LATTICE IDEALS, ARITHMETIC FACTORING, AND LOGIC GÖDEL NUMBERS
A lattice ideal has received much attention in abstract algebra, but little in arithmetic (especially factor theory) and in logic. We open this omission and highlight it's implications for the Gödel numbers which provide the coding to be interpreted as undecidability or incompleteness in arithmetic.
Technically, a lattice ideal, I, of a lattice, L, is a subset such that: (1) if element x, y e I, then x y e I; (2) if x e I and ze L, then x z e I. This is tedious and nongraphic, especially for a simple case.
A lattice ideal can easily be described in, and graphed by, a Hasse Diagram of a lattice:

diagram nodes are lattice elements;
noncyclic edges or paths or "chains" of the lattice connect related elements;
ordering is represented by vertical ascendance;
specifying a lattice ideal:

choose a lattice element (nontrivially higher than rank 0), say, 6;
determine all descending chains from this chosen element;
the sublattice formed by these chains is the ideal determined by the given element.

6-IDEAL IN 30-FACTOR-LATTICE(NODES RED-CODED, IMAGINE RED CHAINS CONNECTING)
30 /\ / \ / | \ / | \ / | \ 6 10 15 |\ / \ /| | \ / \ / | | \/ \/ | | /\ /\ | | / \ / \ | |/ \ / \| 2 3 5 \ | / \ | / \ | / \ | / \ | / 1
6-IDEAL IN FREE 30-FACTOR-LATTICE(SAME CODING)
30 /\ / |\ / | \ / | \ / | \ / | \ 6 10 15 / /|\ \ / / | \ \ / / | \ \ | / | \ \ | / | \ \ | / | \ | | / | \ | 610 615 1015 / \ |\ \ \__________ / \ | \________\_________ | / \______|___________\_______ | | / | \ \| | 2 3 5 61015 / \ /|\ /\ / \ / | \ / \ / \ / | \ / \ / \ / | \ / \ / \ / | \ / \ / \ / | \ / \ / \/ | \/ \ (23)(25) (23) (35) (25)(35) | \ \ / | \ / / | | \ \ / | \ / / | | \ \ / | \ / / | | \ \ / | \ / / | | \ \/ | \/ / | | \ /\ | /\ / | | \/ \ | / \/ | | /\ \ | / /\ | | / \ \ | / / \ | | / \ \ | / / \ | | ___/______\___\_______|_______/_ _/ \ | || / \ \ | / \ | || / \___\_____|_____/______________\__ | ||/ \ | / \ | | 2 3 2 5 3 5 | | | | | | | | | | | | | | | | | | -------------------2 3 5---------------------- | | 1
Behold! The 6-FILTER in LATTICE F(30) is THE FACTORIZATION OF 6! How can we VERIFY THAT THE RED-CODED ELEMENTS ABOVE ARE FACTORS OF 6? Simple -- via COUNTEREXAMPLE. In the TABLE below, if any non-6 TABLE (column) CONTAINS A "1" NOT IN THAT FOR 6, THEN IT IS NOT A FACTOR OF 6. But TRIS NOT THE CASE!
For compression, (2 3) (2 5) X; (2 3) (3 5) Y; (2 5) (3 5) Z. We then have these results.

(EXCEPTING 6) TABLE OF FREE ELEMENTS OF F(30) (NOT IN D(30))
23 25 35 X Y Z 610 615 1015 61015 6 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 1 0 0 1 0 1 1 1 1 1 1 1 0 0 0 0 0 0 1 0 0 0 1 0 1 0 1 0 1 1 1 1 1 1 1 0 0 1 1 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 (2) (2) (2) (3) (3) (3) (5) (5) (5) (4) (6) (BANK)