(A): In "A nonconstructible $Δ_3^1$ set of integers" the following is proved by Robert Solovay using forcing:

Theorem. There is a $Δ_3^1$ set of sets of integers, $X$, which is not constructible from any set of integers $A$.

The following is stated about the proof of the above theorem:

It is amusing to note that the proof uses Cohen's notion of a generic set of integers. This is probably the first application of Cohen's method to set theory yielding an absolute result rather than a relative consistency result.

(B): The paper " Extensions of the measurable choice theorem by means of forcing. Israel J. Math. 14 (1973), 104–114" by Wesley, presents some $ZFC$ results using forcing. The following is taken from its introduction:

Using the method of forcing of set theory, we prove the following two theorems on the existence of measurable choice functions. Let $T$ be the closed unit interval $[0,1]$ and let $m$ be the usual Lebesgue measure defined on the Borel subsets of $T$.

Theorem 1: Let $S⊂T×T$ be a Borel set such that, for all $t∈T$, $S_t=\{x|(t,x)∈S\}$ is countable and nonempty; then there exists a countable series of Lebesgue-measurable functions $f_n:T→T$ such that $S_t=\{f_n(t)|n∈ω\}$ for all $t∈T$.

Theorem 2: Let $W⊂[0,1]×[0,1]$ be a Borel set such that, for each $x∈[0,1]$, $W_x=\{y|(x,y)∈W\}$ is uncountable; then there exists a function $h:[0,1]×[0,1]→W$ with the following properties:

(a) for each $x∈[0,1]$, the function $h(x,⋅)$ is one-one and onto $W_x$ and is Borel measurable;

(b) for each $y,h(⋅,y)$ is Lebesgue measurable;

(c) the function $h$ is Lebesgue measurable.

(A): In "A nonconstructible $Δ_3^1$ set of integers" the following is proved by Robert Solovay using forcing:

Theorem. Assuming the existence of a Ramsey cardinal, there is a $Δ_3^1$ set of sets of integers, $X$, which is not constructible from any set of integers $A$.

The following is stated about the proof of the above theorem:

It is amusing to note that the proof uses Cohen's notion of a generic set of integers. This is probably the first application of Cohen's method to set theory yielding an absolute result rather than a relative consistency result.

(B): The paper " Extensions of the measurable choice theorem by means of forcing. Israel J. Math. 14 (1973), 104–114" by Wesley, presents some $ZFC$ results using forcing. The following is taken from its introduction:

Using the method of forcing of set theory, we prove the following two theorems on the existence of measurable choice functions. Let $T$ be the closed unit interval $[0,1]$ and let $m$ be the usual Lebesgue measure defined on the Borel subsets of $T$.

Theorem 1: Let $S⊂T×T$ be a Borel set such that, for all $t∈T$, $S_t=\{x|(t,x)∈S\}$ is countable and nonempty; then there exists a countable series of Lebesgue-measurable functions $f_n:T→T$ such that $S_t=\{f_n(t)|n∈ω\}$ for all $t∈T$.

Theorem 2: Let $W⊂[0,1]×[0,1]$ be a Borel set such that, for each $x∈[0,1]$, $W_x=\{y|(x,y)∈W\}$ is uncountable; then there exists a function $h:[0,1]×[0,1]→W$ with the following properties:

(a) for each $x∈[0,1]$, the function $h(x,⋅)$ is one-one and onto $W_x$ and is Borel measurable;

(b) for each $y,h(⋅,y)$ is Lebesgue measurable;

(c) the function $h$ is Lebesgue measurable.

The paper " Extensions of the measurable choice theorem by means of forcing. Israel J. Math. 14 (1973), 104–114" by Wesley, presents some $ZFC$ results using forcing. The following is taken from its introduction:

Using the method of forcing of set theory, we prove the following two theorems on the existence of measurable choice functions. Let $T$ be the closed unit interval $[0,1]$ and let $m$ be the usual Lebesgue measure defined on the Borel subsets of $T$.

Theorem 1: Let $S⊂T×T$ be a Borel set such that, for all $t∈T$, $S_t=\{x|(t,x)∈S\}$ is countable and nonempty; then there exists a countable series of Lebesgue-measurable functions $f_n:T→T$ such that $S_t=\{f_n(t)|n∈ω\}$ for all $t∈T$.

Theorem 2: Let $W⊂[0,1]×[0,1]$ be a Borel set such that, for each $x∈[0,1]$, $W_x=\{y|(x,y)∈W\}$ is uncountable; then there exists a function $h:[0,1]×[0,1]→W$ with the following properties:

(a) for each $x∈[0,1]$, the function $h(x,⋅)$ is one-one and onto $W_x$ and is Borel measurable;

(b) for each $y,h(⋅,y)$ is Lebesgue measurable;

(c) the function $h$ is Lebesgue measurable.

(A): In "A nonconstructible $Δ_3^1$ set of integers" the following is proved by Robert Solovay using forcing:

Theorem. There is a $Δ_3^1$ set of sets of integers, $X$, which is not constructible from any set of integers $A$.

The following is stated about the proof of the above theorem:

It is amusing to note that the proof uses Cohen's notion of a generic set of integers. This is probably the first application of Cohen's method to set theory yielding an absolute result rather than a relative consistency result.

(B): The paper " Extensions of the measurable choice theorem by means of forcing. Israel J. Math. 14 (1973), 104–114" by Wesley, presents some $ZFC$ results using forcing. The following is taken from its introduction:

Using the method of forcing of set theory, we prove the following two theorems on the existence of measurable choice functions. Let $T$ be the closed unit interval $[0,1]$ and let $m$ be the usual Lebesgue measure defined on the Borel subsets of $T$.

Theorem 1: Let $S⊂T×T$ be a Borel set such that, for all $t∈T$, $S_t=\{x|(t,x)∈S\}$ is countable and nonempty; then there exists a countable series of Lebesgue-measurable functions $f_n:T→T$ such that $S_t=\{f_n(t)|n∈ω\}$ for all $t∈T$.

Theorem 2: Let $W⊂[0,1]×[0,1]$ be a Borel set such that, for each $x∈[0,1]$, $W_x=\{y|(x,y)∈W\}$ is uncountable; then there exists a function $h:[0,1]×[0,1]→W$ with the following properties:

(a) for each $x∈[0,1]$, the function $h(x,⋅)$ is one-one and onto $W_x$ and is Borel measurable;

(b) for each $y,h(⋅,y)$ is Lebesgue measurable;

(c) the function $h$ is Lebesgue measurable.