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This paper initiates the reverse mathematics of social choice theory, studying Arrow's impossibility theorem and related results including Fishburn's possibility theorem and the Kirman--Sondermann theorem within the framework of reverse…
Ezra Getzler notes in the proof of the main theorem of "The semi-classical approximation for modular operads" that "A proof of the theorem could no doubt be given using [a combinatorial interpretation in terms of a sum over necklaces];…
We give a new simpler proof of a theorem of Jayne and Rogers.
The only widely accepted explanation for the various arrows of time that everywhere and at all epochs point in the same direction is the `past hypothesis': the Universe had a very special low-entropy initial state. We present the first…
We show how G\"odel's first incompleteness theorem has an analog in quantum theory. G\"odel's theorem implies endless opportunities for appending axioms to arithmetic, implicitly showing a role for an agent, namely an agent that asserts an…
Simple and shorter proofs of two Dirac-type theorems involving connectivity are presented.
We give a simple direct proof of Fermat's two squares theorem. Our argument uses no intricate notions or ideas; one might say that it is a proof by careful bookkeeping. As such, the proof may be particularly easy to comprehend by students…
We give a uniform proof of a significant part of the Cachazo-Douglas-Seiberg-Witten conjecture by using topology and geometry of infinite Grassmannians.
We prove a recent conjecture of Sean A. Irvine about a nonlinear recurrence, using mechanized guessing and verification. The theorem-prover Walnut plays a large role in the proof.
If no optimal propositional proof system exists, we (and independently Pudl\'ak) prove that ruling out length $t$ proofs of any unprovable sentence is hard. This mapping from unprovable to hard-to-prove sentences powerfully translates facts…
Here we prove some conjectures on the monotony of combinatorial sequences from the recent preprint of Zhi--Wei Sun.
A short, fairly self-contained proof is given of the Poincar\'e Conjecture. In the previous version there was an error on Page 8. This gap has now been filled.
A proof is given of Rosenthal's \(\ell_1\) theorem.
This somewhat unusual proof for the fact that the reals are uncountable, which is adapted from one of Bourbaki's proofs in "Fonctions d'une variable reelle", may be of some interest.
A detailed and rigorous analysis of G\"odel's proof of his first incompleteness theorem is presented. The purpose of this analysis is two-fold. The first is to reveal what G\"odel actually proved to provide a clear and solid foundation upon…
We present an alternative proof of Perron's theorem, which is probabilistic in nature. It rests on the representation of the Perron eigenvector as a functional of the trajectory of an auxiliary Markov chain.
The Taylor expansion is a widely used and powerful tool in all branches of Mathematics, both pure and applied. In Probability and Mathematical Statistics, however, a stronger version of Taylor's classical theorem is often needed, but only…
We give an infinite number of proofs of Pythagoras theorem.Some can be classified as `self-similar proofs'.
We give a formal Laurent series proof of Andrews's $q$-Dyson Conjecture, first proved by Zeilberger and Bressoud.
This proof of Godel's first incompleteness theorem doesn't require omega-consistency, nor does it refer to codes of negated sentences as in Rosser's. It begins from where Godel's usual proof ends, and stalks it till it ends proving it.