The Löwenheim—Skolem theorem was published in Skolem’s long paper of 1920, with the first section dedicated to the theorem. The second section of the paper contains a proof-theoretical analysis of derivations in lattice theory. The main result, otherwise believed to have been established in the late 1980s, was a polynomial-time decision algorithm for these derivations. Skolem did not develop any notation for the representation of derivations, which makes the proofs of his results hard to follow. Such a formal notation is given here by which these proofs become transparent. A third section of Skolem’s paper gives an analysis for derivations in plane projective geometry. To clear a gap in Skolem’s result, a new conservativity property is shown for projective geometry, to the effect that a proper use of the axiom that gives the uniqueness of connecting lines and intersection points requires a conclusion with proper cases (logically, a disjunction in a positive part) to be proved.

The forgotten parts of Skolem’s first paper on the Löwenheim—Skolem theorem are the perhaps earliest combinatorial analyses of formal mathematical proofs, and at least the earliest analyses with profound results.

There are $\Pi_5$ormulas in the language of the Turing degrees, $\mathcal{D}$ with $\leq, \lor$ and $\land$ that define the relations $\mathbf{x}^{\prime\prime} \leq \mathbf{y}^{\prime\prime}, \mathbf{x}^{\prime\prime}=\mathbf{y}^{\prime\prime}$and so $\mathbf{x} \in \mathbf{L}_{2}(\rm{y}) = {\mathbf{x} \geq \mathbf{y}\, \vert \, \mathbf{x}^{\prime\prime}= \mathbf{y}^{\prime\prime}}$ in any jump ideal containing $\mathbf{0}^{\omega}$. There are also $\Sigma_6 $& $\Pi_6$ and $\Pi_8$ formulas that define the relations $\mathbb{w} = \mathbb{x}^{\prime\prime}$ and $\mathbb{w} = \mathbb{x}^{\prime}$, respectively, in any such ideal $\mathcal{I}$. In the language with just $\leq$ the quantifier complexity of each of these definitions increases by one. On the other hand, no $\Pi_2$ or $\Sigma_2$ formula in the language with just $\leq$ defines $\mathbf{L}_2$ or $\mathbf{x} \in \mathbf{L}_2(\mathbf{y})$. Our arguments and constructions are purely degree theoretic without any appeals to absoluteness considerations, set theoretic methods or coding of models of arithmetic. As a corollary, we see that every automorphism of $\mathcal{I}$ is fixed on every degree above $\mathbf{0}^{\prime\prime}$ and every relation on I that is invariant under double jump or joining with $\mathbf{0}^{\prime\prime}$ is definable over I if and only if it is definable in second order arithmetic with set quantification ranging over sets whose degrees are in $\mathcal{I}$ Similar direct coding arguments show that every hyperjump ideal I is rigid and biinterpretable with second order arithmetic with set quantification ranging over sets with hyperdegrees in $\mathcal{I}$. Analogous results hold for various coarser degree structures.