Simple proofs of $${\mathsf{SCH}}$$ SCH from reflection principles without using better scales

Simple proofs of SCH from reflection principles without using better scales

We give simple proofs of the Singular Cardinal Hypothesis from the Weak Reflection Principle and the Fodor-type Reflection Principle which do not use better scales.

Perfect anti-reflection from first principles

Reducing unwanted reflections through impedance matching, called anti-reflection, has long been an important challenge in optics and electrical engineering. Beyond trial and error optimization, however, a systematic way to realize anti-reflection is still absent. Here, we report the discovery of an analytic solution to this long standing problem. For electromagnetic waves, we find the graded pe...

Extracting simple programs from second order proofs

On reflection principles

Gödel initiated the program of finding and justifying axioms that effect a significant reduction in incompleteness and he drew a fundamental distinction between intrinsic and extrinsic justifications. Reflection principles are the most promising candidates for new axioms that are intrinsically justified. Taking as our starting point Tait’s work on general reflection principles, we prove a serie...

Global Reflection Principles

Reflection Principles are commonly thought to produce only strong axioms of infinity consistent with V = L. It would be desirable to have some notion of strong reflection to remedy this, and we have proposed Global Reflection Principles based on a somewhat Cantorian view of the universe. Such principles justify the kind of cardinals needed for, inter alia , Woodin’s Ω-Logic.1 To say that the un...