Problems in Large Cardinals and their Applications

9972257紫菜钢铁和紫菜从一个非常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。紫菜正在研究关于L.D.的其他一族有限问题。代数和辫子群，使用大的基数公理。在基数可测的假设下，他证明了二元树上强Halpern-Lauchli问题的部分解，并致力于解决全部问题。在所有上述问题中，是否需要一个大的基数还不得而知。这个项目还包括研究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上的确定性公理。研究这个公理是为了它自己的利益，以及它的变体的可能性，从而为连续性假设提供一些启发。众所周知，CH对标准的大型基数公理是免疫的，但不一定对强大的变种免疫。数学中几乎所有的定理都可以从一个简单的基本原理集--关于集合的Zermelo-Fraenkel(ZFC)公理来证明。但是，正如哥德尔在1930年的《S》中首先发现的那样，有些命题既不能从ZFC中证明，也不能从ZFC中反驳。但是，至少对于这些不可判定的陈述中的一些，它们是有解决的希望的。也就是说，当一个人通过“大基数”公理来扩充ZFC时，它们可能被证明，这些公理超出了ZFC的强度，断言非常大的无限数的存在。在某些情况下，不可判定的陈述是关于有限数学的。这个项目涉及研究一些大的基数公理及其解决经典数学中一些研究得很好的公开问题的潜力。由于这些问题中的一些是关于有限世界的，它们不能被排除为具体应用的来源；如果发生这种情况，一个具有高度哲学兴趣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.***