Lifting Recursive Counterexamples to Higher-Order Arithmetic

2019 ◽  
pp. 249-267
Author(s):  
Sam Sanders
Keyword(s):  
Higher Order ◽  
Order Arithmetic

Extension of Lifschitz' realizability to higher order arithmetic, and a solution to a problem of F. Richman

1991 ◽  
Vol 56 (3) ◽  
pp. 964-973 ◽  
Author(s):  
Jaap van Oosten
Keyword(s):  
Higher Order ◽  
Second Order ◽  
Church’S Thesis ◽  
Second Order Arithmetic ◽  
Church's Thesis ◽  
Order Arithmetic ◽  
Image Position

AbstractF. Richman raised the question of whether the following principle of second order arithmetic is valid in intuitionistic higher order arithmetic HAH:and if not, whether assuming Church's Thesis CT and Markov's Principle MP would help. Blass and Scedrov gave models of HAH in which this principle, which we call RP, is not valid, but their models do not satisfy either CT or MP.In this paper a realizability topos Lif is constructed in which CT and MP hold, but RP is false. (It is shown, however, that RP is derivable in HAH + CT + MP + ECT0, so RP holds in the effective topos.) Lif is a generalization of a realizability notion invented by V. Lifschitz. Furthermore, Lif is a subtopos of the effective topos.

scholarly journals HARRINGTON’S PRINCIPLE IN HIGHER ORDER ARITHMETIC

2015 ◽  
Vol 80 (2) ◽  
pp. 477-489 ◽  
Author(s):  
YONG CHENG ◽  
RALF SCHINDLER
Keyword(s):  
Higher Order ◽  
Order Arithmetic ◽  
Image Position

AbstractLet Z2, Z3, and Z4 denote 2nd, 3rd, and 4th order arithmetic, respectively. We let Harrington’s Principle, HP, denote the statement that there is a real x such that every x-admissible ordinal is a cardinal in L. The known proofs of Harrington’s theorem “$Det\left( {{\rm{\Sigma }}_1^1} \right)$ implies 0♯ exists” are done in two steps: first show that $Det\left( {{\rm{\Sigma }}_1^1} \right)$ implies HP, and then show that HP implies 0♯ exists. The first step is provable in Z2. In this paper we show that Z2 + HP is equiconsistent with ZFC and that Z3 + HP is equiconsistent with ZFC + there exists a remarkable cardinal. As a corollary, Z3 + HP does not imply 0♯ exists, whereas Z4 + HP does. We also study strengthenings of Harrington’s Principle over 2nd and 3rd order arithmetic.

On Gödel's theorems on lengths of proofs I: Number of lines and speedup for arithmetics

1994 ◽  
Vol 59 (3) ◽  
pp. 737-756 ◽  
Author(s):  
Samuel R. Buss
Keyword(s):  
Lower Bounds ◽  
Higher Order ◽  
Order Logic ◽  
Use Of Self ◽  
Higher Order Logic ◽  
Order Arithmetic ◽  
Self Reference

AbstractThis paper discusses lower bounds for proof length, especially as measured by number of steps (inferences). We give the first publicly known proof of Gödel's claim that there is superrecursive (in fact, unbounded) proof speedup of (i + l)st-order arithmetic over ith-order arithmetic, where arithmetic is formalized in Hilbert-style calculi with + and • as function symbols or with the language of PRA. The same results are established for any weakly schematic formalization of higher-order logic: this allows all tautologies as axioms and allows all generalizations of axioms as axioms.Our first proof of Gödel's claim is based on self-referential sentences: we give a second proof that avoids the use of self-reference based loosely on a method of Statman.

scholarly journals THE EXOTIC (HIGHER ORDER LÉVY) LAPLACIANS GENERATE THE MARKOV PROCESSES GIVEN BY DISTRIBUTION DERIVATIVES OF WHITE NOISE

2013 ◽  
Vol 16 (03) ◽  
pp. 1350020 ◽  
Author(s):  
LUIGI ACCARDI ◽  
UN CIG JI ◽  
KIMIAKI SAITÔ
Keyword(s):  
White Noise ◽  
Higher Order ◽  
Technical Tool ◽  
Arithmetic Means ◽  
Infinite Dimensional ◽  
Second Derivatives ◽  
Lévy Laplacian ◽  
Order Arithmetic ◽  
Derivatives Of ◽  
Main Technical Tool

We introduce, for each a ∈ ℝ+, the Brownian motion associated to the distribution derivative of order a of white noise. We prove that the generator of this Markov process is the exotic Laplacian of order 2a, given by the Cesàro mean of order 2a of the second derivatives along the elements of an orthonormal basis of a suitable Hilbert space (the Cesàro space of order 2a). In particular, for a = 1/2 one finds the usual Lévy Laplacian, but also in this case the connection with the 1/2-derivative of white noise is new. The main technical tool, used to achieve these goals, is a generalization of a result due to Accardi and Smolyanov5 extending the well-known Cesàro theorem to higher order arithmetic means. These and other estimates allow to prove existence of the heat semi-group associated to any exotic Laplacian of order ≥ 1/2 and to give its explicit expression in terms of infinite dimensional Fourier transform.

