A NEW APPROACH TO INFINITE DECISION-MAKING PROCESS
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Classically, a countably infinite set and its infinite parts are indistinguishable relative to size. This fact has been exploited by several authors (by way of illustration, we focus on [2]) to introduce paradoxes in decision-making over infinitely many alternatives. In this paper, we observe that all such paradoxes arise from fixing the manner in which in-finite processes are to be conceptualised. A standard conceptualisation, reliant upon sequential reasoning, handles infinite, completed processes by means of finite approximations: the ensuing deletion of information concerning achieved completion leads to all the diffculties detected so far. The latter can be eliminated through the introduction of an alterna- tive conceptualisation proposed by Yaroslav Sergeyev (see in particular [11]) in which numerical specification of completions are available. We show how an extension of bounded arithmetic recently axiomatised in [9] can formalise some salient aspects of Sergeyevs approach and provide a logical framework to dissolve infinite decision-making paradoxes.
Abstract
Classically, a countably infinite set and its infinite parts are indistinguishable relative to size. This fact has been exploited by several authors (by way of illustration, we focus on [2]) to introduce paradoxes in decision-making over infinitely many alternatives. In this paper, we observe that all such paradoxes arise from fixing the manner in which in-finite processes are to be conceptualised. A standard conceptualisation, reliant upon sequential reasoning, handles infinite, completed processes by means of finite approximations: the ensuing deletion of information concerning achieved completion leads to all the diffculties detected so far. The latter can be eliminated through the introduction of an alterna- tive conceptualisation proposed by Yaroslav Sergeyev (see in particular [11]) in which numerical specification of completions are available. We show how an extension of bounded arithmetic recently axiomatised in [9] can formalise some salient aspects of Sergeyevs approach and provide a logical framework to dissolve infinite decision-making paradoxes.
참고문헌 (12)
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