Reflection Principles and Absolute Infinity

Reflection Principles#

Reflection principles are associated with attempts to formulate the idea that no one notion, idea, or statement can capture our whole view of the universe of sets. Kurt Gödel described it as follows:

The universe of all sets is structurally indefinable. One possible way to make this statement precise is the following: The universe of sets cannot be uniquely characterized (i.e., distinguished from all its initial segments) by any internal structural property of the membership relation in it which is expressible in any logic of finite or transfinite type, including infinitary logics of any cardinal number. This principle may be considered a generalization of the closure principle.\

All the principles for setting up the axioms of set theory should be reducible to Ackermann's principle: The Absolute is unknowable. The strength of this principle increases as we get stronger and stronger systems of set theory. The other principles are only heuristic principles. Hence, the central principle is the reflection principle, which presumably will be understood better as our experience increases. Meanwhile, it helps to separate out more specific principles which either give some additional information or are not yet seen clearly to be derivable from the reflection principle as we understand it now.

Generally I believe that, in the last analysis, every axiom of infinity should be derivable from the (extremely plausible) principle that \(V\) is indefinable, where definability is to be taken in [a] more and more generalized and idealized sense.

Georg Cantor expressed similar views on absolute infinity: All cardinality properties are satisfied in this number, in which held by a smaller cardinal.