Problems in Large Cardinals and their Applications

9972257Laver Steel和Laver从一个很大的基数证明了一个关于有限左分配(a (bc) = (ab)(ac))代数塔的定理。Laver正在使用大基数公理研究一类关于ld代数和辫群的有限问题。他在二叉树上证明了强Halpern-Lauchli问题的部分解，在可测量基数的假设下，他正在努力解决完整问题。在上述所有问题中，不知道是否需要一个大基数。该项目还涉及研究Woodin的一个非常大的基数公理，它的行为类似于实线的更高基数版本的确定性公理。研究这个公理是为了它自身的利益，也是为了它的各种变体的可能性，从而对连续假设有所启发。已知CH对标准的大基数公理免疫，但不一定对强变体免疫。数学中几乎所有的定理都可以由一个基本原理的简单集合来证明，即Zermelo-Fraenkel （ZFC）公理。但是，正如哥德尔在20世纪30年代首次发现的那样，有些命题既不能被证明也不能被证伪。然而，至少对于其中一些“无法确定”的声明来说，它们还是有解决的希望的。也就是说，当一个人用“大基数”公理来扩充ZFC时，它们可以被证明，这些公理超出了ZFC的强度，它们断言存在非常大的无限数。不可判定陈述在某些情况下是关于有限数学的。该项目包括研究一些大的基本公理，以及它们解决经典数学中一些研究得很好的开放性问题的潜力。因为这些问题中有些是关于有限世界的，所以不能排除它们是具体应用的来源；如果这种情况发生，一个具有高度哲学意义的领域也将具有惊人的实际用途

9972257Laver Steel and Laver proved from a very large cardinal, a theorem abouta tower of finite left-distributive ( a(bc) = (ab)(ac) ) algebras.Laver is studying a family of other finite problems about l.d. algebrasand the braid groups, using large cardinal axioms. He proved apartial solution to the strong Halpern-Lauchli problem on binarytrees, under the assumption of a measurable cardinal, and is workingon solving the full problem. In all the above problems it is notknown whether a large cardinal is needed. The project also involvesstudying a very large cardinal axiom of Woodin that behaves like theaxiom of determinacy on higher cardinality versions of the real line.This axiom is being studied for its own interest and for thepossibility of variants of it shedding some light on the continuumhypothesis. The CH is known to be immune to standard large cardinalaxioms, but not necessarily to strong variants. Almost all theorems in mathematics can be proved from a simpleset of basic principles, the Zermelo-Fraenkel (ZFC) axioms for sets.But, as first discovered by Goedel im the 1930's, there are somestatements that can neither be proved nor disproved from ZFC.However, at least for some of these ``undecidable'' statements thereis hope of their resolution. Namely, they might be proved when oneaugments ZFC by ``large cardinal'' axioms, axioms beyond the strengthof ZFC, which assert the existence of very large infinite numbers.The undecidable statements are in some cases about finite mathematics.The project involves studying some large cardinal axioms and theirpotential to solve some well studied open questions in classical math.Since some of these problems are about the finite world, they cannotbe ruled out as sources of concrete applications; if this happens, anarea of high philosophical interest will also have a strikingpractical use.***