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A non-well-founded primitive recursive tree provably well-founded for co-r.e. sets

Arnold Beckmann
2002 Archive for Mathematical Logic  
We construct by diagonalization a non-well-founded primitive recursive tree, which is well-founded for co-r.e. sets, provable in Σ 0 1 -IND.  ...  It follows that the supremum of order-types of primitive recursive wellorderings, whose well-foundedness on co-r.e. sets is provable in Σ 0 1 -IND, equals the limit of all recursive ordinals ω CK 1 .  ...  Fix a recursive non-well-founded tree T ′ such that P A proves transfinite induction of T ′ for any arithmetical set, P AFound(T ′ , Π 1 0 ), cf. [5] and [6, pp. 280-284] .  ... 
doi:10.1007/s001530100107 fatcat:rxdcxlxa2nfqvkl5cumnz7e4l4

Page 811 of Mathematical Reviews Vol. , Issue 2003B [page]

2003 Mathematical Reviews  
Summary: “We construct by diagonalization a non-well-founded primitive recursive tree which is well-founded for co-r.e. sets, and this well-foundedness is provable in L}-IND.  ...  Christophe Raffalli (F-SAVO-LA2; Le Bourget-du-Lac) 2003b:03080 03F15 03F30 Beckmann, Arnold (1-UCSD; La Jolla, CA) A non-well-founded primitive recursive tree provably well-founded for co-r.e. sets.  ... 

The Logic of Provability [chapter]

Giorgi Japaridze, Dick de Jongh
1998 Studies in Logic and the Foundations of Mathematics  
Let T be a well-speci ed r.e. set of modal formulas. Then T is realistic i T is consistent with S.  ...  However, no results to the e ect that Solovay's theorems hold for broad classes of non-r.e. predicates are known.  ... 
doi:10.1016/s0049-237x(98)80022-0 fatcat:v7s2oocxfncirggz7auo6xntcu

Hilbert's program and the omega-rule

Aleksandar Ignjatović
1994 Journal of Symbolic Logic (JSL)  
In order to investigate the provability with such a modified rule, we define new consistency and provability predicates which are weaker than the usual ones.  ...  We show that Detlefsen's proposal is unacceptable as originally formulated in [1], but that a reasonable modification of the rule he suggest leads to a partial program already studied for many years.  ...  to £ 0 for a primitive recursive well-ordering applied only to primitive recursive predicates suffices to show the consistency of (PA).  ... 
doi:10.2307/2275269 fatcat:qcdh3c7xabeorpqvbqoxbdljnq

Aspects of categorical recursion theory [article]

Pieter Hofstra, Philip Scott
2020 arXiv   pre-print
We present a survey of some developments in the general area of category-theoretic approaches to the theory of computation, with a focus on topics and ideas particularly close to the interests of Jim Lambek  ...  Conclusion We hope that we have shown in this -admittedly biased-overview of categorical recursion theory how various of Lambek's seminal ideas have initiated and inspired numerous strands of research  ...  We also hope to have conveyed to the reader that there are still many interesting unanswered questions and relatively unexplored facets of categorical recursion theory that deserve further investigation  ... 
arXiv:2001.05778v1 fatcat:orhmpipltngtdcnkuq2y7rvqaa

Abstracts of Papers

1966 Journal of Symbolic Logic (JSL)  
Otherwise these axioms get their full intuitive strength only from the interpretation of predicate variables in a level of "sets" exceeding the given axiomatic frame.  ...  Predicate variables in set theory.  ...  The second (slightly modified) step in the classification is to define a r.e. sequence of r.e. sets Co, Ci, .... such that if c e is the degree of C e then a i c , g b and R e £ G(a) = C e e G(b) for all  ... 
doi:10.1017/s0022481200128312 fatcat:b5vdrwqssben7nmq7vyxrwrp2u

Short Proofs for Slow Consistency [article]

Anton Freund, Fedor Pakhomov
2017 arXiv   pre-print
For a large class of natural theories T, Pudlák has shown that the lengths of the shortest proofs of Con( T)n in the theory T itself are bounded by a polynomial in n.  ...  Our argument is proof-theoretic, while previous investigations of slow consistency relied on non-standard models of arithmetic.  ...  Now set f (2x + 3) := max{f 2 (p) + 1 | p is a proof IΣ x + F * ε0 ↓ ⊢ 0 = 1 of length ≤ g(x)}. Clearly f is primitive recursive.  ... 
arXiv:1712.03251v1 fatcat:ez2p2b3ym5ek7fkhgwas7zgv2q

