Mathematical Sciences: Mathematical Logic and Applications

小行星9400889 分析中的许多存在定理都有使用收敛论证的标准证明和使用提升论证产生更强结果的非标准证明。 研究人员最近介绍了一种新的模型理论的方法证明存在定理，其中一个往往可以使一个简单的计算，以获得一个问题的近似解，然后调用一个一般的强迫定理，以获得一个更难的结果，给出一个精确的解决方案。 本项目将继续发展这一方法。 对于纯数学和应用数学中的各种问题，很容易证明存在近似解，但很难证明存在精确解。 这是典型的随机微分方程，优化问题，平衡问题。 凯斯勒的新方法，这是证明了在数学逻辑定理，往往可以大大简化建设的确切解决方案。 这种方法已经产生了一些新的结果，并有可能成为一个有价值的和广泛适用的工具。 ***

9400889 Keisler Many existence theorems in analysis have standard proofs using convergence arguments and nonstandard proofs yielding stronger results using lifting arguments. The investigator has recently introduced a new model-theoretic method of proving existence theorems, where one can often make an easy computation to obtain an approximate solution to a problem, and then invoke a general forcing theorem to get a harder result giving an exact solution. The present project will continue development of this method. For a variety of problems in both pure and applied mathematics, it is easy to show that there are approximate solutions but very difficult to show that there are exact solutions. This is typical for stochastic differential equations, optimization problems, and equilibrium problems. Keisler's new approach to such problems, which is justified by a theorem in mathematical logic, can often dramatically simplify the construction of exact solutions. This method has already produced several new results and has the potential of becoming a valuable and broadly applicable tool. ***