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Page 1799 of Mathematical Reviews Vol. , Issue 85e [page]

1985 Mathematical Reviews  
In the first group, four statements are given about such functions on the Cantor space K = 2”; each statement is true but none is provable in second-order arithmetic (even in ZFC + V = L minus the power  ...  This theorem is applicable to the classical, intuitionistic and minimal systems of formal arithmetic and predicate calculus. M. I.  ... 

Resource Trade-offs in Syntactically Multilinear Arithmetic Circuits

Maurice Jansen, Meena Mahajan, B. V. Raghavendra Rao
2013 Computational Complexity  
Finally, in contrast to the general case, we report closure and stability of coefficient functions for the syntactically multilinear classes studied in this paper.  ...  Along the way we also observe a characterisation of the class NC 1 in terms of a restricted class of planar branching programs of polynomial size.  ...  Acknowledgements The results in this paper were announced in MFCS 2008 and CSR 2009 , in Jansen & Rao (2009 Mahajan & Rao (2008) .  ... 
doi:10.1007/s00037-013-0072-x fatcat:6jg6o6x5k5bkvm6tazo5dtm4fe

Forcing in Proof Theory

Jeremy Avigad
2004 Bulletin of Symbolic Logic  
Here, I argue that forcing also has a place in traditional Hilbert-style proof theory, where the goal is to formalize portions of ordinary mathematics in restricted axiomatic theories, and study those  ...  I will discuss the aspects of forcing that are useful in this respect, and some sample applications.  ...  Restrict induction to Σ 0 1 formulas with parameters, and restrict set existence principles: The Jockusch-Soare low basis theorem.  ... 
doi:10.2178/bsl/1102022660 fatcat:7sr63xqpabcqjdsuazxbi3rude

The Green-Tao Theorem on arithmetic progressions in the primes: an ergodic point of view

Bryna Kra
2005 Bulletin of the American Mathematical Society  
In an amazing fusion of methods from analytic number theory and ergodic theory, Ben Green and Terence Tao showed that for any positive integer k, there exist infinitely many arithmetic progressions of  ...  length k consisting only of prime numbers.  ...  If 1 A − |A|/N U k−1 is large, then the restriction of 1 A to some sufficiently large subset of Z N has many useful arithmetic properties and the average in Equation (4.1) is once again large.  ... 
doi:10.1090/s0273-0979-05-01086-4 fatcat:6gpllt6sv5b2nkfpkaewyttnli

Multiple recurrence and convergence for sequences related to the prime numbers

Nikos Frantzikinakis, Bernard Host, Bryna Kra
2007 Journal für die Reine und Angewandte Mathematik  
In an amazing fusion of methods from analytic number theory and ergodic theory, Ben Green and Terence Tao showed that for any positive integer k, there exist infinitely many arithmetic progressions of  ...  length k consisting only of prime numbers.  ...  If 1 A − |A|/N U k−1 is large, then the restriction of 1 A to some sufficiently large subset of Z N has many useful arithmetic properties and the average in Equation (4.1) is once again large.  ... 
doi:10.1515/crelle.2007.076 fatcat:3nqzyhrpdvc4vpwimlne4uuhke

Embedding arithmetic hyperbolic manifolds

Alexander Kolpakov, Alan W. Reid, Leone Slavich
2018 Mathematical Research Letters  
We prove that any arithmetic hyperbolic n-manifold of simplest type can either be geodesically embedded into an arithmetic hyperbolic (n + 1)-manifold or its universal mod 2 Abelian cover can.  ...  The second author was supported in part by an NSF grant and The Wolfensohn Fund administered through the Institute for Advanced Study. He would also like  ...  Acknowledgements The first and third authors were partially supported by the Italian FIRB project "Geometry and topology of lowdimensional manifolds", RBFR10GHHH.  ... 
doi:10.4310/mrl.2018.v25.n4.a12 fatcat:7gazwcllhrb4hio3fwn45axhsq

The metamathematics of ergodic theory

Jeremy Avigad
2009 Annals of Pure and Applied Logic  
This paper surveys some recent contributions to the metamathematical study of ergodic theory, focusing on the mean and pointwise ergodic theorems and the Furstenberg structure theorem for measure preserving  ...  In particular, I characterize the extent to which these theorems are nonconstructive, and explain how proof-theoretic methods can be used to locate their "constructive content".  ...  For f in L 2 (X), Gerhardy, Towsner, and I provide explicit bounds on n in terms of f , λ 1 , λ 2 , and K .  ... 
doi:10.1016/j.apal.2008.09.001 fatcat:2fqr4futnngo3j62seebfsn5da

Computational reverse mathematics and foundational analysis [article]

Benedict Eastaugh
2018 arXiv   pre-print
Reverse mathematics studies which subsystems of second order arithmetic are equivalent to key theorems of ordinary, non-set-theoretic mathematics.  ...  to Simpson [1988], and predicativism in the extended form due to Feferman and Schütte.  ...  I am indebted to Carl Mummert for his proof of theorem 7.2 and for his patient explanations of the result and related matters.  ... 
arXiv:1807.10022v1 fatcat:z6xtuwgiofdwfptqvaixohsaiu

