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Page 1799 of Mathematical Reviews Vol. , Issue 85e
[page]
1985
Mathematical Reviews
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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
2013
Computational Complexity
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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
2004
Bulletin of Symbolic Logic
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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
2005
Bulletin of the American Mathematical Society
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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
2007
Journal für die Reine und Angewandte Mathematik
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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
2018
Mathematical Research Letters
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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
2009
Annals of Pure and Applied Logic
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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]
2018
arXiv
pre-print
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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
1952
Proceedings of the American Mathematical Society
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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
1952
Proceedings of the American Mathematical Society
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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]
2018
arXiv
pre-print
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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]
2018
arXiv
pre-print
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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
2021
The Australasian Journal of Logic
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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
1967
Transactions of the American Mathematical Society
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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
1967
Transactions of the American Mathematical Society
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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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