Issues in the Foundations of Mathematics

弗里德曼建议继续他的工作，建立独立于通常的数学公理（即，Zermelo-Frankel集合论与选择公理）的一些简单和基本的有限组合语句。 PI最近的工作讨论了断言在满足一定相干条件的每个有限函数集合中，某些元素具有与拉姆齐理论相关的非常强的组合性质的版本。 他建议将这些例子与快速增长的数值函数联系起来，这些数值函数与相关的大基数公理有关。 此外，他还发现了一族转移原理，这些原理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，他已经表明，某些大基数公理相当于转让原则断言，任何陈述的一个简单的一种是真实的功能对自然数是真实的功能对序数。PI建议将这项工作扩展到更强的大基数公理，并制定和研究从遗传有限集到任意集的相关转移原理。 一个独立的陈述是一个数学断言，它不能在通常的数学公理中被证明为真或假。以前已知的独立语句有某些不令人满意的功能，使他们非常典型的正常日常数学断言。弗里德曼发现了一些例子，这些例子在具体性和自然性方面都更接近正常的数学，并建议继续寻找更具体和自然的例子。他的例子也有积极的特点，断言可以证明使用某些研究良好的新公理的数学（所谓的大基数公理），但不是否则。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上下文中的自然扩展。他建议将这种看待这些数学新公理的新方法扩展到更强的数学新公理。

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.