Problems in Large Cardinals and their Applications

小行星9972257 Steel和Laver从一个极大基数出发证明了关于有限左分配（a（bc）=（ab）（ac））代数的塔的一个定理，Laver正在研究一族关于l.d.代数和辫子群，使用大基数公理。 他证明了partial解决方案的强Halpern-Lauchli问题的二叉树，假设一个可衡量的基数，并正在努力解决整个问题。 在上述所有问题中，是否需要一个大基数是未知的。 该项目还包括研究一个非常大的Woodin基数公理，它的行为类似于真实的线的更高基数版本上的确定性公理。这个公理是为了它自己的利益和它的变体的可能性而研究的，它为连续性假设提供了一些光。 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从一组简单的基本原理--Zermelo-Fraenkel（ZFC）集合公理中得到证明。但是，正如Goedel在20世纪30年代首次发现的那样，有些陈述既不能从ZFC中得到证明，也不能从ZFC中得到反证。然而，至少对于这些"不可判定“的陈述中的一些，它们的解决是有希望的。 也就是说，当一个人用“大基数”公理扩充ZFC时，它们可能被证明，这些公理超出了ZFC的强度，它们断言存在非常大的无限数。不可判定的陈述在某些情况下是关于有限数学的。该项目涉及研究一些大基数公理及其解决经典数学中一些研究得很好的开放问题的潜力。由于其中一些问题是关于有限世界的，不能排除它们作为具体应用的来源;如果这种情况发生，一个高度哲学兴趣的领域也将有惊人的实际用途。

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.***