Foundational Investigations into the Infinite/Finite in Mathematics

From Kurt Goedel, we know that there are theorems about the mathematical finite that can only be proved using the mathematical infinite. The Goedel examples are "formal systems for infinitary mathematics are free of contradiction", and are of great importance for foundations of mathematics, but do not conform to "normal mathematical activity". BIG QUESTION 1. Are there statements in standard finite or discrete mathematical contexts which can be proved by using Infinities higher than those available in ZFC, but not within ZFC alone? Skepticism has set in to the mathematics community that strong infinities are not relevant to concrete, discrete/finite mathematical contexts with clear intuitive meaning. Breakthroughs on this Big Question have been achieved under current JTF ID#15557, resulting in "Invariant Maximal Cliques and Incompleteness", submitted for publication. These new examples are of a higher level of simplicity and natural thematic character than what has been achieved earlier by us in our 45 year quest. These new examples - and further examples - will have enduring impact. Friedman is scheduled to speak on March 1, 2012 concerning this work, at the Harvard Mathematics Department - as part of the Brandeis-Harvard-MIT-Northeastern Mathematics Colloquium. These new examples have a clear thematic character suggesting further examples conforming to diverse settings across a wide range of mathematics. A principal outcome of this grant will be a major research monograph provisionally titled "Maximality and Incompleteness". This will be a major upgrade of the paper submitted under JTF ID#15557. Friedman also proposes to investigate two other Big Questions: BIG QUESTION 2. Can we view the usual ZFC axioms of set theory as an extrapolation of unproblematic facts from finite set theory? BIG QUESTION 3. Are there no-algorithms and incompleteness results in the realm of physical systems?