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

2002
*
Archive for Mathematical Logic
*

We construct by diagonalization

doi:10.1007/s001530100107
fatcat:rxdcxlxa2nfqvkl5cumnz7e4l4
*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*A*⊢*Found*(T ′ , Π 1 0 ), cf. [5] and [6, pp. 280-284] . ...##
###
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*. ...##
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The Logic of Provability
[chapter]

1998
*
Studies in Logic and the Foundations of Mathematics
*

Let T be

doi:10.1016/s0049-237x(98)80022-0
fatcat:v7s2oocxfncirggz7auo6xntcu
*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. ...##
###
Hilbert's program and the omega-rule

1994
*
Journal of Symbolic Logic (JSL)
*

In order to investigate the

doi:10.2307/2275269
fatcat:qcdh3c7xabeorpqvbqoxbdljnq
*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). ...##
###
Aspects of categorical recursion theory
[article]

2020
*
arXiv
*
pre-print

We present

arXiv:2001.05778v1
fatcat:orhmpipltngtdcnkuq2y7rvqaa
*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 ...##
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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

doi:10.1017/s0022481200128312
fatcat:b5vdrwqssben7nmq7vyxrwrp2u
*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 ...##
###
Short Proofs for Slow Consistency
[article]

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*. ...

##
###
Implication and analysis in classical frege structures

1987
*
Annals of Pure and Applied Logic
*

We claim f(

doi:10.1016/0168-0072(87)90040-6
fatcat:iwkfjwsjcfehnmndhkyewotb6y
*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. ...##
###
A computable expression of closure to efficient causation

2009
*
Journal of Theoretical Biology
*

An important implication of our formulation is that, by exhibiting an expression in λ-calculus, which is

doi:10.1016/j.jtbi.2008.12.012
pmid:19168079
fatcat:ywosyigg6zejdi7ah66gf6l3ry
*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. ...##
###
Principles of continuous choice and continuity of functions in formal systems for constructive mathematics

1977
*
Annals of Mathematical Logic
*

Namely, Y must be complete, and

doi:10.1016/s0003-4843(77)80003-x
fatcat:jq37ou5dvzeblau7oigj2hyypm
*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. ...##
###
Emil Post
[chapter]

2009
*
Handbook of the History of Logic
*

The biographical essay [Gleiser, 1980] is

doi:10.1016/s1874-5857(09)70016-0
fatcat:erwzbkejovc7vfrx2xbqqmtybe
*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 ...##
###
Principles of programming with complex objects and collection types

1995
*
Theoretical Computer Science
*

The most general operation on

doi:10.1016/0304-3975(95)00024-q
fatcat:hompuhysungrhjokpdphfdgaxe
*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*...##
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INFORMAL PROOF, FORMAL PROOF, FORMALISM

2015
*
The Review of Symbolic Logic
*

In this paper I look at the relevance of these issues

doi:10.1017/s1755020315000234
fatcat:aw2kgf7yifdc7kxgffbcmjqudy
*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. ...##
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ON COMPUTABILITY
[chapter]

2009
*
Philosophy of Mathematics
*

As examples

doi:10.1016/b978-0-444-51555-1.50017-1
fatcat:7fjtsyt6izeizbzm4zgoastvuy
*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. ...##
###
Kripke models and the intuitionistic theory of species

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. ...

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