1   The Continuum Hypothesis  1398  G del showed Con   ZFC     Con   ZFC   CH     1962  Cohen showed Con   ZFC     Con   ZFC     CH     Theorem   Hartogs s Theorem   For every set X  there is a  least  cardinal   such that there is no injection from   to X   We denote this by Hartogs s aleph of X      X     Theorem  For every set X  there is no injection from P   X   into X  We denote this by 2   X     Using Axiom of Choice  well order P   X   and get ordinal 2   X     We have          X     2   X         Notation  Define        0             1                                     0           1   2                           Clearly              Continuum Hypothesis  CH     1     1   Generalised Continuum Hypothesis  GCH                   Why  continuum    Lemma  CH  if and only if   X      X is uncountable   X        means  there is a bijection      Proof      Obvious since   1               Suppose           1  Well order the reals in order type       1        r                   Consider X       r         w 1        This is a subset of the reals of cardinality   1  hence is an uncountable subset which is not in bijection with it     Reminder   G del showed Con   ZFC     Con   ZFC   CH     Cohen showed Con   ZFC     Con   ZFC     CH     Relative consistency proofs   By Completeness Theorem  this means   If there is M   ZFC  then I can construct N   ZFC         CH   Analogy from algebra       L fields     0   1                 1         Axioms of fields  Fields   Let   2       x   x   x   1   1    Note         2   Idea   Start with   and extend   to get F   Fields     2   Construct by algebraic closure   not in the usual sense   here we just mean adding in 2 and then adding everything else that this requires us to add    Obtain     2     Fields     2   This is easy because everything that matters  Fields and   2  is determined by equations  all formulas we need to check are atomic   Definition 1 1   absolute    If M   N and M   N are L structures and   an L formula  then we say   is  absolute between  M and  N if for all x 1       x m   M      M       x 1       x n     N       x 1       x n         If the   direction holds  then we say  upwards absolute   and if the   direction holds  then we say  downwards absolute    Theorem   Substructure Lemma   All atomic formulas are absolute between substructures   WHat if we have models of ZFC  Have      L                 No function symbols nor constant symbols  So  almost nothing is atomic   M   N if and only if M is an L   substructure of N   And  the formulas that we care about are definitely not atomic  but instead very complex   Try to imagine a proof of   If M   ZFC then there is N   M such that N   ZFC   CH   Without loss of generality M     CH  else we are done   For simplicity  let s consider the case   1     2   What can we do to  get rid of   1    Maybe a surjection f         1  Maybe we can form M   f     M to get a smallest model M   f     ZFC   Clearly  in M   f      1 M is not a cardinal anymore   Does that show CH   All sorts of things can happen  Assuming it is actually possible to form this smallest model M   f    there are lots of ways that this might not end up being enough to deduce CH  For example   Maybe   M   f       M  Maybe   2 M is not a cardinal either  A fundamental problem of non absoluteness  Consider       x         z   z   x    which means  x is empty    Consider M   ZFC  Therefore there are e   e   such that M         e   and M     z   z   e     z   e     Consider N     M     e     N is an L   substructure of M  But N         e      even though M           e       So     is not absolute between substructures   Instead of substructures  we will restrict out attention to transitive substructures  in particular  to M transitive     x   x   M   x   M or equivalently x   M   y   x   y   M  such that M   ZFC   Next time  theorems about absoluteness for transitive substructures   Definition   Absolute formula   We say   is absolute for  M if for all x 1       x n   M  we have      M       x 1       x n         x 1       x n   is true       Clearly  if   is absolute  for M and absolute  for N  then it s absolute between M and N   Cohen s proof becomes   If M is a countable transitive set such that M   ZFC  then there is a a countable transitive set N   M such that N   ZFC     CH