An analysis of Spector Classes associated with quasi-inductive definitions

不同的作者以不同的方式回应了塔尔斯基和歌德尔关于真理的不确定性和形式系统的不完全性的结果，这排除了一个人可以在演绎系统中拥有一个完全定义真理的公式的可能性。这些理论为我们可以通过某种公式部分定义真理的可能性敞开了大门。最广为人知的两种方法是由克里普克在70年代的S和古普塔以及贝尔纳普在80年代的S和90年代的S设计的。众所周知，我们可以通过使用两人完全信息无限博弈来描述像克里普克这样的系统中明确为真的句子集。(其中一名球员的策略告诉他们如何移动。制胜策略确保他们永远赢，即使棋盘上有无限多步棋。)“开放游戏”本质上是指一个玩家可以在有限的多个阶段中获胜(尽管另一个玩家--在“封闭”盘中进行游戏--可能必须无限地频繁地比赛才能获胜)。这是众所周知的“确定的”(即其中一个玩家必须有获胜的策略)。克里普克的真理集可以用这样简单的开放博弈来刻画。提出者研究了Gupta和Belnap的循环定义理论，一种形式的“准归纳”定义，如何嵌入到一个理论中，该理论涉及比算术层次中的开放/封闭更复杂的博弈的确定性语句。因此，我们知道存在算术等级的一个层次，其确定性允许发生像古普塔-贝纳普那样的循环定义。我们想要解决的一个数学问题是，给出一个恰当的答案，即准归纳定义理论需要多少无穷大才能起作用。这是通过要求精确的游戏，其确定性允许这样的定义关闭或达到稳定状态(通常不会出现固定点)而变得精确的。第二个数学问题问我们如何才能颠倒第一个问题：假设古普塔和贝尔纳普的循环定义修正理论总是稳定的，那么就分析理论而言，这一理论的力量有多大？广义归纳定义理论和前面提到的真理理论最近都被认为在形式上等同于一个超限计算模型。这是一个令人兴奋的事件，在根本不同的领域之间发生了令人兴奋的“趋同”，因为它们来自完全不同的学科。在这里，我们想象一台标准的图灵机或计算机，被允许运行无限长的时间。这样一台概念性的或“虚拟的”机器能计算出什么？就像普通的图灵机一样，我们可以限定它们的计算能力。对什么是可计算的分析被证明是一类被称为Spector类的集合。该项目将分析与这些游戏相关的实数集的Spector类，或者等价于这些计算模型，或者再次分析这些哲学真理理论。

Various authors have responded in differing ways to the results of Tarski and Goedel on theundefinability of truth and incompleteness of formal systems, which ruled out the possibility that one can have a formula in a deductive system that defines truth completely. These theories left open the door to the possibility that we can partially define truth by some formula. The two most widely known approaches to this were designed by Kripke in the 70's and Gupta and Belnap in the 80's and 90's. It is also well known that we can give a description of the sets of sentences definably true in a system such as Kripke's by using two person perfect information infinite games. (A strategy for one of the players tells them how to move. A winning strategy ensures they will always win even if there are infinitely many moves on the board.) An 'open game' is one essentially in which one player can win in finitely many stages (although the other player - playing in to the `closed' set - may have to play infinitely often in order to win).These are known to be 'determined' (that is one of the players must have a winning strategy). Kripke's truth sets can be characterised by means of such simple open games.The proposer has looked at how Gupta & Belnap's theory of circular definitions, a form of 'quasi-inductive' definition, can be embedded in a theory involving determinacystatements for games of greater complexity than open/closed in the arithmetic hierarchy. We thus know there is a level of the arithmetic hierarchy whose determinacy allows such circular definitions as those of Gupta-Benap to take place. One mathematical question we should like to address is to give a proper answer as to how much infinity is required for thetheory of quasi-inductive definitions to work . This is made precise through asking for precise games whose determinacy allows such definitions to close-off or reach a stable state ( fixed points in general cannot occur). A second mathematical question asks how we can reverse the first question: assuming Gupta and Belnap's revision theory of circular definitions always stabilizes, then how strong is this in terms of theories of analysis?Both the theory of generalised inductive definitions, and the theories of truth mentionedhas been seen quite recently to be formally equivalent with a transfinite computational model.This is an exciting occurrence of `convergence' between radically different areas, coming as they do from quite different disciplines. Here we imagine a standard Turing machine or computer, being allowed to run over tranfinite lengths of time. What could such a conceptual or 'virtual' machine compute? Just as for ordinary Turing machines, we can delimit their computational power. The analysis of what is computable turns out to be one of a class of sets called Spector Classes .The project will analysis the Spector class of sets of real numbers associated with thesegames, or equivalently these computational models, or again, these philosophical theories of truth.