Implication and analysis in classical frege structures

Robert C. Flagg, John Myhill
1987 Annals of Pure and Applied Logic  
We claim f(cO = 0.  ...  A word is in order here concerning the relation between ext and to. ext is a fmitary rule, hence the set of (G6del numbers) of theorems (provable identities) of CL(A)+ ext is r.e.  ...  It is this that forces us to use non-truth functional implications even for the very beginnings of real numbers.  ... 
doi:10.1016/0168-0072(87)90040-6 fatcat:iwkfjwsjcfehnmndhkyewotb6y

A computable expression of closure to efficient causation

Matteo Mossio, Giuseppe Longo, John Stewart
2009 Journal of Theoretical Biology  
An important implication of our formulation is that, by exhibiting an expression in λ-calculus, which is a paradigmatic formalism for computability and programming, we show that there are no conceptual  ...  or principled problems in realizing a computer simulation or model of closure to efficient causation.  ...  In fact, even the constant or the primitive recursive functions (which contain no universal primitive recursive function) have nonr.e. sets of all r.e. indexes.  ... 
doi:10.1016/j.jtbi.2008.12.012 pmid:19168079 fatcat:ywosyigg6zejdi7ah66gf6l3ry

Principles of continuous choice and continuity of functions in formal systems for constructive mathematics

Michael J. Beeson
1977 Annals of Mathematical Logic  
Namely, Y must be complete, and for each a, the set {b E Y : P(a, b)} must be closed as a subset of Y. The resulting principle is LC*(X,Y): H&Va~X3bE YP(a,b)&VaEX{b~Y:P(a,b)} is closed  ...  Let us call a solution b "stable for a" if Ve>038 >0VcEN~(a)3d~N,(b)P(c,d). (Here N~(a) is the neighborhood of a with radius 8.)  ...  It is well-known that (even in HA) there is a recursive binary tree which is well-founded with respect to recursive descending sequences but is still finite.  ... 
doi:10.1016/s0003-4843(77)80003-x fatcat:jq37ou5dvzeblau7oigj2hyypm

Emil Post [chapter]

UrquharT. Alasdair
2009 Handbook of the History of Logic  
The biographical essay [Gleiser, 1980] is a very informative source on Post's life. The description of Post's papers was obtained from the web site of the American Philosophical Society, which is  ...  sets (co-r.e. sets) constitute Π 1 .  ...  In Post's terminology, the set K is a creative set, that is to say, a recursively enumerable set C for which there exists a recursive function f giving a unique positive integer n = f (i) for each basis  ... 
doi:10.1016/s1874-5857(09)70016-0 fatcat:erwzbkejovc7vfrx2xbqqmtybe

Principles of programming with complex objects and collection types

Peter Buneman, Shamim Naqvi, Val Tannen, Limsson Wong
1995 Theoretical Computer Science  
The most general operation on sets, that of structural recursion, is one in which not all programs are well-defined.  ...  Thus rather than developing query languages by extending, for example, relational calculus, we advocate a very powerful paradigm in which a number of well-known languages are to be found as natural sublanguages  ...  Acknowledgement The authors thank Foto Afrati, Dirk Van Gucht, Leonid Libkin, Hermann Puhlmann, Jon Riecke, Dan Suciu, and Steve Vickers for helpful discussions and Paul Taylor for  ... 
doi:10.1016/0304-3975(95)00024-q fatcat:hompuhysungrhjokpdphfdgaxe

INFORMAL PROOF, FORMAL PROOF, FORMALISM

ALAN WEIR
2015 The Review of Symbolic Logic  
In this paper I look at the relevance of these issues for formalism, construed as an anti-platonistic metaphysical doctrine.  ...  Whereas some claim that any correct proof will be underwritten by a fully formal proof, sceptics demur.  ...  is defined in terms of a primitive notion of modality.  ... 
doi:10.1017/s1755020315000234 fatcat:aw2kgf7yifdc7kxgffbcmjqudy

ON COMPUTABILITY [chapter]

Wilfried Sieg
2009 Philosophy of Mathematics  
As examples for classes F Herbrand considers the set E 1 of addition and multiplication, as well as the set E 2 of all primitive recursive functions.  ...  It is not difficult to show that the recursive sets are exactly those that are r.e. and have an r.e. complement.  ... 
doi:10.1016/b978-0-444-51555-1.50017-1 fatcat:7fjtsyt6izeizbzm4zgoastvuy

Kripke models and the intuitionistic theory of species

D.H.J. de Jongh, C. Smorynski
1976 Annals of Mathematical Logic  
For theories HAS + 1-', we must discuss sets F which are preserved under the operation ( )--~ (X )'. Here, ~ is a primitive recursive wel!  ...  See Section 5, below, for references. [] Corollary 1.9, H • AC-NS and HAS have the same provable recursive jiinctions. Remark.  ... 
doi:10.1016/0003-4843(76)90008-5 fatcat:tjegtssx6bhpnltn5cfaypyioy
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