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Number theory and elementary arithmetic
 Philosophia Mathematica
, 2003
"... Elementary arithmetic (also known as “elementary function arithmetic”) is a fragment of firstorder arithmetic so weak that it cannot prove the totality of an iterated exponential function. Surprisingly, however, the theory turns out to be remarkably robust. I will discuss formal results that show t ..."
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Elementary arithmetic (also known as “elementary function arithmetic”) is a fragment of firstorder arithmetic so weak that it cannot prove the totality of an iterated exponential function. Surprisingly, however, the theory turns out to be remarkably robust. I will discuss formal results that show that many theorems of number theory and combinatorics are derivable in elementary arithmetic, and try to place these results in a broader philosophical context. 1
Is P versus NP formally independent
 Bulletin of the European Association for Theoretical Computer Science
, 2003
"... I have moved back to the University of Chicago and so has the web page for this column. See above for new URL and contact informaion. This issue Scott Aaronson writes quite an interesting (and opinionated) column on whether the P = NP question is independent of the usual axiom systems. Enjoy! ..."
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I have moved back to the University of Chicago and so has the web page for this column. See above for new URL and contact informaion. This issue Scott Aaronson writes quite an interesting (and opinionated) column on whether the P = NP question is independent of the usual axiom systems. Enjoy!
Consistency and Gamesin Search of New Combinatorial Principles
, 2004
"... We show that a semantical interpretation of Herbrand's disjunctions can be used to obtain 2 independent sentences whose nature is more combinatorial than the nature of the usual consistency statements. Then we apply this method to Bounded Arithmetic and present 8 1 combinatorial sentences tha ..."
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We show that a semantical interpretation of Herbrand's disjunctions can be used to obtain 2 independent sentences whose nature is more combinatorial than the nature of the usual consistency statements. Then we apply this method to Bounded Arithmetic and present 8 1 combinatorial sentences that characterize all 8 1 sentences provable in S 2 . We use the concept of a two player game to describe these sentences.
The Impact of the Incompleteness Theorems
"... In addition to this being the centenary of Kurt Gödel’s birth, January marked 75 years since the publication (1931) of his stunning incompleteness theorems. Though widely known in one form or another by practicing mathematicians, and generally thought to say something fundamental about the limits an ..."
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In addition to this being the centenary of Kurt Gödel’s birth, January marked 75 years since the publication (1931) of his stunning incompleteness theorems. Though widely known in one form or another by practicing mathematicians, and generally thought to say something fundamental about the limits and potentialities of mathematical knowledge, the actual importance of these results for mathematics is little understood. Nor is this an isolated example among famous results. For example, not long ago, Philip Davis wrote me about what he calls The Paradox of Irrelevance: “There are many math problems that have achieved the cachet of tremendous significance, e.g., Fermat, fourcolor, Kepler’s packing, Gödel, etc. Of Fermat, I have read: ‘the most famous math problem of all time’. Of Gödel, I have read: ‘the most mathematically significant achievement of the 20th century’. … Yet, these problems have engaged the attention of relatively few research mathematicians—even in pure math. ” What accounts for this disconnect between fame and relevance? Before going into the question for Gödel’s theorems, it should be distinguished in one respect from the other examples mentioned, which in any case form quite a mixed bag. Namely, each of the Fermat, fourcolor, and Kepler’s packing problems posed a standout challenge following extended efforts to settle them; meeting the challenge in each case required new ideas or approaches and intense work, obviously of different degrees. By contrast, Gödel’s theorems were simply unexpected, and their proofs, though requiring novel techniques, were not difficult on the scale of things. SetSolomon Feferman is professor of mathematics and philosophy, emeritus, at Stanford University. His email address is
Brief introduction to unprovability
"... Abstract The article starts with a brief survey of Unprovability Theory as of autumn 2006. Then, as an illustration of the subject's modeltheoretic methods, we reprove exact versions of unprovability results for the ParisHarrington Principle and the KanamoriMcAloon Principle using indiscernibles. ..."
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Abstract The article starts with a brief survey of Unprovability Theory as of autumn 2006. Then, as an illustration of the subject's modeltheoretic methods, we reprove exact versions of unprovability results for the ParisHarrington Principle and the KanamoriMcAloon Principle using indiscernibles. In addition, we obtain a short accessible proof of unprovability of the ParisHarrington Principle. The proof employs old ideas but uses only one colouring and directly extracts the set of indiscernibles from its homogeneous set. We also present modified, abridged statements whose unprovability proofs are especially simple. These proofs were tailored for teaching purposes. The article is intended to be accessible to the widest possible audience of mathematicians, philosophers and computer scientists as a brief survey of the subject, a guide through the literature in the field, an introduction to its modeltheoretic techniques and, finally, a modeltheoretic proof of a modern theorem in the subject. However, some understanding of logic is assumed on the part of the readers. The intended audience of this paper consists of logicians, logicaware mathematicians andthinkers of other backgrounds who are interested in unprovable mathematical statements.
Polynomial equations with quantifierprefixes
, 2010
"... A prefixed polynomial equation (or “a polynomial expression with a quantifierprefix”) is an equation P (x1, x2,..., xn) = 0, where P is a polynomial whose variables x1, x2,... xn range over natural numbers, that is preceded by some quantifiers over some or all of its variables x1, x2,..., xn. Poly ..."
