A transfer principle for second-order arithmetic, and applications
A transfer principle for second-order arithmetic, and applications
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DOI：
10.4115/jla.2018.10.8
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发表时间：
2018-04
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影响因子：
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通讯作者：
M. Carl;Asgar Jamneshan
中科院分区：
文献类型：
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作者：
M. Carl;Asgar Jamneshan
在条件集理论中，许多来自泛函分析、概率论或测度论等领域的经典定理被提升到条件框架中，通常应用于数理经济学或优化等领域。这种定理可以通过经典证明的“条件化”来证明的频繁经验表明，一个一般的转移原理是在背景中的，并且制定和证明这样一个转移原理将产生大量有用的经典结果的进一步条件版本，除了提供一个统一的方法来处理已知的结果之外。在本文中，我们提出并证明了这样一个基于二阶算术的转移原理，根据逆向数学的结果，它足以用于大部分经典数学，包括真实的分析，测度论和可数代数，而只排除更遥远的领域，如范畴论，集合论拓扑或不可数集合论，例如，参见\cite{simpson 2009 subsystems}的介绍。然后，该转移原理被用于对数学各个领域中的中心结果的条件版本给出简短而容易的证明，包括以前没有手工证明的定理，例如Peano存在定理，Urysohn引理和Markov-Kakutani不动点定理。此外，我们比较了在一个条件模型中的某些结构的解释与它们在一个标准模型中的含义。
In the theory of conditional sets, many classical theorems from areas such as functional analysis, probability theory or measure theory are lifted to a conditional framework, often to be applied in areas such as mathematical economics or optimization. The frequent experience that such theorems can be proved by `conditionalizations' of the classical proofs suggests that a general transfer principle is in the background, and that formulating and proving such a transfer principle would yield a wealth of useful further conditional versions of classical results, in addition to providing a uniform approach to the results already known. In this paper, we formulate and prove such a transfer principle based on second-order arithmetic, which, by the results of reverse mathematics, suffices for the bulk of classical mathematics, including real analysis, measure theory and countable algebra, and excluding only more remote realms like category theory, set-theoretical topology or uncountable set theory, see e.g. the introduction of \cite{simpson2009subsystems}.This transfer principle is then employed to give short and easy proofs of conditional versions of central results in various areas of mathematics, including theorems which have not been proven by hand previously such as Peano existence theorem, Urysohn's lemma and the Markov-Kakutani fixed point theorem. Moreover, we compare the interpretation of certain structures in a conditional model with their meaning in a standard model.