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Reference; Foundation for analysis without axiom of choice?

Please let me know if I'm misunderstanding something and I hope you explain this with relatively easy words. I am eager to learn, but I have no idea what forcing is like, so i couldn't understand this properly from past posts.

  1. Solovay's model says that "In ZF, there exists a model such that all subsets of $\mathbb{R}$ is measurable." (I believe I'm misunderstanding this.. Since if it is true, then it must be true in ZFC)

Why are Solovay's model and Dependent Choice related? Is this true in ZF+DC, but not true in ZFC? How is that possible?

  1. In the link, Asaf says, $\mathbb{R} = \bigcup_{i\in I} A_i$, where $A_i \subset \mathbb{R}$ and are mutually disjoint, nonempty and $|I|>|\mathbb{R}|$.

Really??

You can see, $|\mathbb{R}|=|\bigcup_{i\in I} A_i|=\sum_{i\in I} |A_i|≧\sum_{i\in I} 1=|I|$.

(I'm not sure if $≧$ part uses AC)

Thus $|I|≦|\mathbb{R}|$. How come $|I|>|\mathbb{R}|$?

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"In ZF, there exists a model such that all subsets of $\mathbb R$ is measurable." There is a slight misunderstanding here. The correct statement is that "There is a model of ZF such that ..." The difference is that to say "In ZF, blah" means that ZF can prove blah. Which is not true in this case. –  Andres Caicedo Oct 11 '12 at 3:06
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up vote 4 down vote accepted

In Solovay's model the following holds: ZF, DC, All sets of real numbers are Lebesgue measurable (and more, much more).

It is a theorem that from ZF+DC+"$\aleph_1\leq|\mathbb R|$" we can prove that there is an unmeasurable set of real numbers, so in Solovay's model we have that there are no sets of real numbers which have size $\aleph_1$. In fact, in Solovay's model every uncountable set of reals is of size continuum, and in some sense the continuum hypothesis holds.

However the continuum can always be mapped onto $\aleph_1$. So if you map the interval $[0,1]$ onto $\aleph_1$, it forms a partition of $[0,1]$ to $\aleph_1$ many parts; add the singletons of $\mathbb R\setminus[0,1]$ and you have a partition of $\mathbb R$ into $2^{\aleph_0}+\aleph_1$ many sets. However since $\aleph_1\nleq2^{\aleph_0}$ we have that $\aleph_1+2^{\aleph_0}>2^{\aleph_0}$.

Your arguments about cardinality and sums fails because without the axiom of choice infinite sums of cardinals play by different rules. The fact there is no injection which chooses a point from every $A_i$ means that $I$ does not have to be of smaller cardinality.

Indeed $\sum_{i\in I}|A_i|\geq\sum_{i\in I} 1$ means that you can choose a point from every $A_i$, but in this case you cannot.

The importance of DC in Solovay's model is that under the assumptions of ZF+DC we can prove a lot of classical analysis and basic measure theory. This means that this is a reasonable model for mathematicians to consider. In fact in his paper from the 1970's about automatic continuity of linear operators Garnir foresaw functional analysis developed in Solovay's model naturally. While I am aware of some functional analysis being done in such models, I don't think it caught on strongly as Garnir hoped.

One more point to make about the paradoxical decomposition to more parts than elements, I should add that currently we do not know of any model of ZF+$\lnot$AC where such decomposition does not exist. Namely, as far as we know, in all models where choice fails there is some set which can be partitioned into more parts than elements.

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I believe there might be a misinterpretation here. Solovay's model is a model of ZF in which the following hold:

  • The axiom of dependent choice;
  • Every set of reals in Lebesgue measurable; and
  • Every set of reals has the Baire property.

Due to the usual construction of a Vitali set from the Axiom of Choice it follows in particular that the full Axiom of Choice fails in this model. Since the Axiom of Choice fails, most of the intuition about cardinal arithmetic falls by the wayside.

(You should also note that the construction of Solovay's model does not begin with ZF or even a model of ZF+DC, but instead with a model of "ZFC + $\exists$ inaccessible cardinal".)

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Last sentence is exactly what i wanted to know. So to be clear, it's not a 'counter-intuitional' example when accepting DC? –  Katlus Oct 10 '12 at 18:49
    
@Katlus: It is just a side-effect; the same could happen when the real numbers are a countable union of countable sets (e.g. Feferman-Levy model) where DC fails. This has nothing to do with DC, but rather something to do with the fact that there are no sets of real numbers of size $\aleph_1$. –  Asaf Karagila Oct 10 '12 at 18:55
    
@Asaf You mean $\aleph_1$ in ZF? –  Katlus Oct 10 '12 at 19:17
    
@Katlus: What do you mean by that? –  Asaf Karagila Oct 10 '12 at 19:26
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@Katlus: No... I didn't say that. I said that in Solovay's model we know that ZF+DC+... holds; to construct Solovay's model we begin with a model of ZFC+Inaccessible cardinal. –  Asaf Karagila Oct 10 '12 at 19:57
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