It is consistent with ZFC that $2^{\aleph\_1}=2^{\aleph\_0}$. This can be gotten easily via forcing; more interestingly, it is a direct consequence of forcing axioms (which also set this value at $\aleph\_2$). However, just because a bijection exists between the subsets of $\omega$ and the subsets of $\omega\_1$ doesn't mean that such a bijection is "nice" in any sense.


My question is, roughly:



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> Can we ever have a "nice" injection from $\mathcal{P}(\omega\_1)$ to $\mathcal{P}(\omega)$, or at least a "nice" surjection from $\mathcal{P}(\omega)$ to $\mathcal{P}(\omega\_1)$?
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Exactly what "nice" here means is a bit up for grabs; my instinct at first is to lean on [generalized descriptive set theory](https://arxiv.org/pdf/1207.4311.pdf) and ask for something like "Borel on $\omega\_1^{\omega\_1}$," but I'm not at all sure that's a good thing to do.


Another take might be along the lines of the following:



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> Is it consistent with ZFC that there is a parameter-freely-definable surjection from $\mathcal{P}(\omega)$ to $\mathcal{P}(\omega\_1)$?
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We can also ask a "local" version of this question:



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> Is it consistent with ZFC that for every $s\subset\omega\_1$ there is a $r\subset\omega$ with $s\in L[r]$?
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While this wouldn't establish the existence of a "nice" correspondence, it would at least establish a more meaningful equivalence than follows immediately from them having the same cardinality. (Note, however, that this would imply that $\omega\_1=\omega\_1^r$ for some real $r$; since this is contradicted by large cardinals, this seems like it might be the wrong question to ask.)



Conflating reals and sets of countable ordinals "nicely"
The technique of almost disjoint forcing was introduced in 



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> [MR0289291 (44 #6482)](http://www.ams.org/mathscinet-getitem?mr=289291). Jensen, R. B.; Solovay, R. M. *Some
>  applications of almost disjoint sets*. In **Mathematical Logic and
>  Foundations of Set Theory** (Proc. Internat. Colloq., Jerusalem, 1968),
>  pp. 84–104, North-Holland, Amsterdam, 1970.
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Fix an almost disjoint family $X=(x\_\alpha:\alpha<\omega\_1)$ of subsets of $\omega$, so $x\_\alpha\cap x\_\beta$ is finite for all $\alpha<\beta<\omega\_1$. Given $s\subseteq\omega\_1$, the poset $\mathbb P\_{X,s}$ adds a real $r$ such that $$s=\{\alpha<\omega\_1:r\cap x\_\alpha\mbox{ is infinite}\}.$$
This shows that $s\in L[X,r]$; if $X\in L$ then in fact $s\in L[r]$. The poset $\mathbb P\_{X,s}$ is ccc, and it follows from standard arguments that if $\mathsf{MA}\_{\omega\_1}$ holds and $\omega\_1=\omega\_1^L$, then $\mathcal P(\omega\_1)=\bigcup\_{r\subseteq\omega}\mathcal P(\omega\_1)^{L[r]}$. Moreover, by taking as $X$ the first almost disjoint family in the standard enumeration of $L$, we see that $X$ is definable without parameters, and this gives us a simple parameter-free injection of $\mathcal P(\omega\_1)$ into $\mathcal P(\omega)$ as long as there is, for instance, a parameter-free definable well-ordering of $\mathcal P(\omega)$. This is consistent with $\mathsf{MA}\_{\omega\_1}+\omega\_1=\omega\_1^L$. 


(In fact, it is a consequence of $\mathsf{BPFA}+\omega\_1=\omega\_1^L$, see my paper with Friedman listed below and available at [my page](https://andrescaicedo.wordpress.com/papers/). For the consistency result just for $\mathsf{MA}$, this is a result of Friedman, see Theorem 8.51 in his class forcing book.) 


In general, in the context of forcing axioms, to get a nice well-ordering of $\mathcal P(\omega\_1)$, it suffices to find a nice well-ordering of $\mathcal P(\omega)$ and apply almost disjoint forcing. If we further arrange that the almost disjoint family we begin with is sufficiently nice, this reflects as well in the complexity and/or parameters of the well-ordering of $\mathcal P(\omega\_1)$. The same holds for a nice injection of $\mathcal P(\omega\_1)$ into $\mathcal P(\omega)$ and similar variants.



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> [MR2895389 (2012m:03123)](http://www.ams.org/mathscinet-getitem?mr=2895389). Caicedo, Andrés Eduardo; Friedman, Sy-David. *$\mathsf{BPFA}$ and projective well-orderings of the reals*. J. Symbolic Logic **76** (2011), no. 4, 1126–1136. 
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I like the description of the poset $\mathbb P\_{X,s}$ in Jech's book: A condition is a function $f\!:d\_f\to2$ whose domain $d\_f$ is a subset of $\omega$ such that $d\_f\cap x\_\alpha$ is finite for all $\alpha\in s$ and $\{n\in d\_f: f(n)=1\}$ is finite. The order is inclusion. 


Note that if $p,q$ are incompatible conditions, then in particular $\{n:p(n)=1\}\ne\{n:q(n)=1\}$. This shows that $\mathbb P\_{X,s}$ is actually Knaster. 


For any $p$ and any $\alpha\in\omega\_1\smallsetminus s$, we can extend $p$ to a condition $q$ whose domain contains $x\_\alpha$ and satisfies $q(n)=0$ for all $n\in x\_\alpha\smallsetminus d\_p$ (since the $x\_\beta$ are almost disjoint). 


Similarly, for any $\alpha\in s$ and any $k\in\omega$, we can extend any condition to ensure that its domain meets $x\_\alpha$ in a set of size at least $k$. 


From this, standard density arguments give us the result.


By the way, it is consistent that $\mathsf{MM}$ holds and there is a parameter-free well-ordering of $\mathcal P(\omega\_1)$, which naturally gives us a nice injection of $\mathcal P(\omega\_1)$ into $\mathcal P(\omega)$. (I am using "nice" loosely here, but the definition of the well-ordering, and therefore of the injection, is actually not too bad.) See



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> [MR2474445 (2009k:03085)](http://www.ams.org/mathscinet-getitem?mr=2474445). Larson, Paul B. [*Martin's maximum and definability in $H(\aleph\_2)$*](https://doi.org/10.1016/j.apal.2008.06.012). Ann. Pure Appl. Logic **156** (2008), no. 1, 110–122.
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