Related papers: Cantor's Non-Equinumerosity Theorems, Inductively
This note presents a unified theorem of the alternative that explicitly allows for any combination of equality, componentwise inequality, weak dominance, strict dominance, and nonnegativity relations. The theorem nests 60 special cases,…
The notion of a tensor captures three great ideas: equivariance, multilinearity, separability. But trying to be three things at once makes the notion difficult to understand. We will explain tensors in an accessible and elementary way…
We consider the logic MSO+U, which is monadic second-order logic extended with the unbounding quantifier. The unbounding quantifier is used to say that a property of finite sets holds for sets of arbitrarily large size. We prove that the…
The theory of addition in the domains of natural (N), integer (Z), rational (Q), real (R) and complex (C) numbers is decidable, so is the theory of multiplication in all those domains. By Godel's Incompleteness Theorem the theory of…
We prove a version of $Z$-set unknotting theorem for uncountable products of real numbers.
Given any polynomial $p$ in $C[X]$, we show that the set of irreducible matrices satisfying $p(A)=0$ is finite. In the specific case $p(X)=X^2-nX$, we count the number of irreducible matrices in this set and analyze the arising sequences…
Godel's First Incompleteness Theorem is generalized to definable theories, which are not necessarily recursively enumerable, by using a couple of syntactic-semantic notions, one is the consistency of a theory with the set of all true…
If $F$ and $G$ are iterated function systems, then any infinite word $W$ in the symbols $F$ and $G$ induces a limit set. It is natural to ask whether this Cantor set can also be realized as the limit set of a single $C^{1 + \alpha}$…
Starting with Zhang's theorem on the infinitude of prime doubles, we give an inductive argument that there exists an infinite number of prime $k$-tuples for at least one admissible set $\mathcal{H}_k=\{h_1,\ldots,h_k\}$ for each $k$.
The rather unintuitive nature of quantum theory has led numerous people to develop sets of (physically motivated) principles that can be used to derive quantum mechanics from the ground up, in order to better understand where the structure…
We prove that any finite set of real numbers can be split into two parts, one part being highly non-additive and the other highly non-multiplicative.
Quantum theory combines density matrices, Born probabilities, tensor-product composites, positive-operator-valued measures (POVMs), and quantum channels. In a finite-dimensional causal operational theory, we prove that two postulates…
We give examples of calculi that extend Gentzen's sequent calculus LK by unsound quantifier inferences in such a way that (i) derivations lead only to true sequents, and (ii) proofs therein are non-elementarily shorter than LK-proofs.
This paper discusses limitations of reflexive and diagonal arguments as methods of proof of limitative theorems (e.g. G\"odel's theorem on Entscheidungsproblem, Turing's halting problem or Chaitin-G\"odel's theorem). The fact, that a formal…
The Kochen-Specker theorem shows the impossibility for a hidden variable theory to consistently assign values to certain (finite) sets of observables in a way that is non-contextual and consistent with quantum mechanics. If we require…
According to mathematical constructivism, a mathematical object can exist only if there is a way to compute (or "construct") it; so, what is non-computable is non-constructive. In the example of the quantum model, whose Fock states are…
Let $L$ be a countable language. We characterize, in terms of definable closure, those countable theories $\Sigma$ of $\mathcal{L}_{\omega_1, \omega}(L)$ for which there exists an $S_\infty$-invariant probability measure on the collection…
Answering a question of Kaye, we show that the compositional truth theory with a full collection scheme is conservative over Peano Arithmetic. We demonstrate it by showing that countable models of compositional truth which satisfy the…
This work presents theorems which state (i) Z is a proper subset for any bijection f between A and Z, where Z is contained in P(A), A is a non-finite set and |Z|=|A|, and (ii) being Z a proper subset of P(A) nothing affirms or denies that…
We present a new explanation for a quantum eraser. Mathematical description of the traditional explanation needs quantum-superposition states. However, the phenomenon can be explained without quantum-superposition states by introducing…