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Accessible independence results for Peano arithmetic
 Bulletin of the London Mathematical Society
, 1982
"... Recently some interesting firstorder statements independent of Peano Arithmetic (P) have been found. Here we present perhaps the first which is, in an informal sense, purely numbertheoretic in character (as opposed to metamathematical or combinatorial). The methods used to prove it, however, are c ..."
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Recently some interesting firstorder statements independent of Peano Arithmetic (P) have been found. Here we present perhaps the first which is, in an informal sense, purely numbertheoretic in character (as opposed to metamathematical or combinatorial). The methods used to prove it, however, are combinatorial. We also give another independence result (unashamedly combinatorial in character) proved by the same methods. The first result is an improvement of a theorem of Goodstein [2]. Let m and n be natural numbers, n> 1. We define the base n representation of m as follows: First write m as the sum of powers of n. (For example, if m = 266, n = 2, write 266 = 2 8 + 2 3 + 2 1.) Now write each exponent as the sum of powers of n. (For example, 266 = 2 23 + 2 2 + 1 +2 1.) Repeat with exponents of exponents and so on until the representation stabilizes. For example, 266 stabilizes at the representation 2 * +l + 2 2 + l +2 l. We now define the number Gn(m) as follows. If m = 0 set Gn(m) = 0. Otherwise set Gn(m) to be the number produced by replacing every n in the base n representation of m by n +1 and then subtracting 1. (For example, G2(266) = 3 33+1 + 3 3 + 1 +2). Now define the Goodstein sequence for m starting at 2 by So, for example, m0 = m, mx = G2{m0), m2 = G^mJ, m3 = G^m2),.... 266O = 266 = 2 22+1 + 2 2+1 + 2 X = 3 33+1 + 3 3+1 + 2 ~ 1O 38 2662 = 4 44+1 + 4 4+1 + l ~ 10 616 2663 = 5 s5+1 + 5 5+1 ~ 10 10 000. Similarly we can define the Goodstein sequence for m starting at n for any n> 1. THEOREM 1. (i) (Goodstein [2]) Vm 3/c mk = 0. More generally for any m, n> 1 the Goodstein sequence for m starting at n eventually hits zero. (ii) Vm 3k mk = 0 (formalized in the language of first order arithmetic) is not provable
Combinatorial Principles Weaker than Ramsey's Theorem for Pairs
"... We investigate the complexity of various combinatorial theorems about linear and partial orders, from the points of view of computability theory and reverse mathematics. We focus in particular on the principles ADS (Ascending or Descending Sequence), which states that every infinite linear order ha ..."
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We investigate the complexity of various combinatorial theorems about linear and partial orders, from the points of view of computability theory and reverse mathematics. We focus in particular on the principles ADS (Ascending or Descending Sequence), which states that every infinite linear order has either an infinite descending sequence or an infinite ascending sequence, and CAC (ChainAntiChain), which states that every infinite partial order has either an infinite chain or an infinite antichain. It is wellknown that Ramsey's Theorem for pairs (RT
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.
On the Independence of Goodstein's Theorem
"... In this undergraduate thesis the independence of Goodstein's Theorem from Peano arithmetic (PA) is proved, following the format of the rst proof, by Kirby and Paris. All the material necessary for its understanding is developed, beginning with the foundations of set theory, followed by ordinal nu ..."
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In this undergraduate thesis the independence of Goodstein's Theorem from Peano arithmetic (PA) is proved, following the format of the rst proof, by Kirby and Paris. All the material necessary for its understanding is developed, beginning with the foundations of set theory, followed by ordinal numbers and a proof of Goodstein's Theorem, and concluded with basic model theory and the independence proof. Other relevant information, such as an outline for an alternative independence proof and an application to dynamical systems is also included. Contents 1.