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A prefixed polynomial equation (or “a polynomial expression with a quantifierprefix”) is an equation P (x1, x2,..., xn) = 0, where P is a polynomial whose variables x1, x2,... xn range over natural numbers, that is preceded by some quantifiers over some or all of its variables x1, x2,..., xn. Polynomial equations P (x1, x2,..., xn) = 0 without quantifiers are also called Diophantine. We study prefixed polynomial equations and produce a range of unprovable statements of this form: a polynomial expression equivalent to PH 2 (strength: 1Con(IΣ1)); a polynomial expression equivalent to PH 3 (strength: 1Con(IΣ2)); a polynomial expression equivalent to PH (strength: 1Con(PA)); a polynomial expression equivalent to KT (strength: 1Con(ATR0)); a polynomial expression for KTr·log (a phase transition between EFAprovability and ATR0unprovability); a polynomial expression for the Graph Minor Theorem (strength: at least 1Con(Π 1 1CA0)); a polynomial expression for planar GMT (a phase transition between EFAprovability and the full strength of GMT); a polynomial expression that knows all values of all polynomials; a polynomial expression that knows all values of all BAFterms; a polynomial expression equivalent to Proposition E of Boolean Relation Theory (strength: 1Con(ZFC + {nMahlo}n∈ω). This early draft only has a long version (which will shrink in later drafts!) In this early draft, the lengths of polynomials haven’t yet been reasonably minimised. So far we only aimed for the very first very coarse polynomials. The results below will be improved before the end of summer 2010: all polynomials will be shortened, and the polynomial of Chapter 7 for 1Con(ZFC + {nMahlo cardinals}n∈ω) will be shortened very considerably. A second (not yet typed) layer of this project is to write expressions that allow exponentiation x y and logarithm to be mentioned. In this more expressive setup the answers become quite short: often 14 lines. We estimate that even Proposition E will fit well onto 13 lines or less. The currently growing part of the draft is the theory of seeds (polynomial equations of minimal length in their
Labib Haddad
, 2007
"... A stroll taken around the landscape of Ramsey’s Theory. One way, “Down from infinite to finite”, then, another way, “Up from disorder to order”. An exposé made at the “Rencontres arithmétique et combinatoire”, SaintEtienne, june 2006. 1 Introduction. Ramsey’s theorems At any given time, in a given ..."
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A stroll taken around the landscape of Ramsey’s Theory. One way, “Down from infinite to finite”, then, another way, “Up from disorder to order”. An exposé made at the “Rencontres arithmétique et combinatoire”, SaintEtienne, june 2006. 1 Introduction. Ramsey’s theorems At any given time, in a given meeting, some of the pairs of persons have already shaken hands, others have not. Pick any group of six persons. You are sure to find one of the two following (not exclusive) situations: Either three of them have already shaken hands
Labib Haddad
, 2006
"... A stroll taken around the landscape of Ramsey’s Theory. One way, “Down from infinite to finite”, then, another way, “Up from disorder to order”. An exposé made at the “Rencontres arithmétique et combinatoire”, SaintEtienne, june 2006. 1 Introduction. Ramsey’s theorems At any given time, in a given ..."
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A stroll taken around the landscape of Ramsey’s Theory. One way, “Down from infinite to finite”, then, another way, “Up from disorder to order”. An exposé made at the “Rencontres arithmétique et combinatoire”, SaintEtienne, june 2006. 1 Introduction. Ramsey’s theorems At any given time, in a given meeting, some of the pairs of persons have already shaken hands, others have not. Pick any group of six persons. You are sure to find one of the two following (not exclusive) situations: Either three of them have already shaken hands
On Ramseytype Positional Games
, 2007
"... Beck introduced the concept of Ramsey games by studying the game versions of Ramsey and van der Waerden theorems. We contribute to this topic by investigating games corresponding to structural extensions of Ramsey and van der Waerden theorems—the theorem of Brauer, structural and restricted Ramsey t ..."
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Beck introduced the concept of Ramsey games by studying the game versions of Ramsey and van der Waerden theorems. We contribute to this topic by investigating games corresponding to structural extensions of Ramsey and van der Waerden theorems—the theorem of Brauer, structural and restricted Ramsey theorems.
Book Review: Lorenz J. Halbeisen: “Combinatorial Set Theory.”
"... Combinatorics is that area of mathematics where we abstract that much from any structural properties of the objects under study that the Pigeonhole Principle is of prominent use. This principle says that if 5 pigeons sit in 3 holes, then at least one hole is occupied by more that 1 pigeon, or more g ..."
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Combinatorics is that area of mathematics where we abstract that much from any structural properties of the objects under study that the Pigeonhole Principle is of prominent use. This principle says that if 5 pigeons sit in 3 holes, then at least one hole is occupied by more that 1 pigeon, or more generally: there can be no injection f: A → B if the cardinality of the set A is strictly bigger than the cardinality of the set B. The Pigeonhole Principle yields Ramsey’s Theorem [5]. It’s infinite version tells us that for all n and k ∈ N, if every subset of N with n elements is colored by one of the colors 1, 2,..., k, then there is an infinite X ⊂ N and some color c ∈ {1, 2,..., k} such that every subset of X with n elements is colored by c. Ramsey Theory studies (finite and infinite) variants and generalizations of Ramsey’s Theorem and their applications. An old result which exploits Ramsey theoretic methods, even though it actually predates F. Ramsey’s formulation of his theorem, is a theorem of I. Schur [6] according to which for every n> 1 and for every sufficiently large prime p, the