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A quantifier-free first-order formula is built from atomic formulas using logical connectives, with no quantifiers. An embedding of first-order structures preserves and reflects its truth, by mathematical induction on the first-order formula. Both positive and negated atomic information are needed for this conclusion in a relational language.

The quantifier rank of a first-order formula is the maximum nesting depth of quantifiers. Atomic formulas have rank zero; negation preserves rank, binary Boolean connectives take the maximum, and either quantifier increases rank by one. It is not the total number of quantifiers.

A finite collection of first-order formulas has a closed unbounded class of ordinals whose levels in a definable continuous hierarchy agree with its union for all parameters in the level. Close the collection under subformulas, bound the least witness levels for existential instances on each set-sized stage by Axiom schema of replacement, and iterate these bounds countably. Continuity supplies the closed stages, and induction on formulas proves agreement. The Tarski-Vaught test describes the same witness criterion. For a hierarchy of length an uncountable regular cardinal, bounds stay below that cardinal when every stage has smaller cardinality.

For transitive sets $M,X$ with $X\in M$, a single first-order formula $\Phi (X)$ recognises the nonzero limit levels $L_{\alpha }(X)$. Require finite relation closure for set-theoretic coding, transitivity of $X$, that every element belongs to a coded relative constructible stage, and that every represented stage index has a larger represented index. Correct history codes have downward-closed indices with no maximum, hence form a nonzero limit ordinal $\alpha$. The exhaustion assertion and transitivity then give $M={\bigcup }_{\xi <\alpha }{L}_{\xi }(X)$. Conversely every limit level has the required closure, correct histories cofinal in its index, and exhaustion. This formulation does not assume ZF for an arbitrary input $M$ and does not replace finite coding closure with an unjustified internal constructibility assertion.

The satisfaction relation says that the finite code $e$ of a first-order formula is true in a set-sized first-order structure $A$ at assignment $\vec{a}$. It is defined uniformly in set theory by recursion on the finite syntax tree: evaluate atomic relations, Boolean operations and quantifiers over the given domain. A code includes only finitely many subformulas, and their truth relations are sets. This does not define a truth predicate for the ambient universe, whose domain is not a set. Finite relation closure for set-theoretic coding gives an explicit closure criterion ensuring that these truth relations are computed correctly in a transitive set.

Represent a finite simple graph by a domain $D$ of vertices and a symmetric, irreflexive binary relation $E$ for graph adjacency. Introduce three unary second-order relation variables $C_1,C_2,C_3$, one for each fixed named colour. Define the following abbreviations for first-order formulas in this expanded first-order signature:Then the existential second-order logic sentenceasserts that every vertex receives exactly one of the three colours and adjacent vertices receive different colours. The first clause gives coverage and pairwise disjointness of the three subsets; the second makes each subset an independent set. Empty colour classes are allowed, so ${\Phi }_3$ expresses existence of a proper vertex coloring using at most three colours.

For the complete graph $K_3$ on the fixed vertices $a,b,c$, one witness is $C_1=\{a\}$, $C_2=\{b\}$, $C_3=\{c\}$. Each vertex satisfies exactly one predicate, and each of the three edges joins different predicates. Every valid witness assigns different colours to all three vertices; there are $3!=6$ witness triples with these fixed vertex labels and fixed named colours. We are counting assignments, without identifying them under colour permutations or graph isomorphisms. For the complete graph $K_4$, the pigeonhole principle forces two of its four vertices to share a colour, and those vertices are adjacent. Thus it has no witness triple. This calculation illustrates a complexity certificate without requiring the reader to reconstruct a machine computation.

Reproduce the diagram by running python ../second-order-coloring-witness.py from the repository's _media directory. The script checks all witness triples for these two complete graphs before drawing them.

Second-order logic extends first-order logic by allowing existential quantification and universal quantification over second-order relation variables, as well as over individual objects. In the relational presentation used here, an individual variable $x$ ranges over a domain $D$, while a second-order relation variable $R$ of arity $k$ denotes a subset of $D^k$. Unary second-order relation variables denote subsets of $D$; binary ones denote binary relations. Presentations with function variables can instead describe their graphs by second-order relation variables, with first-order formulas requiring a unique output for every input.

For a graph with vertices $D$ and fixed graph adjacency $E$, the first-order sentence $\forall x\exists yE(x,y)$ says that every vertex has a neighbour. It selects individual vertices while $E$ retains its given interpretation. By contrast,selects an entire subset $P$ and says that it is a nonempty independent set. Here $P$ is a unary second-order relation variable, rather than a symbol in the input first-order signature. This particular property also has a simple first-order sentence on simple graphs; the example illustrates the change in what can be selected, rather than proving a separation in expressive power.

