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In the paper we introduce a weak set theory $\mathsf{H}_{<\omega}$ . Aformalization of arithmetic on finite von Neumann ordinals gives an embeddingof arithmetical language into this theory. We show that $\mathsf{H}_{<\omega}$proves a natural arithmetization of its own Hilbert-style consistency. Unlikesome previous examples of theories proving their own consistency,$\mathsf{H}_{<\omega}$ appears to be sufficiently natural.  The theory $\mathsf{H}_{<\omega}$ is infinitely axiomatizable and provesexistence of all individual hereditarily finite sets, but at the same time allits finite subtheories have finite models. Therefore, our example avoids thestrong version of G\"odel second incompleteness theorem (due to Pudl\'ak) thatasserts that no consistent a theory interpreting Robinson's arithmetic$\mathsf{Q}$ proves its own consistency. To show that $\mathsf{H}_{<\omega}$proves its own consistency we establish a conservation result connecting Kalmarelementary arithmetic $\mathsf{EA}$ and $\mathsf{H}_{<\omega}$.  We also consider the version of $\mathsf{H}_{<\omega}$ over higher orderlogic denoted $\mathsf{H}^{\omega}_{<\omega}$. It has the same non-G\"odelianproperty as $\mathsf{H}_{<\omega}$ but happens to be more attractive from atechnical point of view. In particular, we show that$\mathsf{H}^{\omega}_{<\omega}$ proves a $\Pi_1$ sentence $\varphi$ of thepredicate-only version of arithmetical language iff $\mathsf{EA}$ proves that$\varphi$ holds on the superexponential cut.

A weak set theory that proves its own consistency