Related papers: The Essence of Intuitive Set Theory
We investigate an extension of ZFC set theory (in an extended language) that stipulates the existence of a proper class of indiscernibles over the universe. One of the main results of the paper shows that the purely set-theoretical…
The Fluctuation Theorem gives an analytical expression for the probability of observing second law violating dynamical fluctuations, in nonequilibrium systems. At equilibrium statistical mechanical fluctuations are known to be ensemble…
In what follows, essentially two things will be accomplished: Firstly, it will be proven that a version of the Arzel\`a--Ascoli theorem and the Fr\'echet--Kolmogorov theorem are equivalent to the axiom of countable choice for subsets of…
We define a function by refining Stern's diatomic sequence. We name it the {\it assembly function}. It is strictly increasing continuous. The first and the second main theorems are on an action to the function. The third theorem is on…
The concept of input-to-state stability (ISS) proposed in the late 1980s is one of the central notions in robust nonlinear control. ISS has become indispensable for various branches of nonlinear systems theory, such as robust stabilization…
Input-to-state stability (ISS) unifies global asymptotic stability with respect to variations of initial conditions with robustness with respect to external disturbances. First, we present Lyapunov characterizations for input-to-state…
Mathematicians still use Naive Set Theory when generating sets without danger of producing any contradiction. Therefore their working method can be considered as a consistent inference system with an experience of over 100 years. My…
We formulate and explore two basic axiomatic systems of typefree subjective probability. One of them explicates a notion of finitely additive probability. The other explicates a concept of infinitely additive probability. It is argued that…
This essay addresses the implications of integrated information theory (IIT) for free will. IIT is a theory of what consciousness is and what it takes to have it. According to IIT, the presence of consciousness is accounted for by a maximum…
Physics is introduced as a semantics of a formal set theory.
Choice functions constitute a simple, direct and very general mathematical framework for modelling choice under uncertainty. In particular, they are able to represent the set-valued choices that typically arise from applying decision rules…
We define the empiric stochastic stability of an invariant measure in the finite-time scenario, the classical definition of stochastic stability. We prove that an invariant measure of a continuous system is empirically stochastically stable…
We propose a unifying picture where the notion of generalized entropy is related to information theory by means of a group-theoretical approach. The group structure comes from the requirement that an entropy be well defined with respect to…
Following the definition of intuitionistic fuzzy n-norm [ 3 ], we have introduced the definition of intuitionistic fuzzy norm (in short IFN) over a linear space and there after a few results on intuitionistic fuzzy normed linear space and…
In set theory without the axiom of regularity, we consider a game in which two players choose in turn an element of a given set, an element of this element, etc.; a player wins if its adversary cannot make any next move. Sets that are…
Tarski gave a general semantics for deductive reasoning: a formula a may be deduced from a set A of formulas iff a holds in all models in which each of the elements of A holds. A more liberal semantics has been considered: a formula a may…
A concept of "guessability" is defined for sets of sequences of naturals. Eventually, these sets are thoroughly characterized. To do this, a nonstandard logic is developed, a logic containing symbols for the ellipsis as well as for…
Information theory is a mathematical theory of learning with deep connections with topics as diverse as artificial intelligence, statistical physics, and biological evolution. Many primers on information theory paint a broad picture with…
A question is proposed whether or not set theory is consistent.
The theory of optimal choice sets is a solution theory that has a long and well-established tradition in social choice and game theories. Some of important general solution concepts of choice problems when the set of best alternatives does…