The equivalence of Axiom (∗)+ and Axiom (∗)++

Journal of Mathematical Logic (forthcoming)
  Copy   BIBTEX


Asperó and Schindler have completely solved the Axiom [Formula: see text] vs. [Formula: see text] problem. They have proved that if [Formula: see text] holds then Axiom [Formula: see text] holds, with no additional assumptions. The key question now concerns the relationship between [Formula: see text] and Axiom [Formula: see text]. This is because the foundational issues raised by the problem of Axiom [Formula: see text] vs. [Formula: see text] arguably persist in the problem of Axiom [Formula: see text] vs. [Formula: see text]. The first of our two main theorems is that Axiom [Formula: see text] is equivalent to Axiom [Formula: see text], and as a corollary we show that Axiom [Formula: see text] fails in all the known models of [Formula: see text]. This suggests that [Formula: see text] actually refutes Axiom [Formula: see text]. Our second main theorem is that the [Formula: see text] Conjecture holds assuming [Formula: see text]. This is the strongest partial result known on this conjecture which is one of the central open problems of [Formula: see text]-theory and [Formula: see text]-logic. These results identify a fundamental asymmetry between the Continuum Hypothesis and any axiom which is both [Formula: see text]-expressible and which implies [Formula: see text], on the basis of generic absoluteness for the simplest of the nontrivial sentences of Third-Order Number Theory. These are the [Formula: see text]-sentences with no parameters. Such sentences are those which simply assert the existence of a set [Formula: see text] for which some property involving only quantification over [Formula: see text] holds.



    Upload a copy of this work     Papers currently archived: 94,385

External links

Setup an account with your affiliations in order to access resources via your University's proxy server

Through your library

Similar books and articles

Specializing trees and answer to a question of Williams.Mohammad Golshani & Saharon Shelah - 2020 - Journal of Mathematical Logic 21 (1):2050023.
Ordinal definability and combinatorics of equivalence relations.William Chan - 2019 - Journal of Mathematical Logic 19 (2):1950009.
Collapsing the cardinals of HOD.James Cummings, Sy David Friedman & Mohammad Golshani - 2015 - Journal of Mathematical Logic 15 (2):1550007.
The mouse set theorem just past projective.Mitch Rudominer - forthcoming - Journal of Mathematical Logic.


Added to PP

12 (#1,108,968)

6 months
12 (#310,009)

Historical graph of downloads
How can I increase my downloads?

Author's Profile

W. Hugh Woodin
Harvard University

Citations of this work

No citations found.

Add more citations

References found in this work

No references found.

Add more references