Issues in the Foundations of Mathematics

弗里德曼建议继续他的工作，建立一些简单和基本的有限组合语句独立于通常的数学公理(即具有选择公理的Zermelo-Frankel集合论)的工作。PI最近的工作讨论了断言在每个满足一定相干性条件的有限函数集合中，某些元素具有与Ramsey理论有关的非常强的组合性质的观点。他建议将这些例子与与相关大型基数公理相关的快速增长的数值函数联系起来。此外，他还发现了一族迁移原理，它们在有限集合论和超限集合论之间建立了一种新的形式关系。特别是，他证明了某些大的基数公理等价于转移原理，即任何简单的关于自然数上的函数的陈述都是关于序数上的函数的真实的。PI建议将这项工作扩展到更强的大型基数公理，并制定和研究从遗传有限集到任意集的相关转换原理。独立陈述是一种数学断言，在通常的数学公理中无法证明其对或错。以前已知的独立陈述具有某些不令人满意的特征，这使得它们与正常的日常数学断言非常不典型。弗里德曼发现了一些在具体性和自然性方面更接近正常数学的例子，并建议继续寻找更具体和自然的例子。他的例子也有一个积极的特点，即断言可以用某些经过充分研究的数学新公理(所谓的大型基数公理)来证明，但不能用其他方式来证明。PI还建立了一种全新的方式来产生这些新的公理--它们可以被认为是已知事实在整数语境中的自然延伸。他建议将这种看待数学新公理的新方式扩展到更强大的数学新公理。

Friedman proposes to continue his work establishing the independence from the usual axioms of mathematics (i.e., Zermelo-Frankel set theory with the axiom of choice) of some simple and basic finite combinatorial statements. Recent work of the PI discusses versions that assert that in every collection of finite functions satisfying a certain coherence condition, some element has a very strong combinatorial property related to Ramsey theory. He proposes to relate these examples to the fast growing numerical functions associated with the relevant large cardinal axioms. In addition, he has discovered a family of transfer principles which establish a new kind of formal relationship between finite set theory and transfinite set theory. In particular, he has shown that certain large cardinal axioms are equivalent to transfer principles asserting that any statement of a simple kind that is true about the functions on the natural numbers is true about the functions on the ordinals. The PI proposes to extend this work to stronger large cardinal axioms, and also to formulate and investigate related transfer principles from the hereditarily finite sets to arbitrary sets. An independent statement is a mathematical assertion which cannot be proved true or false within the usual axioms for mathematics. The previously known independent statements have certain unsatisfactory features which make them very atypical of normal everyday mathematical assertions. Friedman has discovered some examples which are much closer to normal mathematics in terms of both concreteness and naturality and proposes to continue the search for more concrete and natural examples. His examples also have the positive feature that the assertions can be proved using certain well studied new axioms for mathematics (so called large cardinal axioms), but not otherwise. The PI also has established an entirely new way in which these new axioms arise - they can be thought of as the natural extension of known facts in the context of the inte gers. He proposes to extend this new way of looking at these new axioms for mathematics to yet stronger new axioms for mathematics.