Our account does not rob the mathematicians of their science, by disproving the actual existence of the infinite in the direction of increase, in the sense of the untraversable.In point of fact they do not need the infinite and do not use it.…
For all finite sets this gives us the usual definition of "the same size".What does it tell us about the size of infinite sets?…
This is also the difference between the arrival rate and the effective arrival rate, where the arrival rate roughly corresponds to the rate at which customers arrive at the store, whereas the effective arrival rate corresponds to the rate at which customers enter the store.However, in a system with an infinite size and no loss, the two are equal.…
Matter cannot quite reach the speed of light, as this would require an infinite amount of energy.In relativity physics, the concept of rapidity replaces the classical idea of speed.…
Hence this definition has the advantage of enabling a function, defined as a set of ordered pairs, to have a type only 1 higher than the type of its arguments.This definition works only if the set of natural numbers is infinite.This is the case in NF, but not in type theory or in NFU.…
One of the few general approaches is through the Hasse principle.Infinite descent is the traditional method, and has been pushed a long way.The depth of the study of general Diophantine equations is shown by the characterisation of Diophantine sets as equivalently described as recursively enumerable.…
In case m = n, the value of the product is the same as that of the single factor x m, If m > n, the product is the empty product, with the value 1.One may also consider products of infinitely many terms; these are called infinite products.…
In particular, no first-order theory with an infinite model can be categorical.Thus there is no first-order theory whose only model has the set of natural numbers as its domain, or whose only model has the set of real numbers as its domain.…
Phase space is a continuous space containing an infinite number of distinct physical states within any small region.In order to connect the probability density in phase space to a probability distribution over microstates, it is necessary to somehow partition the phase space into blocks that are distributed representing the different states of the system in a fair way.…
Reaching the next quarter, he must then cover the next eighth of the distance, then the next sixteenth, and so on.There are thus an infinite number of steps that must first be accomplished before he could reach the bus.Expressed this way, the dichotomy paradox is very much analogous to that of Achilles and the tortoise.…
Once they are filled (by the hotel's redistributed occupants), the remaining empty rooms form the shape of a pyramid exactly identical to the original shape.Thus, the process can be repeated for each infinite set.Doing this one at a time for each coach would require an infinite number of steps, but by using the prior formulas, a guest can determine what his room "will be" once his coach has been reached in the process, and can simply go there immediately.…
The soul identifies itself with each successive form, becoming thus tied to illusion.During this evolution of forms thinking also increases, until in human form thinking becomes infinite.Although in human form the soul is capable of conscious divinity, all the impressions that it has gathered during evolution are illusory ones, creating a barrier for the soul to know itself.…
In this case the corresponding question is: does an infinite open cluster exist?That is, is there a path of connected points of infinite length "through" the network?By Kolmogorov's zero-one law, for any given p, the probability that an infinite cluster exists is either zero or one.…
It is not meaningful to assign a value to the king relative to the other pieces, as it cannot be captured or exchanged.In this sense, its value could be considered infinite.…
If both input and output are real numbers, this ordered pair can be viewed as the Cartesian coordinates of a point on the graph of the function.But no picture can exactly define every point in an infinite set.…
In transfinite arithmetic, inverse operations of subtraction and division with infinite quantities are prohibited because they lead to contradictions.If actual infinites could exist in reality, there would be no way to stop these operations from being performed.…
The wide belt of the universe has no need for finger-rings.I am one with the infinite and need no other assurance.…
One problem with this is that the number of policies can be extremely large, or even infinite.Another is that variance of the returns might be large, in which case a large number of samples will be required to accurately estimate the return of each policy.…
An extension of the previously described hidden Markov models with Dirichlet priors uses a Dirichlet process in place of a Dirichlet distribution.This type of model allows for an unknown and potentially infinite number of states.It is common to use a two-level Dirichlet process, similar to the previously described model with two levels of Dirichlet distributions.…
The logarithm hits every complex number except possibly one number, which implies that the first function will hit any value other than 0 an infinite number of times.Similarly, an entire function that does not hit a particular value will hit every other value an infinite number of times.…