366 research outputs found
Busy beavers gone wild
We show some incompleteness results a la Chaitin using the busy beaver
functions. Then, with the help of ordinal logics, we show how to obtain a
theory in which the values of the busy beaver functions can be provably
established and use this to reveal a structure on the provability of the values
of these functions
Unprovability and phase transitions in Ramsey theory
The first mathematically interesting, first-order arithmetical example of incompleteness was given in the late seventies and is know as the Paris-Harrington principle. It is a strengthened form of the finite Ramsey theorem which can not be proved, nor refuted in Peano Arithmetic. In this dissertation we investigate several other unprovable statements of Ramseyan nature and determine the threshold functions for the related phase transitions.
Chapter 1 sketches out the historical development of unprovability and phase transitions, and offers a little information on Ramsey theory. In addition, it introduces the necessary mathematical background by giving definitions and some useful lemmas.
Chapter 2 deals with the pigeonhole principle, presumably the most well-known, finite instance of the Ramsey theorem. Although straightforward in itself, the principle gives rise to unprovable statements. We investigate the related phase transitions and determine the threshold functions.
Chapter 3 explores a phase transition related to the so-called infinite subsequence principle, which is another instance of Ramsey’s theorem.
Chapter 4 considers the Ramsey theorem without restrictions on the dimensions and colours. First, generalisations of results on partitioning α-large sets are proved, as they are needed later. Second, we show that an iteration of a finite version of the Ramsey theorem leads to unprovability.
Chapter 5 investigates the template “thin implies Ramsey”, of which one of the theorems of Nash-Williams is an example. After proving a more universal instance, we study the strength of the original Nash-Williams theorem. We conclude this chapter by presenting an unprovable statement related to Schreier families.
Chapter 6 is intended as a vast introduction to the Atlas of prefixed polynomial equations. We begin with the necessary definitions, present some specific members of the Atlas, discuss several issues and give technical details
Exponential prefixed polynomial equations
A prefixed polynomial equation is an equation of the form , where is a polynomial whose variables range over the
natural numbers, preceded by quantifiers over some, or all, of its variables.
Here, we consider exponential prefixed polynomial equations (EPPEs), where
variables can also occur as exponents. We obtain a relatively concise EPPE
equivalent to the combinatorial principle of the Paris-Harrington theorem for
pairs (which is independent of primitive recursive arithmetic), as well as an
EPPE equivalent to Goodstein's theorem (which is independent of Peano
arithmetic). Some new devices are used in addition to known methods for the
elimination of bounded universal quantifiers for Diophantine predicates
How unprovable is Rabin's decidability theorem?
We study the strength of set-theoretic axioms needed to prove Rabin's theorem
on the decidability of the MSO theory of the infinite binary tree. We first
show that the complementation theorem for tree automata, which forms the
technical core of typical proofs of Rabin's theorem, is equivalent over the
moderately strong second-order arithmetic theory to a
determinacy principle implied by the positional determinacy of all parity games
and implying the determinacy of all Gale-Stewart games given by boolean
combinations of sets. It follows that complementation for
tree automata is provable from - but not -comprehension.
We then use results due to MedSalem-Tanaka, M\"ollerfeld and
Heinatsch-M\"ollerfeld to prove that over -comprehension, the
complementation theorem for tree automata, decidability of the MSO theory of
the infinite binary tree, positional determinacy of parity games and
determinacy of Gale-Stewart games are all
equivalent. Moreover, these statements are equivalent to the
-reflection principle for -comprehension. It follows in
particular that Rabin's decidability theorem is not provable in
-comprehension.Comment: 21 page
Phase Transitions for Gödel Incompleteness
Gödel's first incompleteness result from 1931 states that there are true assertions about the natural numbers which do not follow from the Peano axioms. Since 1931 many researchers
have been looking for natural examples of such assertions and breakthroughs have been obtained in the seventies by Jeff Paris (in part jointly with Leo Harrington and Laurie Kirby) and Harvey Friedman who produced first mathematically interesting
independence results in Ramsey theory (Paris) and well-order and well-quasi-order theory (Friedman).
In this article we investigate Friedman style principles of combinatorial well-foundedness for the ordinals below epsilon_0. These principles state that there is a uniform bound on the length of decreasing sequences of ordinals which satisfy an elementary recursive growth rate condition with respect to their Gödel numbers.
For these independence principles we classify (as a part of a general research program) their phase transitions, i.e. we classify exactly the bounding conditions which lead from
provability to unprovability in the induced combinatorial
well-foundedness principles.
As Gödel numbering for ordinals we choose the one which is induced naturally from Gödel's coding of finite sequences from his classical 1931 paper on his incompleteness results.
This choice makes the investigation highly non trivial but rewarding and we succeed in our objectives by using an intricate and surprising interplay between analytic combinatorics and the theory of descent recursive functions.
For obtaining the required bounds on count functions for ordinals we use a classical 1961 Tauberian theorem by Parameswaran which apparently is far remote from Gödel's theorem
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