Object binding, the brain’s ability to bind the many features that collectively represent an object into a coherent whole, is central to human cognition. It groups low-level perceptual features into high‑level object representations, stores those objects efficiently and compositionally in memory, and supports human reasoning about individual object instances. While prior work often imposes object-centric attention (e.g., Slot Attention) explicitly to probe these benefits, it remains unclear whether this ability naturally emerges in pre-trained Vision Transformers (ViTs). Intuitively, they could: recognizing which patches belong to the same object should be useful for downstream prediction and thus guide attention. Motivated by the quadratic nature of self-attention, we hypothesize that ViTs represent whether two patches belong to the same object, a property we term *IsSameObject*. We decode *IsSameObject* from patch embeddings across ViT layers using a quadratic similarity probe, which reaches over 90\% accuracy. Crucially, this object-binding capability emerges reliably in DINO, CLIP, and ImageNet-supervised ViTs, but is markedly weaker in MAE, suggesting that binding is not a trivial architectural artifact, but an ability acquired through specific pretraining objectives. We further discover that *IsSameObject* is encoded in a low-dimensional subspace on top of object features, and that this signal actively guides attention. Ablating *IsSameObject* from model activations degrades downstream performance and works against the learning objective, implying that emergent object binding naturally serves the pretraining objective. Our findings challenge the view that ViTs lack object binding and highlight how symbolic knowledge of “which parts belong together” emerges naturally in a connectionist system.

Abstract We unify Gödel’s first incompleteness theorem (1931), Tarski’s undefinability theorem (1933), Gödel-Carnap’s diagonal lemma (1934) and Rosser’s (strengthening of Gödel’s first) incompleteness theorem (1936), whose proofs resemble much and use almost the same technique.

Abstract There is a long-standing debate in the logico-philosophical community as to if/why the Gödelian sentences of a consistent and sufficiently strong theory are true ($\gamma $ is a Gödelian sentence of $T$ when $\gamma $ is equivalent to the $T$-unprovability of $\gamma $ inside $T$). The prevalent argument seems to be something like the following: since every one of the Gödelian sentences of such a theory is equivalent to the consistency statement of the theory, even provably so inside the theory, the truth of those sentences follows from the consistency of the theory in question; so, Gödelian sentences of consistent and sufficiently strong theories are true. In this paper, we critically examine this argument and present necessary and sufficient conditions for the truth of Gödelian sentences (and Rosserian sentences) of consistent and sufficiently strong arithmetical theories.

The multiplicative theory of a set of numbers (which could be natural, integer, rational, real or complex numbers) is the first-order theory of the structure of that set with (solely) the multiplication operation (that set is taken to be multiplicative, i.e., closed under multiplication). In this paper we study the multiplicative theories of the complex, real and (positive) rational numbers. These theories (and also the multiplicative theories of natural and integer numbers) are known to be decidable (i.e., there exists an algorithm that decides whether a given sentence is derivable form the theory); here we present explicit axiomatizations for them and show that they are not finitely axiomatizable. For each of these sets (of complex, real and [positive] rational numbers) a language, including the multiplication operation, is introduced in a way that it allows quantifier elimination (for the theory of that set).

Gödel–Rosser's Incompleteness Theorem is generalized by showing Πn+1-incompleteness of any Σn+1-definable extension of Peano Arithmetic which is either Σn-sound or n-consistent. The optimality of this result is proved by presenting a complete, Σn+1-definable, Σn−1-sound, and (n−1)-consistent theory for any n>0⁠. Though the proof of the incompleteness theorem for Σn+1-definable theories using the Σn-soundness assumption is constructive, it is shown that there is no constructive proof for the Incompleteness Theorem for Σn+1-definable theories using the n-consistency assumption, when n>2⁠.

Abstract Gödel's second incompleteness theorem is proved for Herbrand consistency of some arithmetical theories with bounded induction, by using a technique of logarithmic shrinking the witnesses of bounded formulas, due to Z. Adamowicz [Herbrand consistency and bounded arithmetic, Fundamenta Mathematicae, vol. 171 (2002), pp. 279–292]. In that paper, it was shown that one cannot always shrink the witness of a bounded formula logarithmically, but in the presence of Herbrand consistency, for theories IΔ0 + Ωm with m ≥ 2, any witness for any bounded formula can be shortened logarithmically. This immediately implies the unprovability of Herbrand consistency of a theory T ⊇ IΔ0 + Ω2 in T itself. In this paper, the above results are generalized for Δ0 + Ω1. Also after tailoring the definition of Herbrand consistency for IΔ0 we prove the corresponding theorems for IΔ0. Thus the Herbrand version of Gödel's second incompleteness theorem follows for the theories IΔ0 + Ω1 and IΔ0.

The problem of Π1–separating the hierarchy of bounded arithmetic has been studied in the article. It is shown that the notion of Herbrand consistency, in its full generality, cannot Π1–separate the theory IΔ0+⋀jΩj from IΔ0; though it can Π1–separate IΔ0+Exp from IΔ0. Namely, we show the unprovability of the Herbrand consistency of IΔ0 in the theory IΔ0+⋀jΩj. This partially extends a result of L. A. Kołodziejczyk who showed that for a finite fragment S⊆IΔ0, the Herbrand consistency of S+Ω1 is not provable in IΔ0+⋀jΩj.

Panoramic images are a new type of visual data that provide many new possibilities as compared to the classic planar images. Tele-presence and virtual navigation are examples of such interesting applications. In order to enable smooth navigation across different view points, we propose a method for aligning cubic-panorama image datasets by using the concept of epipolar geometry. Unlike existing methods, which are limited to a pair of panoramas, our approach is accurate and applicable on datasets with large number of panoramic images. Also for the first time, a measure is introduced in order to enable objective evaluations.