. by defining k CA(f) j 8l 9g k (f(l; x; u) 0 0) since the latter can be reduced to the former (relative to G n A for n 2) by coding l; x together and applying comprehension without number parameters to this pair. Here we can use Godel s [4] translation or any other of the various negative translations. For a systematical treatment of negative translations see [16] This last assertion is not stated in the formulation of the theorem in [13] but does follow immediately from its proof. Whether one has here 9u k or 8u k ....
Godel, K., Zur intuitionistischen Arithmetik und Zahlentheorie. Ergebnisse eines Mathematischen Kolloquiums, vol. 4 pp. 34--38 (1933).
Realizability: A Historical Essay - van Oosten (2000) (1 citation) (Correct)
....information effectively) He started to think about this in 1940 2 . In order to appreciate the originality of his thinking, one should recall that the formal system of intuitionistic arithmetic HA did not exist at the time [Well, there is a system closely resembling HA in Godel s paper [28]. Kleene appears to have been at least initially unaware of this, for although his 1945 paper gives the reference, the retrospective survey stresses that Heyting Arithmetic [ does not occur as a subsystem readily separated out from Heyting s full system of intuitionistic mathematics , and ....
K. Godel. Zur intuitionistischen arithmetik und zahlentheorie. Ergebnisse eines mathematisches Kolloquiums, 4:34--38, 1932.
Proof Interpretations and the Computational Content of Proofs - Kohlenbach (2002) (1 citation) (Correct)
....# where not every formula is provably equivalent to a prenex one. In the next lecture we will introduce a translation of the classical variant WE PA # of WE HA # (i.e. WE HA # plus the tertium non datur schema A # A) into WE HA # , the so called negative translation A ## A # due to [44]. We will see that the composition of # and D, A ## (A # ) D provides a very subtle constructive interpretation of A which faithfully reflects the proof theoretic and computational strength of A (in contrast to the no counterexample interpretation of (a prenex normal form of) A which in ....
....translations of classical logic as well as many theories based on classical logic into their intuitionistic variant. All these translations A ## A # have in common that A # is (or is intuitionistically equivalent to) a negative formula. The first such translation is due to Godel [44] (although G. Gentzen independently discovered a similar translation) There is some preceding work by Kolmogorov [88] and Glivenko [41] Two further variants of Godel s translation are due to Kuroda [99] and it his one of these which we will adopt here: Definition 7.1 Let A be a formula in a ....
Godel, K., Zur intuitionistischen Arithmetik und Zahlentheorie. Ergebnisse eines Mathematischen Kolloquiums, vol. 4 pp. 34--38 (1933). BIBLIOGRAPHY 99
On the Arithmetical Content of Restricted Forms of.. - Kohlenbach (1998) (Correct)
.... Pi 0 k CA which are used in given proofs of sentences 8u 1 8v tu B ar (u; v) as discussed in the introduction. Definition 3. 1 Pi 0 k CA(f) j 9g 1 8x 0 Gamma gx = 0 0 8u 0 1 9u 0 2 : 9 (d) u 0 k Gamma f(x; u) 0 0 Delta Delta : 10 8 Here we can use Godel s [4] translation or any other of the various negative translations. For a systematical treatment of negative translations see [17] 9 This last assertion is not stated in the formulation of the theorem in [13] but does follow immediately from its proof. 10 Whether one has here 9u 0 k or 8u ....
Godel, K., Zur intuitionistischen Arithmetik und Zahlentheorie. Ergebnisse eines Mathematischen Kolloquiums, vol. 4 pp. 34--38 (1933).
Realizability: A Historical Essay - van Oosten (2000) (Correct)
....information effectively) He started to think about this in 1940 2 . In order to appreciate the originality of his thinking, one should recall that the formal system of intuitionistic arithmetic HA did not exist at the time [Well, There is a system closely resembling HA in Godel s paper [26]. Kleene appears to have been at least initially unaware of this, for although his 1945 paper gives the reference, the retrospective survey stresses that Heyting Arithmetic [ does not occur as a subsystem readily separated out from Heyting s full system of intuitionistic mathematics , and ....
K. Godel. Zur intuitionistischen arithmetik und zahlentheorie. Ergebnisse eines mathematisches Kolloquiums, 4:34--38, 1932.
Higman's Lemma in Type Theory - Fridlender (1997) (Correct)
....lemma in terms of ordinals. This proof is a translation to English of the proof of Schutte and Simpson [SS85] which would then be the first constructive proof of Higman s lemma. Independently of Simpson s results, Stolzenberg noticed that as a consequence of Godel s double negation translation [God33] and Friedman s A translation [Fri78] Higman s lemma had a constructive proof. Moreover, the translations provided a method to build a constructive proof from a classical one. Murthy [Mur90] applied the method to Nash Williams proof, obtaining another constructive proof of the lemma, which was ....
K. Godel. Zur intuitionistischen Arithmetik und Zahlentheorie. Ergebnisse eines mathematischen Kolloquiums, 4:34--38, 1933. English:
Higman's Lemma in Type Theory - Fridlender (1997) (Correct)
....the lemma in terms of ordinals. This proof is a translation to English of the proof of Schutte and Simpson [SS85] which would then be the first constructive proof of Higman s lemma. Independently of Simpson s results, Stolzenberg noticed that as a consequence of Godel s double negation translation [God33] and Friedman s A translation [Fri78] Higman s lemma had a constructive proof. Moreover, the translations provided a method to build a constructive proof from a classical one. Murthy [Mur90] applied the method to Nash Williams proof, obtaining another constructive proof of the lemma, which was ....
K. Godel. Zur intuitionistischen arithmetik und zahlentheorie. Ergebnisse eines mathematischen Kolloquiums, 4:34--38, 1933. English: [God65].
Samsara - Crossley (2005) (Correct)
No context found.
Kurt Godel. Zur intuitionistischen Arithmetik und Zahlentheorie. Ergebnisse eines mathematischen Kolloquiums, 4:34--38, 1933. Reprinted in [31], pages 286--295.
Proof Mining in Subsystems of Analysis - Oliva (2003) (Correct)
No context found.
K. Godel. Zur intuitionistischen Arithmetik und Zahlentheorie. Ergebnisse eines Mathematischen Kolloquiums, 4:34--38, 1933.
Elimination of Skolem functions for monotone formulas in analysis - Kohlenbach (Correct)
No context found.
Godel, K., Zur intuitionistischen Arithmetik und Zahlentheorie. Ergebnisse eines Mathematischen Kolloquiums, vol. 4 pp. 34--38 (1933).
Arithmetizing Proofs in Analysis - Kohlenbach (1 citation) (Correct)
No context found.
Godel, K., Zur intuitionistischen Arithmetik und Zahlentheorie. Ergebnisse eines Mathematischen Kolloquiums, vol. 4 pp. 34--38 (1933).
Analyzing Proofs in Analysis - Kohlenbach (Correct)
No context found.
Godel, K., Zur intuitionistischen Arithmetik und Zahlentheorie. Ergebnisse eines Mathematischen Kolloquiums, vol. 4 pp. 34--38 (1933).
Online articles have much greater impact More about CiteSeer.IST Add search form to your site Submit documents Feedback
CiteSeer.IST - Copyright Penn State and NEC