Abstract
Since the 1980’s work on Projective Determinacy and AD^{L(R)} the concept of Woodin cardinal has become to be seen as central in the theory of large cardinals
and inner model theory. The use by Woodin himself of a background assumption
in many arguments that the universe contains unboundedly many such cardinals
again draws attention to the centrality of this concept.
As is well known the Reflection Principles dating to a more classicial era only provide large cardinals consistent with V = L, and not the wherewithal for such theorems on absoluteness under set forcing that Woodin has proven.
We discuss here a reflection principle derived from weak subcompactness that implies the existence of a proper class of measurable Woodin cardinals  thus providing adequate background assumptions for many of Woodin’s absoluteness results in his work
As is well known the Reflection Principles dating to a more classicial era only provide large cardinals consistent with V = L, and not the wherewithal for such theorems on absoluteness under set forcing that Woodin has proven.
We discuss here a reflection principle derived from weak subcompactness that implies the existence of a proper class of measurable Woodin cardinals  thus providing adequate background assumptions for many of Woodin’s absoluteness results in his work
Original language  English 

Title of host publication  Foundations of Mathematics 
Subtitle of host publication  Logic at Harvard: Essays in Honor of W. Hugh Woodin's 60th Birthday, March 2729 2015, Harvard University, Cambridge, MA 
Editors  Andres Eduardo Caicedo, James Cummings, Peter Koellner, Paul B Larson 
Publisher  American Mathematical Society 
Pages  161176 
Number of pages  16 
ISBN (Electronic)  9781470440794 
ISBN (Print)  9781470422561 
DOIs  
Publication status  Published  30 May 2017 
Publication series
Name  Contemporary Mathematics 

Publisher  American Mathematical Society 
Volume  690 
ISSN (Print)  02714132 
ISSN (Electronic)  10983627 
Keywords
 Reflection
 Woodin cardinals
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Professor Philip D Welch
 School of Mathematics  Professor of Pure Mathematics
 Pure Mathematics
 Set Theory and Logic
Person: Academic , Member, Group lead