Related papers: Adding a random real number and its effect on Mart…
Martin's Axiom for $\sigma$-centered partial orders implies that there is a cosmic space with non-coinciding dimensions.
This paper presents a new representation of natural numbers and discusses its consequences for computability and computational complexity. The paper argues that the introduction of the first Peano axiom in the traditional definition of…
We prove that a random choice rule satisfies Luce's Choice Axiom if and only if its support is a choice correspondence that satisfies the Weak Axiom of Revealed Preference, thus it consists of alternatives that are optimal according to some…
Study on pre- and postselected quantum system indicates that ``product rule'' and ``sum rule'' for elements of reality should be abandoned. We show that this so-called non-partial realism can refute arguments against hidden variables in a…
We present axioms for the real numbers by omitting the field axioms and then derive the field properties of the real numbers. We prove all our theorems constructively.
Randomization has been applied to Hamiltonian simulation in a number of ways to improve the accuracy or efficiency of product formulas. Deterministic product formulas are often constructed in a symmetric way to provide accuracy of even…
David Aspero asks on the possibility of having Forcing axiom FA_{aleph_2}(K), where K is the class of forcing notions preserving stationarity of subsets of aleph_1 and of aleph_2. We answer negatively, in fact we show the negative result…
In this article we introduce the operations of insertion and deletion working in a random-context and semi-conditional manner. We show that the conditional use of rules strictly increase the computational power. In the case of…
The hidden-variable question is whether or not various properties --- randomness or correlation, for example --- that are observed in the outcomes of an experiment can be explained via introduction of extra (hidden) variables which are…
In this paper, we introduce a partial order on rings with involution, which is a generalization of the partial order on the set of projections in a Rickart *-ring. We prove that a *-ring with the natural partial order form a sectionally…
We prove the following conjecture, raised by Aaronson and Ambainis in 2008: Let $f:\{-1,1\}^n \rightarrow [-1,1]$ be a multilinear polynomial of degree $d$. Then there exists a variable $x_i$ whose influence on $f$ is at least…
We present two ways in which the model $L({\mathbb R})$ is canonical assuming the existence of large cardinals. We show that the theory of this model, with {\em ordinal} parameters, cannot be changed by small forcing; we show further that a…
We generalize current known distribution results on Shanks--R\'enyi prime number races to the case where arbitrarily many residue classes are involved. Our method handles both the classical case that goes back to Chebyshev and function…
We introduce probability estimation, a broadly applicable framework to certify randomness in a finite sequence of measurement results without assuming that these results are independent and identically distributed. Probability estimation…
We study a large class of stochastic $p$-Laplace Allen-Cahn equations with singular potential. Under suitable assumptions on the (multiplicative-type) noise we first prove existence, uniqueness, and regularity of variational solutions.…
We present a coherent collection of finite mathematical theorems some of which can only be proved by going well beyond the usual axioms for mathematics. The proofs of these theorems illustrate in clear terms how one uses the well studied…
Let $Z_3$ denote $3^{rd}$ order arithmetic. Let Harrington's Principle, HP, denote the statement that there is a real $x$ such that every $x$--admissible ordinal is a cardinal in $L$. In this paper, assuming there exists a remarkable…
Martin's remarkable proof of $\mathbf{\Pi}^1_2$-determinacy from an iterable rank-into-rank embedding highlighted the connection between large cardinals and determinacy. In this paper, we isolate a large cardinal object called a measurable…
It is shown that any denumerable list L to which Cantor's diagonal method was applied is incomplete. However, this doesn't allow us to affirm that the cardinality of the real numbers of the interval [0, 1] is greater than the cardinality of…
Following Laczkovich we consider the partially ordered set $\iB_1(\RR)$ of Baire class 1 functions endowed with the pointwise order, and investigate the order types of the linearly ordered subsets. Answering a question of Komj\'ath and…