Forcing Iterations Using Models as Side Conditions

The story of infinity in mathematics started seriously with G. Cantor, who showed that there are different infinities, and one can compare them using one-to-one correspondences in the same way we compare two baskets of apples and oranges. A very early realisation in this new-born topic was that there are more real numbers than natural numbers (the "counting numbers"), this was accompanied with other surprising facts such as the size of the set of rational numbers (fraction of natural numbers) is the same as the one of the natural numbers. These then raised the question whether the following statement, Continuum Hypothesis (CH), is true: The size of the real line is the smallest infinity greater than the infinity of natural numbers. CH puzzled mathematicians for a while until K. Godel demonstrated that we cannot disprove CH based on our foundation and our rules of inference (called ZFC). Godel`s result does not imply that CH is true. The next step was to try the other direction to show that the Continuum Hypothesis might be false. Though the attempts made by mathematicians invented new research directions and new mathematical topics, it was still unknown until P. Cohen showed that CH cannot be proven, and thus that CH is undecidable (in ZFC). Not only was it not the end of the story, but also a revolution in Cantor`s Set Theory paradise. Since then many other problems have been solved using Cohen`s method, called "forcing". Basically, forcing allows us to extend a mathematical universe (i.e., model of ZFC) by adding certain generic objects for certain partial orders. This also led to the discovery of the so-called forcing axioms, which state that for certain partial orders the universe is already "partially complete", i.e., that we already have partial generic objects. Such axioms help us to build structures using partial information, but there are some constraints preventing us doing this when our final structure is supposed to be too big. In 2014, A new direction was opened by I. Neeman to attack this hard problem. In our project we aim to lift up or adapt some consequences of well-known forcing axioms to higher infinities. To this end we need to develop Neeman`s method case by case using a new approach initiated by B. Velickovic. For example, a combinatorial principle which holds true under PFA (a very nice forcing axiom) is usually about the counting phenomenon of objects related to the first uncountable cardinal. We want to extend it to the second uncountable cardinal, but we first have to find its correct formulation and an accessible method to prove it. We are also going to solve certain technical problems which we expect will open up new paths to higher forcing axioms.

We are pleased to share with you the results of our latest project, which has delved into some intricate areas of mathematics. Through our research efforts we've contributed to several papers published in prestigious journals and conference proceedings. The subject of the project is essentially G. Cantor's infamous question of how many real numbers there are, and the Forcing Axioms, which are additional axioms for mathematics that can solve many problems left undecidable by traditional axiomatisations. There is a metamathematical question as to what the answer to Cantor's question would be. We now know, thanks to K. Godel and P. Cohen, that it is not decidable in mathematics, so it is a metamathematical question. A solid approach to this is forcing axioms, which t are strong enough to decide certain problems that remain undecidable. The forcing axioms usually contradict Cantor's original conjecture, and are sometimes able to determine exactly the number of points on the real line. One of our key papers, published in the Proceedings of the American Mathematical Society, offers a new perspective on a mathematical axiom called the Proper Forcing Axiom (PFA). The paper introduces a new idea related to the PFA and presents a novel proof of one of its consequences, the Mapping Reflection Principle, which implies, in particular, that there is exactly one infinity between the size of the natural numbers and that of the real numbers. This consequence has important implications for the understanding of complex mathematical structures of the size of the first uncountable infinity, especially when dealing with the Cantor continuum problem. In another paper, published in the Journal of Symbolic Logic, we address a challenging and highly technical question posed by a colleague. Our results shed light on how to make certain mathematical structures indestructible, even in scenarios where traditional methods fail. This demonstrates the importance of considering alternative approaches to research in mathematical logic, as they can lead to unexpected breakthroughs.