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from Part I - Let’s Be Independent
Published online by Cambridge University Press: 28 September 2020
Forcing was invented to prove the independence of CH. We have already indicated that this is going to be done through adding ‘new’ subsets of ω, to the extent that we shall have more than ω1 of them. What do we mean by ω1 here? We know that whatever M[G] is going to understand by ω1 is simply some countable ordinal (as any ordinal that is in M[G] is countable) so what we are aiming for here is that M[G] ╞ (2ω > ω1), so (ω1)M[G], i.e. the ω1 that we mean here is the first ordinal in M[G] which is not countable from the point of view of M[G]. This means that there is no function in M[G] which surjectively maps ω onto ω1. This discussion contains the proof of the following:
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