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    WLC is Wrong: The Argument for Actual Infinities

    Quick Preamble on the Kalaam Cosmological Argument

    As a part of an argument for the existence of God (known as the Kalaam Cosmological Argument), the Christian apologist WLC has argued that an actual infinity as realized in Nature is physically impossible. He argues that the the consequences of physical infinities are illustrated by Hilbert's Hotel. ODN now has many, many other threads on the Kalaam Cosmological argument (e.g. Clive's well-known thread against the KCA), and I encourage readers unfamiliar with the Kalaam Cosmological argument and Hilbert's Hotel to quickly peruse those threads before continuing on this one.

    The purpose of this thread is to conclusively demonstrate the falsity of WLC's assertion by constructing an explicit counter-example. This will be done by creating a physical system that:

    a.) Explicitly obeys the laws of physics.
    b.) Contains an actual infinity, and thus is a physical analog of Hilbert's Hotel.
    c.) In a finite amount of time, fully accommodates an additional "guests" (particle) into it's infinite hotel even though all of its rooms (occupied points) were previously full.

    Note that I am not arguing that such a system exists in the real world, the only point that I am making is that this system could exist in the physical world and if it did, it would break no physical laws and thus cause no physical contradiction. For simplicity, I'm choosing to work with Newtonian mechanics as my model for physical laws, but in my footnote at the bottom, I address why it's unlikely that there's any practical obstructions when incorporating more modern physical principles, such as quantum mechanics or relativity. (Note that I've previously explained that if you understand that quantum gravity might disallow the argument I'm giving here, then you've actually already ceded other premises of the KCA, but I leave this discussion for other threads. I'll ignore the full implications for the KCA, other than noting that if this argument is correct, then at least one central premise of the KCA must be false.)



    Challenge to support a claim.: If you disagree with me that the following system obeys the laws of physics, you need to explicitly explain to me which physical law has been broken in the following system. Note that "it violates common sense" is not a physical law.






    The Argument for the Possibility of Actual Infinities


    Let's begin with definitions for the Newtonian system:

    Def 1: A particle is an object with zero extent (it's a point), but has dynamics (it's a function of time, and so it can move around). At the first instant in time, the only things needed to specify the particle is its starting position, x0, and starting velocity, v0. I'm assuming no forces, so the time evolution of each particle is governed by the equation, x(t) = x0 + v0 t, for the position of the particle at all later times. This uniquely defines the point each particle sits at for all times t, such that t ≥ 0.

    Def 2: The momentum, p, of a particle is defined to be p = m v0, where m is the mass of a particle. The energy, E, of a particle is defined by E = 1/2 m v02.

    Ax 3: In Newtonian mechanics, the only criterion for the system to be physical is that in any finite volume with particles, all of the masses of each particle must be greater than zero, the total energy of the system must be finite, and the the total momentum must be finite. Thus if the system starts off with a well-defined set of initial positions and velocities, and in each finite region of space there's a finite amount of energy and momentum, then the system has not generated any physical contradiction according to classical mechanics. In other words, the system is entirely described with physical laws.

    So that's basic 100 level Newtonian mechanics.


    The Scenario: Hilbert's Hotel in a Finite Amount of Space

    Suppose this hypothetical universe starts out with an infinite number of particles. Again, classical mechanics has no objections to this (in fact, the consistency of infinite particles is used thoroughly in statistical mechanics to approximate systems of large particles), quite literally, the only question is "How many particles, where are they, and how fast are they moving?" As long as you can specify this data whilst keeping finite momentum and energy, there's no physical contradiction. So let's just pick their positions (Remember, I'm not doing this in real time, I'm saying that my initial time, this system happens to be in this configuration).

    Let's start all of them off in a finite interval, say [1,2). For the sake of the argument, let's take some specific length scale, for instance a meter, and just call it "1" in that length scale. To do this, start the particles off in the following positions (known as a geometric series):

    x1 = 1
    x2 = 1 + 1/2 = 1.5
    x3 = 1 + 1/2 + 1/4 = 1.75
    .
    .
    .
    xn = xn-1 + (1/n)2 = 2[1 - (1/2)n ]

    Here's a graphical representation of the positions of the first three particles:

    Click image for larger version. 

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    It's easy to verify that all particles will be at a point greater than or equal to 1 but always less than 2 (i.e. no particle sits at point "2", just arbitrarily close to it, on the interval known as [1,2) by mathematicians). However, each particle is also never lying on top of one another; for each n, each particle is at a distinct point. This is an infinite collection of points where each particle is located, but in a 1 unit distance interval.



