Gödel’s program in set theory

Abstract Gödel proved in the 1930 s in his famous Incompleteness Theorems that not all statements in mathematics can be proven or disproven from the accepted $$\operatorname {ZFC} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mo>ZFC</mml:mo> </mml:math> axioms. A few years later he showed the celebrated result that Cantor’s Continuum Hypothesis is consistent. Afterwards, Gödel raised the question whether, despite the fact that there is no reasonable axiomatic framework for all mathematical statements, natural statements, such as Cantor’s Continuum Hypothesis, can be decided via extending $$\operatorname {ZFC} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mo>ZFC</mml:mo> </mml:math> by large cardinal axioms. While this question has been answered negatively, the problem of finding good axioms that decide natural mathematical statements remains open. There is a compelling candidate for an axiom that could solve Gödel’s problem: $$\mathsf {V = Ultimate-L}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>V</mml:mi> <mml:mo>=</mml:mo> <mml:mi>Ultimate</mml:mi> <mml:mo>-</mml:mo> <mml:mi>L</mml:mi> </mml:mrow> </mml:math> . In addition, due to recent results the $$\textsf{Sealing}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>Sealing</mml:mi> </mml:math> scenario has gained a lot of attention. We describe these candidates as well as their impact and relationship.