An Extension of Mercer's Theorem

R. T. Leslie, E. R. Love
1952 Proceedings of the American Mathematical Society  
The reduction consists in indicating, as is proved by Hurwitz [5] , that, if both Agnew also shows in a few lines that there is no need to assume that (i") is bounded.  ...  These cases are of sufficient intrinsic interest to merit separate statements, and in them the shape of Mercer's original theorem is clearly seen.  ...  If (ünk) is regular and triangular, \a"n\ ¡£d> 1/2 for all sufficiently large n, and n n E I ank I -» 1, E ankSk ~* S, *=1 *-l where (i") is bounded, then i"->i.  ... 
doi:10.2307/2031902 fatcat:zrdhh2olbra5bm4rwbrhpgbdce

An extension of Mercer's theorem

R. T. Leslie, E. R. Love
1952 Proceedings of the American Mathematical Society  
The reduction consists in indicating, as is proved by Hurwitz [5] , that, if both Agnew also shows in a few lines that there is no need to assume that (i") is bounded.  ...  These cases are of sufficient intrinsic interest to merit separate statements, and in them the shape of Mercer's original theorem is clearly seen.  ...  If (ünk) is regular and triangular, \a"n\ ¡£d> 1/2 for all sufficiently large n, and n n E I ank I -» 1, E ankSk ~* S, *=1 *-l where (i") is bounded, then i"->i.  ... 
doi:10.1090/s0002-9939-1952-0047159-6 fatcat:none2rbjeze5dcrvpriv2g34za

Embedding arithmetic hyperbolic manifolds [article]

Alexander Kolpakov, Alan W. Reid, Leone Slavich
2018 arXiv   pre-print
We prove that any arithmetic hyperbolic n-manifold of simplest type can either be geodesically embedded into an arithmetic hyperbolic (n+1)-manifold or its universal mod 2 Abelian cover can.  ...  The second author was supported in part by an NSF grant and The Wolfensohn Fund administered through the Institute for Advanced Study.  ...  ., Trieste, Italy, and the organizers of the "Advanced School on Geometric Group Theory and Low-Dimensional Topology: Recent Connections and Advances" held at the I.C.T.P. in June 2016, which allowed them  ... 
arXiv:1703.10561v6 fatcat:ohygrnionbg7hirs2ptbu3r2am

The modal logic of arithmetic potentialism and the universal algorithm [article]

Joel David Hamkins
2018 arXiv   pre-print
With respect to sentences, however, the validities of a model lie between S4 and S5, and these bounds are sharp in that there are models realizing both endpoints.  ...  For a model of arithmetic to validate S5 is precisely to fulfill the arithmetic maximality principle, which asserts that every possibly necessary statement is already true, and these models are equivalently  ...  Since every extension is ∆ 0 -elementary, it follows that Σ 1 statements true in M are preserved to N , and so N satisfies every assertion in the Σ 1 type of a in M .  ... 
arXiv:1801.04599v3 fatcat:6om3ywyx25gabeiucqxbegti2y

Arithmetic Formulated in a Logic of Meaning Containment

Ross Brady
2021 The Australasian Journal of Logic  
We argue in favour of the latter as it better captures the key logical concepts of meaning and truth in arithmetic.  ...  We assess Meyer's formalization of arithmetic in his [21], based on the strong relevant logic R and compare this with arithmetic based on a suitable logic of meaning containment, which was developed in  ...  As for classical recapture, MC does not include the LEM, and we instead prove the simple consistency of the arithmetic MC and, as a result, show that the DS is admissible and finally add it to the logic  ... 
doi:10.26686/ajl.v18i5.6920 fatcat:lsuzih6pdvgn7exiq4tujpg67u

The forcing method and the upper semilattice of hyperdegrees

S. K. Thomason
1967 Transactions of the American Mathematical Society  
Denote least upper bounds and greatest lower bounds in P by u and n, respectively.  ...  Finally, <j> is lattice-preserving if it is both u-preserving and n-preserving. Theorem 14.  ...  By Corollary 1 and Theorem 17, A = hyp (B') for some B. By relativizing the proof of Theorem 18, obtain sets A0, Ax <h B' such that A0 u Ax =k B'.  ... 
doi:10.1090/s0002-9947-1967-0219421-0 fatcat:szevyqze3vhcririkelqz7g3ni

The Forcing Method and the Upper Semilattice of Hyperdegrees

S. K. Thomason
1967 Transactions of the American Mathematical Society  
Denote least upper bounds and greatest lower bounds in P by u and n, respectively.  ...  Finally, <j> is lattice-preserving if it is both u-preserving and n-preserving. Theorem 14.  ...  By Corollary 1 and Theorem 17, A = hyp (B') for some B. By relativizing the proof of Theorem 18, obtain sets A0, Ax <h B' such that A0 u Ax =k B'.  ... 
doi:10.2307/1994363 fatcat:zh3ozfnkvre4npz3cdjvxno2bq
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