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If $V_{\kappa }\models \mathrm{ZFC}$, it has the standard natural numbers and standard finite proofs. Every arithmetical sentence, including a formal consistency statement, therefore has the same truth value in $V_{\kappa }$ and the universe.

Gentzen's consistency argument associates finite arithmetic derivations with effective ordinal notations below epsilon zero. An alleged contradiction admits an effective proof reduction preserving the contradictory end-sequent and strictly decreasing the assigned ordinal. Termination would therefore produce a normal contradiction, excluded by inspection of the logical rules and true atomic axioms. This explains why the relevant transfinite induction principle implies formal consistency statement. For a finite slow-well-ordering formulation one additionally needs the controlled-termination miniaturization theorem, rather than assuming that ordinary proof reductions have linearly growing size. The reduction program is discussed in math.stanford.edu/~feferman/papers/reductive.pdf .

The consistency implication needs more than the fact that the finite tree statement is unprovable in Peano arithmetic. Its mechanism is a concrete encoding of ordinal descent by bad finite-tree sequences, combined with the controlled-termination form of Gentzen ordinal reduction for arithmetic.

First define the natural-sum ordinal rank of a finite rooted tree. A leaf has rank zero, and a tree with immediate child subtrees $T_1,\dots ,T_m$ haswhere $\#$ is Hessenberg natural sum. By induction every such rank is below epsilon zero. Conversely, if $\alpha ={\sum }_{j=1}^r{\omega }^{{\alpha }_j}{c}_j$ is in Cantor normal form, represent it by a root with $c_j$ copies of the recursively constructed tree for ${\alpha }_j$. The leaf represents zero. Denote the resulting canonical tree by $\tau (\alpha )$ and its vertex count by $\Vert \alpha \Vert$.

We claim that a rooted-tree homeomorphic embedding $T⪯U$ implies $o(T)\le o(U)$. If the roots match, the source child subtrees embed into distinct target branches. By induction their ranks do not decrease; exponentiation and Hessenberg natural sum preserve these comparisons, and additional target branches only increase the sum. If the source root maps below the host root, its entire image lies in an immediate subtree $V$ of $U$. Induction gives $o(T)\le o(V)$, whileThis also holds when $o(V)=0$. Thus the claim follows by induction on the host tree. In particular,

Consequently Friedman's finite form of Kruskal's theorem supplies slow well-ordering below epsilon zero: for each $k$, descending ordinal sequences with $\Vert {\alpha }_i\Vert \le k+i$ have uniformly bounded length. Otherwise their canonical trees would be arbitrarily long admissible bad sequences. The notation system and its comparisons are primitive recursive, and there are only finitely many notations of bounded norm.

The proof-theoretic input is the following precise miniaturization lemma: for the standard effective Cantor normal form notation system below ${\epsilon }_0$, slow well-ordering with linear norm control implies the formal consistency statement for Peano arithmetic. The Friedman–Smith controlled-termination theorem, stated in its general ordinal-analysis form in Section 2.2 of formal.hknu.ac.kr/Publi/kruskalJKMS.pdf , supplies this implication over arithmetical comprehension theory ${\mathrm{ACA}}_0$, whose first-order part is conservative over Peano arithmetic. This is an established proof-theoretic lemma, separate from the elementary tree-rank calculation above.

Here is why that lemma is a consistency result. Gentzen ordinal reduction for arithmetic assigns an alleged arithmetic contradiction an effective ordinal below ${\epsilon }_0$. Reducing cuts and induction inferences preserves the contradictory end-sequent while strictly decreasing the assigned ordinal. A terminal normal derivation of a contradiction is excluded by inspection of the rules. The miniaturization argument converts this reduction/induction analysis into finite controlled descents: a contradictory proof would yield arbitrarily long descending notation sequences with a common initial allowance and linear norm control. Slow well-ordering excludes these. The control conversion is substantive; ordinary proof reductions must not simply be asserted to have linear-size growth. Gentzen's ordinal-reduction framework is discussed in math.stanford.edu/~feferman/papers/reductive.pdf .

Combining the coding with that precise termination lemma givesThus the finite combinatorial principle proves that no PA proof of a contradiction exists in the specified arithmetic metatheory. If consistent Peano arithmetic proved FFF, it would also prove its own formal consistency statement: the displayed implication is arithmetical and the first-order conservativity of ${\mathrm{ACA}}_0$ transfers it to PA. This contradicts the Gödel second incompleteness theorem. It is the consistency implication, rather than unprovability by itself, that explains the strength of the finite tree principle.

For a first-order theory $R$ extending ZFC, let $C_R$ be its set of formal consequences and let $\mathrm{Cons}$ denote the class of formal consistency statements $\mathrm{Con}(Q)$ for recursively axiomatized extensions $Q$ of ZFC. Using Gödel numbering to code proofs and theories, these objects and the following comparison are definable in the base theory ZFC.

The consistency-strength preorder isThus every consistency assertion provable in $T$ is also provable in $S$. Its strict part is

Let $\mathrm{IC}$ be the sentence asserting that a strongly inaccessible cardinal exists, and begin withDefine the iterated consistency progressionThe construction is effective, so every $T_n$ and $T_{\infty }$ is a recursively axiomatized first-order theory extending ZFC.

Because $T_{n+1}$ extends $T_n$, every theorem of $T_n$, including every formal consistency statement it proves, is a theorem of $T_{n+1}$; hence $T_n{\le }_{\mathrm{Cons}}{T}_{n+1}$. The theory $T_{n+1}$ proves $\mathrm{Con}(T_n)$ by construction, whereas a consistent $T_n$ cannot prove its own consistency by Gödel second incompleteness theorem. ThereforeLikewise $T_{\infty }$ extends every $T_n$ and contains $\mathrm{Con}(T_n)$ as an axiom already at stage $n+1$, while $T_n$ does not prove it. Consequentlyassuming the stated consistency hypotheses.

Primitive recursive arithmetic formalizes primitive recursive functions, their defining equations, and quantifier-free induction. It is a weak arithmetic base for coding finite syntax and checking effective proof reductions. Assertions that a combinatorial principle implies formal consistency statement must specify such a base or another arithmetic metatheory.

Use the unary Cantor normal form tree norm, counting one root and recursively counting all repeated exponent subterms. There are only finitely many notations of bounded norm. This finite principle bounds all descending sequences whose norms grow at most linearly with the position. Friedman's finite form of Kruskal's theorem implies it through the natural-sum ordinal rank of a finite rooted tree. The Friedman–Smith miniaturization theorem turns this controlled termination principle for standard epsilon-zero notations into the formal consistency statement for Peano arithmetic; the technical ordinal-analysis result is needed in addition to the tree coding. Section 2.2 of formal.hknu.ac.kr/Publi/kruskalJKMS.pdf records the general miniaturization and soundness theorem.