KURT GÖDEL’S FIRST STEPS IN LOGIC: FORMAL PROOFS IN ARITHMETIC AND SET THEORY THROUGH A SYSTEM OF NATURAL DEDUCTION

2018 ◽  
Vol 24 (3) ◽  
pp. 319-335
Author(s):  
JAN VON PLATO
Keyword(s):  
Linear System ◽  
Set Theory ◽  
Natural Deduction ◽  
Higher Order ◽  
Order Logic ◽  
Formal Proofs ◽  
Higher Order Logic ◽  
Order Arithmetic

AbstractWhat seem to be Kurt Gödel’s first notes on logic, an exercise notebook of 84 pages, contains formal proofs in higher-order arithmetic and set theory. The choice of these topics is clearly suggested by their inclusion in Hilbert and Ackermann’s logic book of 1928, the Grundzüge der theoretischen Logik. Such proofs are notoriously hard to construct within axiomatic logic. Gödel takes without further ado into use a linear system of natural deduction for the full language of higher-order logic, with formal derivations closer to one hundred steps in length and up to four nested temporary assumptions with their scope indicated by vertical intermittent lines.

scholarly journals On the Uniform Weak König’s Lemma

1999 ◽  
Vol 6 (11) ◽  
Author(s):  
Ulrich Kohlenbach
Keyword(s):  
Finite Type ◽  
Binary Tree ◽  
Higher Order ◽  
Elimination Process ◽  
Primitive Recursive Arithmetic ◽  
Classical Mathematics ◽  
Order Arithmetic ◽  
Primitive Recursive ◽  
Weak König’S Lemma ◽  
König’S Lemma

The so-called weak K¨onig's lemma WKL asserts the existence of an infinite<br />path b in any infinite binary tree (given by a representing function f). Based on<br />this principle one can formulate subsystems of higher-order arithmetic which<br />allow to carry out very substantial parts of classical mathematics but are PI^0_2-conservative<br />over primitive recursive arithmetic PRA (and even weaker fragments of arithmetic). In [10] we established such conservation results relative to finite type extensions PRA^omega of PRA (together with a quantifier-free axiom of choice schema). In this setting one can consider also a uniform version UWKL of WKL which asserts the existence of a functional Phi which selects uniformly in a given infinite binary tree f an infinite path Phi f of that tree.<br />This uniform version of WKL is of interest in the context of explicit mathematics as developed by S. Feferman. The elimination process in [10] actually can be used to eliminate even this uniform weak K¨onig's lemma provided that PRA^omega only has a quantifier-free rule of extensionality QF-ER instead of the full axioms (E) of extensionality for all finite types. In this paper we show that in the presence of (E), UWKL is much stronger than WKL: whereas WKL remains to be Pi^0_2 -conservative over PRA, PRA^omega +(E)+UWKL contains (and is conservative over) full Peano arithmetic PA.

scholarly journals Measure theory and higher order arithmetic

2015 ◽  
Vol 143 (12) ◽  
pp. 5411-5425 ◽  
Author(s):  
Alexander P. Kreuzer
Keyword(s):  
Measure Theory ◽  
Higher Order ◽  
Order Arithmetic

Non-constructive rules of inference

2018 ◽  
Author(s):  
A.P. Hazen
Keyword(s):  
Natural Number ◽  
Infinite Number ◽  
Deductive System ◽  
Higher Order ◽  
The Other ◽  
Natural Numbers ◽  
Substitutional Quantification ◽  
Deductive Systems ◽  
Formal Systems ◽  
Order Arithmetic

For some theoretical purposes, generalized deductive systems (or, ‘semi-formal’ systems) are considered, having rules with an infinite number of premises. The best-known of these rules is the ‘ω-rule’, or rule of infinite induction. This rule allows the inference of ∀nΦ(n) from the infinitely many premises Φ(0), Φ(1),… that result from replacing the numerical variable n in Φ(n) with the numeral for each natural number. About 1930, in part as a response to Gödel’s demonstration that no formal deductive system had as theorems all and only the true formulas of arithmetic, several writers (most notably, Carnap) suggested considering the semi-formal systems obtained, from some formulation of arithmetic, by adding this rule. Since no finite notation can provide terms for all sets of natural numbers, no comparable rule can be formulated for higher-order arithmetic. In effect, the ω-rule is valid just in case the relevant quantifier can be interpreted substitutionally; looked at from the other side, the validity of some analogue of the ω-rule is the essential mathematical characteristic of substitutional quantification.

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