Topic Archive: reflection principles
In analogy with the ancient views on potential as opposed to actual infinity, set-theoretic potentialism is the philosophical position holding that the universe of set theory is never fully completed, but rather has a potential character, with greater parts of it becoming known to us as it unfolds. In this talk, I should like to undertake a mathematical analysis of the modal commitments of various specific natural accounts of set-theoretic potentialism. After developing a general model-theoretic framework for potentialism and describing how the corresponding modal validities are revealed by certain types of control statements, which we call buttons, switches, dials and ratchets, I apply this analysis to the case of set-theoretic potentialism, including the modalities of true-in-all-larger-Vβ, true-in-all-transitive-sets, true-in-all-Grothendieck-Zermelo-universes, true-in-all-countable-transitive-models and others. Broadly speaking, the height-potentialist systems generally validate exactly S4.3 and the height-and-width-potentialist systems generally validate exactly S4.2. Each potentialist system gives rise to a natural accompanying maximality principle, which occurs when S5 is valid at a world, so that every possibly necessary statement is already true. For example, a Grothendieck-Zermelo universe Vκ, with κ inaccessible, exhibits the maximality principle with respect to assertions in the language of set theory using parameters from Vκ just in case κ is a Σ3-reflecting cardinal, and it exhibits the maximality principle with respect to assertions in the potentialist language of set theory with parameters just in case it is fully reflecting Vκ < V.
This is current joint work with Øystein Linnebo, in progress, which builds on some of my prior work with George Leibman and Benedikt Löwe in the modal logic of forcing. Comments and questions can be made on the speaker’s blog.
One of the most fruitful research area in set theory is the study of the so-called Reflections Principles. Roughly speaking, a reflection principle is a combinatorial statement of the following form: given a structure S (e.g. stationary sets, tree, graphs, groups …) and a property P of the structure, the principle establishes that there exists a smaller substructure of S that satisfies the same property P. There is a tension between large cardinals axioms and the axiom of constructibility V=L at the level of reflection: on the one hand, large cardinals typically imply reflection properties, on the other hand L satisfies the square principles which are anti-reflection properties. Two particular cases of reflection received special attention, the reflection of stationary sets and the tree property. We will discuss the interactions between these principles and a version of the square due to Todorcevic. This is a joint work with Menachem Magidor and Yair Hayut.
We will present an argument for reflecting the large cardinal axiom I_0 from marginally stronger large cardinals. This will involve presenting some of the theory of inverse limits, which R. Laver first studied in the context of reflecting large cardinals at this level. Along the way we will see many local reflection results below I_0 and state a strong form of reflection which is useful in other contexts.
Reflection principles are classical objects in proof theory and the areas studying Gödel’s Incompleteness. Reflection principles based on provability predicates were introduced in the 1930s by Rosser and Turing, and later were explored by Feferman, Kreisel & Levi, Schmerl, Artemov, Beklemishev and others.
We study reflection principles of Peano Arithmetic involving both Proof and Provability predicates. We find a classification of these principles and establish their linear ordering with respect to their metamathematical strength.