6 Novembre 2023

Juliette Kennedy et Jouko Väänänen (Helsinki)

Historicising Extended Constructibility

In the 1930s Kurt Gödel defined a remarkable so-called
"constructible" subuniverse (inner model) L of the universe of set theory,
one which satisfied the Axiom of Choice and the Continuum Hypothesis,
both of whose consistency was still open at the time.

In his 1946 talk in the Princeton Bicentennial Conference,
Gödel compared constructibility to the concept of Turing computability,
and asked whether "absolute" notions of definability
and even of provability can be developed,
along the lines of what Turing did for computability,
i.e. "not depending on the formalism chosen."

One way to implement Gödel's suggestion is to study
his constructible hierarchy L, which centers around first order logic,
but enhancing the expressive power of first order logic.

In this talk we explain this choice of implementation
and discuss some of the inner models of set theory
that were discovered in the process.

Gödel's inner model L cannot have large cardinals,
e.g. it cannot have measurable cardinals.
This led to a search for bigger inner models
which would still be in line with Gödel's L
in manifesting absoluteness, but which contain large cardinals.
The inner models we discuss in this talk meet these demands to some extent.