On Games on Non-Wellfounded Sets and Stationary Sets

In this thesis we study a few games related to non-wellfounded and stationary sets. Games have turned out to be an important tool in mathematical logic ranging from semantic games defining the truth of a sentence in a given logic to for example games on real numbers whose determinacies have important effects on the consistency of certain large cardinal assumptions. The equality of non-wellfounded sets can be determined by a so called bisimulation game already used to identify processes in theoretical computer science and possible world models for modal logic. Here we present a game to classify non-wellfounded sets according to their branching structure. We also study games on stationary sets moving back to classical wellfounded set theory. We also describe a way to approximate non-wellfounded sets with hereditarily finite wellfounded sets. The framework used to do this is domain theory. In the Banach-Mazur game, also called the ideal game, the players play a descending sequence of stationary sets and the second player tries to keep their intersection stationary. The game is connected to precipitousness of the corresponding ideal. In the pressing down game first player plays regressive functions defined on stationary sets and the second player responds with a stationary set where the function is constant trying to keep the intersection stationary. This game has applications in model theory to the determinacy of the Ehrenfeucht-Fraisse game. We show that it is consistent that these games are not equivalent.

Measurement of Particle Production and Inclusive Differential Cross Sections in pbar{p} Collisions at sqrt{s}=1.96 TeV

We report a set of measurements of particle production in inelastic pbar{p} collisions collected with a minimum-bias trigger at the Tevatron Collider with the CDF II experiment. The inclusive charged particle transverse momentum differential cross section is measured, with improved precision, over a range about ten times wider than in previous measurements. The former modeling of the spectrum appears to be incompatible with the high particle momenta observed. The dependence of the charged particle transverse momentum on the event particle multiplicity is analyzed to study the various components of hadron interactions. This is one of the observable variables most poorly reproduced by the available Monte Carlo generators. A first measurement of the event transverse energy sum differential cross section is also reported. A comparison with a Pythia prediction at the hadron level is performed. The inclusive charged particle differential production cross section is fairly well reproduced only in the transverse momentum range available from previous measurements. At higher momentum the agreement is poor. The transverse energy sum is poorly reproduced over the whole spectrum. The dependence of the charged particle transverse momentum on the particle multiplicity needs the introduction of more sophisticated particle production mechanisms, such as multiple parton interactions, in order to be better explained.

Measurement of particle production and inclusive differential cross sections in pp̅ collisions at √s=1.96 TeV

We report a set of measurements of particle production in inelastic pbar{p} collisions collected with a minimum-bias trigger at the Tevatron Collider with the CDF II experiment. The inclusive charged particle transverse momentum differential cross section is measured, with improved precision, over a range about ten times wider than in previous measurements. The former modeling of the spectrum appears to be incompatible with the high particle momenta observed. The dependence of the charged particle transverse momentum on the event particle multiplicity is analyzed to study the various components of hadron interactions. This is one of the observable variables most poorly reproduced by the available Monte Carlo generators. A first measurement of the event transverse energy sum differential cross section is also reported. A comparison with a Pythia prediction at the hadron level is performed. The inclusive charged particle differential production cross section is fairly well reproduced only in the transverse momentum range available from previous measurements. At higher momentum the agreement is poor. The transverse energy sum is poorly reproduced over the whole spectrum. The dependence of the charged particle transverse momentum on the particle multiplicity needs the introduction of more sophisticated particle production mechanisms, such as multiple parton interactions, in order to be better explained.

First Measurement of the tt̅ Differential Cross Section dσ/dMtt̅ in pp̅ Collisions at √s=1.96 TeV

We present a measurement of the $\ttbar$ differential cross section with respect to the $\ttbar$ invariant mass, dSigma/dMttbar, in $\ppbar$ collisions at $\sqrt{s}=1.96$ TeV using an integrated luminosity of $2.7\invfb$ collected by the CDF II experiment. The $\ttbar$ invariant mass spectrum is sensitive to a variety of exotic particles decaying into $\ttbar$ pairs. The result is consistent with the standard model expectation, as modeled by \texttt{PYTHIA} with \texttt{CTEQ5L} parton distribution functions.

Weak Ehrenfeucht-Fraïssé Games

In this paper we define a game which is played between two players I and II on two mathematical structures A and B. The players choose elements from both structures in moves, and at the end of the game the player II wins if the chosen structures are isomorphic. Thus the difference of this to the ordinary Ehrenfeucht-Fra¨ıss´e game is that the isomorphism can be arbitrary, whereas in the ordinary EF-game it is determined by the moves of the players. We investigate determinacy of the weak EF-game for different (the length of the game) and its relation to the ordinary EF-game.

Playing Games on Sets and Models

The most prominent objective of the thesis is the development of the generalized descriptive set theory, as we call it. There, we study the space of all functions from a fixed uncountable cardinal to itself, or to a finite set of size two. These correspond to generalized notions of the universal Baire space (functions from natural numbers to themselves with the product topology) and the Cantor space (functions from natural numbers to the {0,1}-set) respectively. We generalize the notion of Borel sets in three different ways and study the corresponding Borel structures with the aims of generalizing classical theorems of descriptive set theory or providing counter examples. In particular we are interested in equivalence relations on these spaces and their Borel reducibility to each other. The last chapter shows, using game-theoretic techniques, that the order of Borel equivalence relations under Borel reduciblity has very high complexity. The techniques in the above described set theoretical side of the thesis include forcing, general topological notions such as meager sets and combinatorial games of infinite length. By coding uncountable models to functions, we are able to apply the understanding of the generalized descriptive set theory to the model theory of uncountable models. The links between the theorems of model theory (including Shelah's classification theory) and the theorems in pure set theory are provided using game theoretic techniques from Ehrenfeucht-Fraïssé games in model theory to cub-games in set theory. The bottom line of the research declairs that the descriptive (set theoretic) complexity of an isomorphism relation of a first-order definable model class goes in synch with the stability theoretical complexity of the corresponding first-order theory. The first chapter of the thesis has slightly different focus and is purely concerned with a certain modification of the well known Ehrenfeucht-Fraïssé games. There we (me and my supervisor Tapani Hyttinen) answer some natural questions about that game mainly concerning determinacy and its relation to the standard EF-game