So in the past few weeks I have been studying a small amount of descriptive set theory from Kanamori's "The Higher Infinite" and I have some questions.
First and foremost I acknowledge that I am really bad at analysis.
Going from here, to give some context, most of the stuff I have seen are basically about the Baire space ($^\omega\omega$), and I know that $^\omega\omega \cong \mathbb{R} \setminus \mathbb{Q}$ and this gives us the fact that the Borel subsets of the Baire space are in someway the Borel subsets of $\mathbb{R}$ and the null subsets of both $^\omega\omega$ and $\mathbb{R}$ can be put to a bijection modulo $\mathbb{Q}$.(since the isomorphism above also preserves measure in the sense that null sets go to null sets and vice versa. [although I haven't checked it myself.]) So in a measure theoretic sense, the Baire space behaves somehow like $\mathbb{R}$.
But the same cannot be said for the Cantor space as it is null in $\mathbb{R}$. So this is one key differnce (in the measure theoretic sense atleast) between the different notions of "real" that we have in set theory.
Another problem arises from the fact that topologically speaking, the relation between the Baire space and $\mathbb{R}$ is like the relation between cucumbers and pickles.(They are very different.) For example the Baire space has a basis of clopen sets (and hence is totally disconnected) but $\mathbb{R}$ is connected. $\mathbb{R}$ is locally compact but the Baire space isn't. The Baire space has dimension zero but $\mathbb{R}$ is not dimension zero. And many more that I don't know.
So here I want to ask my question, seeing the difference between all these different interpretations of the notion of "real" and noting that in set theory spaces like the Baire space or the Cantor space or even $P(\omega)$ are used more widely than $\mathbb{R}$ itself(atleast I haven't encountered many direct uses of $\mathbb{R}$.),
- How are the theorems proved in set theory relevant to modern analysis?
(Meaning how are the results put in terms of $\mathbb{R}$?)
This one is very specific:
- How are regular mathematical statements like RH, interpreted as $\Sigma^1_2$ or $\Pi^1_2$ relations (and hence are absolute)?(I actually heard this from one of my professors.)
The reason I am asking this is that RH really depends on the topology of $\mathbb{R}$ since it is about the analytical continuation of the $\zeta$ function to $\mathbb{C}$.
This one might be a little bit off-topic but my curiosity doesn't let me not ask it:
- Have there been instances in which the Baire space, Cantor space or $P(\omega)$ were used as spaces imitating $\mathbb{R}$, to prove algebraic statements about $\mathbb{R}$ through a translation? By algebraic I mean any statement that goes beyond the topology of $\mathbb{R}$ and making use of operators like $+$ and $\cdot$ or maybe much more complicated ones.
EDIT I:
I realize that my first question is somewhat broad.(Although I would really appreciate any general answer.) So here I will try to make it a little bit more specific.
One of my main concerns is the projective hierarchy. Since the projective hierarchy depends on the closed subsets of $^k(^\omega\omega)$ and projections, and since topologically $^k(^\omega\omega) \cong {^\omega\omega}$ but $^k\mathbb{R} \not \cong \mathbb{R}$, the closed subsets are very different and might behave differently, so I can narrow down my question partially to:
- How are the results about the projective hierarchy put in terms of $\mathbb{R}$?(e.g. the Ihoda[now Judah]-Shelah theorem on the measurability of $\Delta^1_2$ sets of reals)
Best Answer
What follows is a long string of comments I hope might help to put your questions in context.
All the spaces considered are Polish (separable and completely metrizable). The derived Borel spaces $(X,B(X))$ are called standard Borel spaces. One of the main and basic theorems is that any two uncountable, standard Borel spaces are isomorphic. Going back to the topological spaces, any for any two uncountable Polish spaces $X,Y$ there exists a Borel map $f:X\to Y$ with a Borel inverse.
That actually show that the Borel sets of any two of the aforementioned spaces behave the same. With a little more care, you can calculate the complexity of each of those isomorphisms and, for instance, if the preimage of an open subset of $Y$ by $f$ is a $\Sigma^0_\xi$ subset of $X$, then the Borel sets of $Y$ are brought back to Borel sets of $X$ of at most $\xi$ levels up in complexity (namely, if $f : {X} \rightarrow {Y}$ is $\Sigma_{\xi}^{0}$-measurable and $P$ is a $\Sigma_{\eta}^{0}$ then $f^{-1}[P]$ is $\Sigma_{\xi+\eta}^{0}$; see [Moschovakis, 1G.7]). Moreover, each level of the projective hierarchy is preserved by such an isomorphism [ibid., 1G.1].
Concerning measure, note that any two measure spaces of the form $(X,B(X),\mu)$ where $(X,B(X))$ is standard Borel and $\mu$ is a probability measure, are isomorphic. And since Lebesgue measure on $\mathbb{R}$ is $\sigma$-finite, it is equivalent to a probability measure (i.e. has the same null sets). Therefore all the questions of $\mu$-measurability have the same answer across the whole range of Polish spaces. Incidentally, as in your argument of the similarity between the Baire space and the reales, $^\omega\omega$ is also homeomorphic to a co-countable subset of the Cantor space: Just take the set of sequences of with infinitely many ones.
Concerning RH, one form of the statement depends on the topology, but it can be proved that it is equivalent to another one involving only the counting of primes. And actually, that version is $\Pi^0_1$ (a fortiori, talking only about the first order theory of natural numbers), and hence it is absolute.