    Causing the Hilbert "Paradox" in a Finite Amount of Time


    Now for the alleged paradox. It's been incorrectly conjectured many times now that it would take an infinite amount of time to accommodate the Hilbert Hotel, and this might be resolution to the paradox; it's also been stated that you could never physically accommodate another particle in Hilbert's Hotel (the infinite points on the interval [1,2) which occupy a particle). Let's prove that neither of these assertions are correct. Suppose I give you another particle, let's say at point 0. Let's show that it is physically possible to accommodate the new particle inside of "infinite hotel" in a finite amount of time, and saliently with finite total energy and finite total momentum on the interval. In other words, where no physical contradiction has occurred Specifically, we'll start off with an infinite number of particles occupying an infinite number of points inside the finite interval [1,2) with a single particle lying outside that interval, and then evolve the system forward a unit of time until there's now be "infinity + 1" particles inside of the finite interval, but also with all of the same previous states occupied.

    We also control the initial velocities of these particles, so after a given interval of time, we can uniquely pick each particles next position and that specifies the initial velocity. It's rather simple to solve the set of evolution equations xn(t) = xn vn. Thus, for our setup, we want each particle to move forward by xn -> xn+1 after the allotted finite period of time. So, we require that each particle moves forward by an amount xn+1 - xn = 1/n^2. This means that each velocity needs to be:

    vn = (1/n)2

    So after 1 second, each particle will have moved forward to the next position, particle n moves to particle (n+1)'s spot. This completely defines the system, and the total energy & momentum can trivially be evaluated (let's assume they all have the same mass, m) as

    E = Σ0 1/2 m vn2 = 2/3 m [unit distance]^2/[unit time]^2,

    which is finite (I used the above formula for computing infinite geometric series, which can also be found in the Wikipedia article). The particle at x=0 also has a velocity (which is just 1 [unit distance]/[unit time]), so add 1/2m to get the total energy, including the particle that starts off outside of the "hotel." The total momentum can be more easily calculated with the same formula, and is simply 3 m [unit length]/[unit time].

    So at the instant t=1, the new particle has moved to point 1, particle 1 have moved to point 1.5, particle 2 has moved to 1.75, and so on. Each particle has moved to the next previously occupied point in our set. This means that Hilbert's so-called "paradox" has been realized in a finite amount of time and for a completely physically sensible system.



    In summary: Hilbert's Hotel can be physically realized and everything can happen in a finite amount of time and in a physically sensible manner. Start at t=0, there's an infinite number of points with particles occupying specific points in the interval [1,2), plus one particle outside at x=0. Evolve forward a finite amount of time, i.e. to time t=1, and now an "infinity+1" of particles are sitting inside of the interval [1,2) and occupying the exact same points as were previously occupied. Each particle still has it's own unique point in space, there's no overlaps, and the newest particle was added to the bunch in a finite amount of time, using a finite amount of energy and momentum, not assuming "successive infinities," and the "absurdities" regarding the occupations are as real here as in Hilbert's Hotel. Nevertheless, this is entirely physically possible in classical mechanics and is completely allowed by classical laws of physics. The conclusion? The so-called "paradoxes" that WLC asserts invalidate infinity are essential features, not irresolvable bugs. This is simply how infinite sets work. Your intuition simply has to be modified for how you think about infinite sets.






    N.B. I am also quite certain that I can setup the same system for an infinite number of non-interacting quantum particles (Take it in QFT, where this is easiest to describe and the non-relativistic limit requires it to be valid in QM, too), and where the probability of the transition to including the "infinity + 1" particle should be finite. I may work out this example to make the point really concrete, but it would appear that at first glance, there are zero obstructions to causing this in otherwise completely sensible physical systems. It's also worth pointing out that the Fock space of a free QFT explicitly includes the case of an infinite number of particles (as previously mentioned in other WLC/KCA threads), so literally every assumption there would be unimpeachable (i.e. definitely physically allowed) from the standpoint of QFT, the currently most complete description of Nature until we develop quantum gravity.

    Also, I have notably neglected gravity, but this shouldn't impose any problems. In principle I could add gravity to my evolution equations, only now I'd need to add as a postulate that there's finite mass for every finite region and then setting up the equations to end up at the nth -> (n+1)th positions, for all n. It seems laborious, but not impossible.
    Last edited by GoldPhoenix; July 7th, 2015 at 01:09 PM.
    "Those who can make you believe absurdities, can make you commit atrocities." --Voltaire

 

 

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