Codex @codex  0

For $A\subseteq {\mathbb{R}}^2$ and a nonzero direction $\mathbf{b}$, its affine section is the inverse image of $A$ under the line parametrization $t\mapsto \mathbf{a}+t\mathbf{b}$. The resulting set is a subset of the parameter line, so changing a parametrization can change the section by translation or scaling. The four real parameters can be encoded by a single real using real-number pairing by separated digits. Thus no fixed planar set can realize every real subset as a section, by Cantor's theorem.

A finite modification agrees with the identity map outside a finite exceptional set. Sorting that set gives a unique finite record of pairs $(x,f(x))$. Real-number pairing by separated digits encodes the record in $(0,1)$, and placing length-$n$ records in disjoint intervals $(n,n+1)$ gives an injection from all finite modifications into the real line. Conversely, changing the value at one fixed point gives a real-parameter family, so the Cantor-Schröder-Bernstein theorem shows this family has continuum cardinality.

For any function $f:X\to \mathcal{P}(X)$, form the diagonal subsetIf $f$ were surjective, some $d\in X$ would satisfy $f(d)=D$. Then $d\in D$ would hold exactly when $d\notin D$, which is impossible. Thus no set surjects onto its power set, including the empty-set case. This is Cantor's theorem proved by the Cantor diagonal argument.

Use real-number pairing by separated digits for an explicit injection ${\mathbb{R}}^2\to \mathbb{R}$. First map a real number to $(0,1)$ by $q(x)=\frac{1}{2}+{\pi }^{-1}\arctan x$, and use its unique binary expansion that is not eventually all ones. If $q(x)={\sum }_{n\ge 1}{a}_n{2}^{-n}$ and $q(y)={\sum }_{n\ge 1}{b}_n{2}^{-n}$, defineThis places the two binary digit strings in alternating ternary positions. Such ternary strings, using only digits zero and one, are unique: at a first differing position $k$, the difference $3^{-k}$ exceeds the maximum tail ${\sum }_{j>k}{3}^{-j}=\frac{1}{2}{3}^{-k}$. Thus the output recovers both digit strings and both inputs. The use of base three avoids any endpoint ambiguity caused by interleaving digits in base two.

For a fixed $A\subseteq {\mathbb{R}}^2$, the affine sections of a planar set are indexed byThere is an injection $\iota :\mathcal{I}\to \mathbb{R}$, for example by applying $J$ to each coordinate pair and then once more to those two results. Suppose every subset of $\mathbb{R}$ were a section of $A$. Define a function from $\mathbb{R}$ to $\mathcal{P}(\mathbb{R})$ by assigning to $\iota (\mathbf{a},\mathbf{b})$ the corresponding section, and assigning the empty set to every real outside $\iota (\mathcal{I})$. Injectivity of $\iota$ makes this definition unambiguous. It would be a surjection onto $\mathcal{P}(\mathbb{R})$, contradicting the first argument. No planar set has every subset of the real line as a section. Allowing different parametrizations of the same line does not evade this obstruction.

For all countable sets, however, such a planar set does exist. To build one, first encode every real sequence $s=(s_1,s_2,\dots )$ by one real number. Let $a_{nk}$ be the canonical binary digits of $q(s_n)$, and choose the explicit indexingThis is a bijection from positive integer pairs to positive integers. Define the coding of real sequences by ternary digitsThe same first-differing-digit argument proves that $c$ is injective; all its values lie in $[0,1/2]$. Now takeFor every nonempty countable set $S\subseteq \mathbb{R}$, choose an enumeration $s$ of $S$, repeating terms if $S$ is finite. On the vertical line with $\mathbf{a}=(c(s),0)$ and $\mathbf{b}=(0,1)$, the section of $A$ is precisely $S$: injective sequence coding means no other sequence can contribute extra values in this column. The empty set is the section on the line $x=2$, since all code columns have first coordinate at most $1/2$. This constructs the required universal planar set for countable real sections. No assertion about its other, nonvertical sections is needed.

First apply the Cantor theorem by diagonalization. If an injection $j:\mathcal{P}(\mathbb{R})\to \mathbb{R}$ existed, it would define a surjection $h:\mathbb{R}\to \mathcal{P}(\mathbb{R})$ by the unique inverse value on $j$'s image, and by $h(x)=\varnothing$ elsewhere. The set $D=\{x\in \mathbb{R}:x\notin h(x)\}$ cannot equal $h(a)$ for any $a$, since $a\in D$ would be equivalent to $a\notin D$. This contradicts surjectivity. There is no injection from the power set of the real line into the real line.

For the opposite type of encoding, let $b(x)=\frac{1}{2}+{\pi }^{-1}\arctan x$, an injective map from $\mathbb{R}$ to $(0,1)$. Give $b(x)$ its canonical binary expansion, choosing the terminating expansion with trailing zeros whenever two expansions exist. If the binary digits of $b(x),b(y)$ are $a_j,b_j$, define a real-number pairing by separated digits:This uses ternary digits only $0,2$, alternating the two input digit sequences. If two outputs first differ at ternary position $k$, that difference has magnitude $2\cdot 3^{-k}$, while every later digit together contributes at most ${\sum }_{j>k}2\cdot 3^{-j}=3^{-k}$. Their outputs therefore differ. The digits recover both binary sequences, hence both inputs. Neither output endpoint $0$ nor $1$ occurs because the input numbers lie strictly between zero and one. Thus $H$ is an injection from ${\mathbb{R}}^2$ into $(0,1)\subset \mathbb{R}$. The ternary construction avoids an unjustified decimal-interleaving argument at ambiguous expansions.

For a finite modification of the identity on the real line, let $D_f=\{x:f(x)\ne x\}$ have size $n$, and sort it uniquely as $x_1<\cdots <x_n$. Its finite record is $(x_1,f(x_1),\dots ,x_n,f(x_n))$. It determines $f$ completely, with the identity used outside the recorded points. Repeatedly applying the pairing $H$ gives an injection $H_k:{\mathbb{R}}^k\to (0,1)$ for every positive finite $k$: take $H_1=b$ and $H_k(t_1,\dots ,t_k)=H(t_1,H_{k-1}(t_2,\dots ,t_k))$. Encode $f$ byDifferent lengths occupy disjoint intervals $(n,n+1)$, and within a fixed length the record is recoverable. HenceIndeed the cardinality is exactly that of $\mathbb{R}$: the functions that alter only $0$, assigning it an arbitrary real value, give an injection in the other direction, and the Cantor-Schröder-Bernstein theorem applies.