On the restricted ordinal theorem

Abstract

The proposition that a decreasing sequence of ordinals necessarily terminates has been given a new, and perhaps unexpected, importance by the rôle which it plays in Gentzen's proof of the freedom from contradiction of the “reine Zahlentheorie.” Gödel's construction of non-demonstrable propositions and the establishment of the impossibility of a proof of freedom from contradiction, within the framework of a certain type of formal system, showed that a proof of freedom from contradiction could be found only by transcending the axioms and proof processes of that formal system. Gentzen's proof succeeds by utilising transfinite induction to prove that certain sequences of reduction processes, enumerated by ordinals less than ε (the first ordinal to satisfy ε = ὡ) are finite. Were it possible to prove the restricted ordinal theorem, that a descending sequence of ordinals, less than ε, is finite, in Gentzen's “reine Zahlentheorie,” then it would be possible to determine a contradiction in that number system. In his paper, Gentzen proves the theorem of transfinite induction, which he requires, by an intuitive argument. There is also a method of reducing transfinite induction, for ordinals less than ε, to a number-theoretic principle given by Hilbert and Bernays, and a similar method by Ackermann. None of these proofs of transfinite induction is finitist.

Inductive Complexity of Goodstein’s Theorem

“…Checking the Goodstein Theorem involves the computation of complete hereditary representations for increasing base values for non-negative integers [3,11]. In order to handle and store hereditary representations of integers and compute new integers by bumping up the base value we use arrays and their encoding as developed by Dinneen [12].…”

“…We recursively repeat this procedure to create a 7+ j*6 -element array, in which all integers are smaller than the encoded base value ~B=5+B. This array then represents the complete hereditary representation at base b and has length 7+j*6 and all integers are encoded according to (3). The array is organized in RPN and can easily be executed in further steps necessary to advance the calculation to the next element in the Goodstein sequence of any given integer seed n. We illustrate this with the integer 266 and base B=2.…”

“…In mathematical logic, Goodstein's Theorem [3] is a statement about the natural numbers which states that every Goodstein sequence eventually terminates at 0. In 1982, Kirby & Paris showed [4] that Goodstein's Theorem is unprovable in Peano arithmetic (PA) but can be proven in stronger systems, such as second order arithmetic.To fix notation we work in the standard model of PA and recall [15] that a 1 -sentence is any sentence of the form where P is a function that can proven to be recursive in PA, that is, there is a Turing machine TM that computes P, together with a proof in PA that TM halts on every input.…”

“…Following Calude [9,10] we can associate with each 2 -sentence an inductive Turing machine of first order. In case of the Goodstein Theorem, let n be any non-negative integer and let , k ≥1, denote the k th term of the Goodstein Sequence [3,11] of n defined as:…”

Goodstein sequences for prominent ordinals up to the ordinal ofΠ11−CA0

“…Goodstein sequences provide examples for strictly mathematical statements which are true (by Goodstein, see [Goo44]) but (according to Kirby and Paris, see [KP82]) not provable in PA. In the 80s several attempts have been made to define Goodstein principles capturing larger complexities using Π 1 2 -logic.…”