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Mathematics
Set Theory, Logic, Probability, Statistics
Does one need V not equal to L in order to do Cohen's forcing?
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[QUOTE="SSequence, post: 6434825, member: 601088"] Unfortunately I have no idea about forcing whatsoever, so I don't think I can add much to help you. Very very vaguely I think its about building various models (that satisfy the axioms of set theory). But how, I don't have any idea. Given some degree of (informal) familiarity with L, a couple of quite basic comments. One thing to note is that by default to get V=L (as a model) one must add an extra axiom (constructibility) to ZF. Quite roughly the axiom is about adding sets in a particular manner at each stage ##L_\alpha## to ##L_{\alpha+1}## while ensuring that the whole of V satisfies the set theory axioms. This sounds about right to me. More specifically, it seems to me (pure speculation on my part and it may be wrong) that to move from CH being satisfied in a model to ~CH (in another model), one would definitely have to add many more real numbers. Also seemingly, the new reals would have to added in such a way so that none of the new reals that are added are able to code a well-order for the ##\omega_1## of the original model. ========================= Generally speaking, it is complicated though. One complication is that the notion of ordinals (such as ##\omega_1##) isn't absolute between various models. In others words one can have two different models (both satisfying the the set-theory axioms) such that ##\omega_1## of first model is completely different from ##\omega_1## of second model. Why I think it might matter (even for L) is that it makes the notion of L in itself "relative" in a certain sense I think. Because if one has two different models for "ZF+constructibility" with different ##\omega_1##'s, one would have more reals in one model compared to the other one (both models satisfying CH ofc). ========================= I don't have much more to add, so hopefully is was at least a little bit helpful. [/QUOTE]
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Does one need V not equal to L in order to do Cohen's forcing?
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