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I disagree with this statement. Consider $9$ points in the plane, called $x(i,j)$ where $(i,j)$ is in ${ 0,1,2 }$. There are $12$ triples $((i_1, j_1), (i_2, j_2), (i_3, j_3))$ such that $i_1 + i_2 + i_3 \equiv 0 \mod 3$ and $j_1 + j_2 + j_3 \equiv 0 \mod 3$. Let $L$ be the set of such triples.

Consider the following statement:

Suppose that, for all $((i_1, j_1), (i_2, j_2), (i_3, j_3)) \in L$, the points $x(i_1, j_1)$, $x(i_2, j_2)$ and $x(i_3, j_3)$ are colinear. Then either all of the $x(i,j)$ are colinear, or else two of the $x(i,j)$ are equal to each other.

It seems to me that this statement is an incidence theorem. It is true in $\mathbb{RP}^2$ and false in $\mathbb{CP}^2$, both of which obey Pappus theorem. I learned this example from Kiran Kedlaya.

To see this over $\mathbb{R}$, note that a counterexample to this claim is also a counterexample to the Sylvester-Gallai theorem. Over $\mathbb{C}$, the flexes of any cubic curve form a counter-example, as discussed on the above linked Wikipedia page. More precisely, I believe that the claim is true in $K\mathbb{P}^2$ if and only if $K$ does not contain a root of $x^2+x+1$.

More generally, I would consider an incidence theorem to be a first order statement about points and lines in $\mathbb{RP}^2$ where what we are allowed to say is that a given point does or does not lie on a given line. We can easily turn such a statement into an algebraic claim about $\mathbb{R}$.

By a result of TarkskiTarski, any true statement of this form follows from (1) the field axioms (2) the axioms that $\mathbb{R}$ has an ordering $\leq$ obeying the standard properties and (3) the "polynomial mean intermediate value theorem": for any polynomial $f \in \mathbb{R}[t]$, if $f(a)<0$ and $f(b)>0$, then there exists $c \in (a,b)$ such that $f(c)=0$.

For example, to prove that $x^2+x+1$ has no roots in $\mathbb{R}$, just note that $x^2+x+1 = (x+1/2)^2+3/4 \geq 3/4$. Here we have used the field axioms (many times) and the basic properties of $\leq$.

Pappus theorem encodes the commutativity of multiplication, and I will believe you that the other field axioms can be deduced from it as well. However, it certainly doesn't include the properties related to inequalities. So, if I rig up an algebraic statement (like the above) whose proof requires the order properties of $\mathbb{R}$, you won't be able to prove it from Pappus theorem. I haven't actually worked this out, but presumably if you apply Tarski's algorithm to the above claim, it will give you a proof which, at some point, involves dividing by $x^2+x+1$ for some unknown quantity $x$.

I'll mention that the axioms of an oriented matroid are an attempt to systematize the properties of $\mathbb{RP}^2$ deducible from the order properties of $\mathbb{R}$. It might be true that every true incidence theorem in $\mathbb{RP}^2$ is deducible from Pappus theorem, plus the axiom that the set of points of the plane can be equipped with the structure of an oriented matroid of rank $3$, where the bases are the noncolinear triples.

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I disagree with this statement. Consider $9$ points in the plane, called $x(i,j)$ where $(i,j)$ is in ${ 0,1,2 }$. There are $12$ triples $((i_1, j_1), (i_2, j_2), (i_3, j_3))$ such that $i_1 + i_2 + i_3 \equiv 0 \mod 3$ and $j_1 + j_2 + j_3 \equiv 0 \mod 3$. Let $L$ be the set of such triples.

Consider the following statement:

Suppose that, for all $((i_1, j_1), (i_2, j_2), (i_3, j_3)) \in L$, the points $x(i_1, j_1)$, $x(i_2, j_2)$ and $x(i_3, j_3)$ are colinear. Then either all of the $x(i,j)$ are colinear, or else two of the $x(i,j)$ are equal to each other.

It seems to me that this statement is an incidence theorem. It is true in $\mathbb{RP}^2$ and false in $\mathbb{CP}^2$, both of which obey Pappus theorem. I learned this example from Kiran Kedlaya.

To see this over $\mathbb{R}$, note that a counterexample to this claim is also a counterexample to the Sylvester-Gallai theorem. Over $\mathbb{C}$, the flexes of any cubic curve form a counter-example, as discussed on the above linked Wikipedia page. More precisely, I believe that the claim is true in $K\mathbb{P}^2$ if and only if $K$ does not contain a root of $x^2+x+1$.

More generally, I would consider an incidence theorem to be a first order statement about points and lines in $\mathbb{RP}^2$ where what we are allowed to say is that a given point does or does not lie on a given line. We can easily turn such a statement into an algebraic claim about $\mathbb{R}$.

By a result of Tarkski, any true statement of this form follows from (1) the field axioms (2) the axioms that $\mathbb{R}$ has an ordering $\leq$ obeying the standard properties and (3) the "polynomial mean value theorem": for any polynomial $f \in \mathbb{R}[t]$, if $f(a)<0$ and $f(b)>0$, then there exists $c \in (a,b)$ such that $f(c)=0$.

For example, to prove that $x^2+x+1$ has no roots in $\mathbb{R}$, just note that $x^2+x+1 = (x+1/2)^2+3/4 \geq 3/4$. Here we have used the field axioms (many times) and the basic properties of $\leq$.

Pappus theorem encodes the commutativity of multiplication, and I will believe you that the other field axioms can be deduced from it as well. However, it certainly doesn't include the properties related to inequalities. So, if I rig up an algebraic statement (like the above) whose proof requires the order properties of $\mathbb{R}$, you won't be able to prove it from Pappus theorem. I haven't actually worked this out, but presumably if you apply Tarski's algorithm to the above claim, it will give you a proof which, at some point, involves dividing by $x^2+x+1$ for some unknown quantity $x$.

I'll mention that the axioms of an oriented matroid are an attempt to systematize the properties of $\mathbb{RP}^2$ deducible from the order properties of $\mathbb{R}$. It might be true that every true incidence theorem in $\mathbb{RP}^2$ is deducible from Pappus theorem, plus the axiom that the set of points of the plane can be equipped with the structure of an oriented matroid of rank $3$, where the bases are the noncolinear triples.