I found this pop math article saying that there was a paper published last year that proved that the cardinalities of the reals and naturals are equal. Is this true or is it a misinterpretation of the result? If it is true, I'm dying to know what the bijection between the two sets is.
Set Theory – Is There a Bijection Between the Reals and Naturals?
model-theoryreal numbersset-theory
Related Solutions
I assume you know the general theorem that, using the axiom of choice, every set can be well ordered. Given that, I think you're asking how hard it is to actually define the well ordering. This is a natural question but it turns out that the answer may be unsatisfying.
First, of course, without the axiom of choice it's consistent with ZF set theory that there is no well ordering of the reals. So you can't just write down a formula of set theory akin to the quadratic formula that will "obviously" define a well ordering. Any formula that does define a well-ordering of the reals is going to require a nontrivial proof to verify that it's correct.
However, there is not even a formula that unequivocally defines a well ordering of the reals in ZFC.
The theorem of "Borel determinacy" implies that there is no well ordering of the reals whose graph is a Borel set. This is provable in ZFC. The stronger hypothesis of "projective determinacy" implies there is no well ordering of the reals definable by a formula in the projective hierarchy. This is consistent with ZFC but not provable in ZFC.
Worse, it's even consistent with ZFC that no formula in the language of set theory defines a well ordering of the reals (even though one exists). That is, there is a model of ZFC in which no formula defines a well ordering of the reals.
A set theorist could tell you more about these results. They are in the set theoretic literature but not in the undergraduate literature.
Here is a positive result. If you work in $L$ (that is, you assume the axiom of constructibility) then a specific formula is known that defines a well ordering of the reals in that context. However, the axiom of constructibility is not provable in ZFC (although it is consistent with ZFC), and the formula in question does not define a well ordering of the reals in arbitrary models of ZFC.
A second positive result, for relative definability. By looking at the standard proof of the well ordering principle (Zermelo's proof), we see that there is a single formula $\phi(x,y,z)$ in the language of set theory such that if we have any choice function $F$ on the powerset of the reals then the formula $\psi(x,y) = \phi(x,y,F)$ defines a well ordering of the reals, in any model of ZF that happens to have such a choice function. Informally, this says that the reason the usual proof can't explicitly construct a well ordering is because we can't explicitly construct the choice function that the proof takes as an input.
Once you have the bijection $P:2^\mathbb{N}\cong\mathbb{R}$, you can build your desired bijection as follows.
First, note that $\mathbb{N}\times 2^\mathbb{N}\cong 2^\mathbb{N}$, essentially by the bijection that associates $(n,A)$ with $\{n\}\cup (n+1+A)$, except that this will miss the empty set on the right, but we can fix this by composing with a map witnessing $2^\mathbb{N}-\{\varnothing\}\cong 2^\mathbb{N}$, such as shifting on a fixed countable subset.
Now simply observe that $$\mathbb{R}^\mathbb{R}\cong (2^\mathbb{N})^{(2^\mathbb{N})}\cong 2^{(\mathbb{N}\times 2^\mathbb{N})} \cong 2^{(2^\mathbb{N})} \cong 2^\mathbb{R}, $$
which provides the desired bijection. The first map is conjugation by $P$, simply composing with $P^{-1}$ and $P$ before and after; the second map is an easy exercise in parenthesis rearranging; the third map applies the observation above; and the final map applies $P$.
Best Answer
This could have been written clearer. I think the culprit is the section:
If read quickly, this suggests that $\mathfrak{p}$ and $\mathfrak{t}$ refer to the cardinality of the set of reals and the set of naturals, respectively. This is not the case, though.
So what sort of thing are $\mathfrak{p}$ and $\mathfrak{t}$, then?
$\mathfrak{p}$ and $\mathfrak{t}$ are what are known as cardinal characteristics of the continuum (CCCs) - cardinals which are (i) known to be uncountable, and (ii) measure how big a set of reals has to be to have some "universality" property.
For example, one simple CCC is the dominating number, $\mathfrak{d}$: this is the smallest cardinality of a set $F$ of functions $\mathbb{N}\rightarrow\mathbb{N}$ such that for each $g:\mathbb{N}\rightarrow\mathbb{N}$ there is some $f\in F$ such that $f(n)>g(n)$ for all but finitely many $n$ (we say $f$ dominates $g$). Clearly $\mathfrak{d}$ is at most continuum (since that's how many functions $\mathbb{N}\rightarrow\mathbb{N}$ there are in the first place), and it's also uncountable: if $f_i:\mathbb{N}\rightarrow\mathbb{N}$ for $i\in\mathbb{N}$, the function $$h(i)=\sum_{j\le i}f_j(i)=f_1(i)+f_2(i)+...+f_i(i)$$ is not dominated by any of the $f_i$s.
Another simple CCC is the bounding number, $\mathfrak{b}$. This is "dual" to $\mathfrak{d}$ (in a sense that can be made precise): $\mathfrak{b}$ is the smallest size of any family $G$ of functions $\mathbb{N}\rightarrow\mathbb{N}$ such that no single $f$ dominates all functions in $G$. Again, $\mathfrak{b}$ is clearly at most continuum, and is uncountable since any countably many functions can be dominated by a single function (think about the construction of the $h$ above).
Now cardinal arithmetic is notoriously badly behaved - even basic facts about it tend to be undecidable in ZFC. For example, ZFC doesn't even prove that $\kappa<\lambda \implies 2^\kappa<2^\lambda$. So it's really exciting to see ZFC-provable facts about infinite cardinalities; conversely, it's important to understand when certain questions can't be resolved in ZFC alone. In this context, what we care about is comparing CCCs. We can think about it this way: the two trivial CCCs are $\omega_1$ ("the smallest size of an uncountable set of reals") and $2^{\aleph_0}$ ("the smallest size of a set containing all the reals"); and in between we have the interesting CCCs. Of course, if $\omega_1=2^{\aleph_0}$ then the whole picture collapses; this is the continuum hypothesis, and it's consistent with ZFC. At the far other end, it's known that we can separate certain CCCs - e.g. that it is consistent with ZFC that $\mathfrak{b}<\mathfrak{d}$. (An interesting topic is separating multiple CCCs simultaneously - see this MO question.)
This leaves open:
As an example of the latter, ZFC proves that $\mathfrak{b}\le\mathfrak{d}$ - we can't have $\mathfrak{b}>\mathfrak{d}$ (this is a good exercise). More broadly, the collection of disprovable inequalities between many CCCs (not including $\mathfrak{p}$ and $\mathfrak{t}$, though) is summed up in Cichon's diagram. Malliaris and Shelah proved a result of the former kind - showing that two CCCs were in fact equal. My understanding is that this type of result is much, much rarer even in general, and of course in this particular case it was extremely surprising (see Shelah's quote in the linked article).
Of course, I haven't tried to define $\mathfrak{p}$ and $\mathfrak{t}$; the definitions are there, but they're a bit technical, and more to the point it's hard to see why someone would care. A good analysis of them is not something I can fit into an MSE answer; but hopefully what I've written explains a bit about where this sort of thing can come from!