On Sep 27, 6:03 pm, Arturo Magidin <magi...@member.ams.org> wrote: |While Goedel was explicit in his model construction (from the |assumption that ZF is consistent, he takes a model for ZF and |constructs an "internal" model in which ZF+C holds), forcing works |differently; I'm not sure it would be accurate to say Cohen "found an |interpretation in which AC is false" in the same sense as Goedel's |work does.
Goedel's result does give an obvious interpretation (being true in his class L) which makes AC true. Cohen's proof is not so obvious about it, but could be construed as also giving an interpretation in which it is false.
The tricky thing about the operation in his proof is that he adjoins a "generic" set to a specific model (the minimal model, assuming there is one), which doesn't give us an explicit such set. But the relation of "forcing" which he defines gives us an interpretation of sentences (making them true if they are true for any such model, whatever the generic set was). It's less of an interpretation in the sense that ~A being forced is not the same as A not being forced, while ~A being true in L is just the same as A not being true in L.
There's a more recent style of forcing proof, which uses what are called Boolean-valued models. Given a Boolean-valued model, and a sentence, the sentence has a value which is some element of a Boolean algebra (and can be other than 1=true or 0=false). Cohen's construction in effect supplies a value to each sentence.
I've wondered now and then whether there was a construction that worked in the opposite direction. Despite what I wrote in another message, the way AC was proved consistent was by trimming down the universe to something small enough to make AC true. Cohen then made AC false sometimes by adding to a model in which it is true an element that breaks it. But our usual intuition is that AC is made true by allowing the universe to be big enough to include all those choice sets and so on. I've wondered if there was a way to start with a universe in which AC fails and somehow fluff it up into one in which AC holds.
Cohen mentions in relation to his proof the issue that in order to be able to add a set, in the way that he does, one has to have a universe that doesn't already contain all the possibilities. (What he adds is in principle an actual set that happened not to be in the minimal mode.) Perhaps one wants to trim down the universe in which AC fails maybe to some equivalent countable universe first, or maybe just define some idealized interpretation of set existence which makes AC true anyway.
My vague thought was to make the sets of the new universe be something like imagined maximum elements of partially ordered sets satisfying the conditions of Zorn's lemma. But I don't know if such a construction works out at all.