The usefulness of second-order logic is that one can describe a whole proposed solution, such as a graph coloring, and then impose local constraints on it. Syntax alone does not specify which proposed relations are available: full semantics for second-order logic permits all external subsets of $D^k$, whereas Henkin semantics supplies designated ranges. The second-order encoding of three-colorability makes this distinction concrete.

A first-order formula is syntactically ${\Sigma }_1$ when it is an unbounded existential quantifier block followed by a bounded formula in set theory. Equivalence provable in a specified theory gives the corresponding modulo-theory notion. Existential witnesses show upward absoluteness between transitive models containing the parameters.

A Skolem expansion adjoins a Skolem function for every existential first-order formula. Iterating through the successively enlarged languages, and taking their union, gives a language with witness functions even for first-order formulas involving previously added functions. Starting from a language of size $\kappa$, the resulting language has size at most $\kappa +{\aleph }_0$. A hull closed under its functions is a Skolem hull and is elementary in the expanded structure by the Tarski-Vaught test.

For each existential first-order formula $\exists y\phi (y,\mathbf{x})$ in a structure, a Skolem function selects a witness when one exists, with an arbitrary default when none exists. An expansion containing these functions satisfiesA subset closed under all such functions is elementary in the original language by the Tarski-Vaught test. Simultaneously choosing these functions is performed in the ordinary classical metatheory with choice.

A first-order formula $A$ is stable when intuitionistic first-order logic proves $\neg \neg A\to A$. Double negations and logical falsity are stable. Logical conjunction of stable first-order formulas is stable; a logical implication is stable when its consequent is stable; and universal quantification preserves stability. For logical implication, from $\neg \neg (A\to B)$ and $A$, assuming $\neg B$ contradicts the first hypothesis, giving $\neg \neg B$ and then $B$. For universality, obtain $\neg \neg A(t)$ from $\neg \neg \forall xA(x)$ for arbitrary $t$, then use stability and generalize.

A structure embedding is an injective map respecting constants and operations and preserving and reflecting relations. Equivalently it preserves all atomic formulas and their negations. It identifies the source with a substructure of the target. An elementary embedding additionally preserves every first-order formula.

An $L$-substructure of a first-order structure $A\subseteq B$ has a nonempty domain containing all named constants and closed under the functions of $B$, and each relation of $A$ is the restriction of the corresponding relation of $B$. Consequently every quantifier-free formula with parameters in $A$ has the same truth value in $A$ and $B$. This is weaker than being an elementary substructure, which preserves all first-order formulas.

Expand an infinite first-order structure by Skolem functions, choose distinct elements $b_n$, and fix a nonprincipal ultrafilter $\mathcal{U}$ on $\mathbb{N}$. For each finite subset $F$ of the required index order, form the ultrapower over ${\mathbb{N}}^F$ using the ordered Fubini product of ultrafilters. Pullback along coordinate projections gives coherent elementary maps by the Łoś theorem. In their directed limit of elementary embeddings, the classes of the coordinate functions $\mathbf{n}\mapsto b_{n_i}$ are distinct and form an order-indiscernible sequence; every first-order formula on increasing coordinates has the same nested ultrafilter test. Their Skolem hull gives an Ehrenfeucht-Mostowski model. An index-order automorphism sends a term in the generators to the same term in the transported generators; indiscernibility makes this well defined. Uniqueness is in the specified Skolem expansion, not necessarily in its reduct. If generators must respect a definable infinite order, first produce an increasing sequence by an ultrapower of finite increasing chains.

Below any uncountable strongly inaccessible cardinal $\kappa$, the worldly cardinals contain a closed unbounded subset of $\kappa$ and hence have cardinality $\kappa$. The structure $V_{\kappa }$ models ZFC. Reflect each finite group from an enumeration of all first-order formulas to a closed unbounded subset of $\kappa$, using $|V_{\alpha }|<\kappa$ and regularity for the witness bounds. The countable intersection remains closed unbounded, so its ranks are elementary substructures of $V_{\kappa }$. Intersect with the closed unbounded subset of infinite cardinal numbers below $\kappa$. Uncountability is essential: the regular strong limit $\omega$ has no worldly cardinal below it.

A first-order formula is ZF-equivalent bounded if ZF proves it equivalent, with the same free variables, to a syntactically bounded formula in set theory. It need not itself have only bounded quantifiers. Between transitive models of ZF, use the bounded equivalent and the axioms in each model to prove set-theoretic absoluteness. The syntactic bounded-formula result requires only transitivity; the provable-equivalence extension additionally requires the theory used for the equivalence.