Because the physical universe exists, it is either true that it has done so for an infinite amount of time, or that the physical universe came into being a finite time ago. If one can demonstrate the absurdity of the physical universe existing for an infinite amount of time by showing the incoherence of an infinite temporal regress (that is, the model that the universe has existed for an infinite amount of time), then it becomes most reasonable to assume that the universe began to exist. It is important at this point to make a distinction between two varieties of infinites, actual infinites and theoretical infinites. Theoretical infinites are things which do not exist in reality and are merely conjectures of the mind, like possible worlds and other abstract objects. Mathematicians use these all the time in equations, as do theoretical physicists. Actual infinities, by contrast, are infinities that could exist in the physical world. Some examples of these include, actually infinite amounts of space in the universe, actually infinite amounts of elementary particles, or an actually infinite number of previous seconds. Many contemporary beginning-less models of the universe assume that actual infinites can exist. Some of these include the Caroll-Chen model, and the Hartle-Hawking model when followed to its necessarily implications. Therefore, by showing the absurdity of an actual infinite one can disprove these models while also showing the incoherence of an infinite temporal regress. If an infinite temporal regress is absurd, then the only other option is that the universe began to exist a finite time ago. If the universe came into being a finite time ago, then there is good reason to think that everything that comes into existence has a cause. This is because everything that one observes, namely, the physical universe and everything in it, appear to abide by this premise.

Though there have been entire books written about the existence of actual infinities, only the more straightforward examples will be examined. The first is a simple thought experiment. Imagine that some person is keeping track of every time both Neptune and Uranus complete a full rotation around the sun. Because Uranus completes (approximately) 2 full rotations around the sun for every 1 rotation that Neptune does, it seems reasonable to think that the rotations of Uranus will always equal approximately 2 times what those of Neptune are. We can generalize this expression mathematically to “x = 2y”, where the rotations of Uranus are “x” and the rotations of Neptune are “y”. If this person continues watching Uranus and Neptune rotate around the sun, they will quickly find that after 100 of Neptune’s rotations, Uranus will have 200. After 1,000 of Neptune’s rotations, Uranus will have 2,000. Even after 100,000,000,000 of Neptune’s rotations, Uranus will have 200,000,000,000. As Neptune and Uranus keep rotating, this gap will just keep getting bigger and bigger, because the orbits of Uranus will always be double what those of Neptune are.

Now, notice what happens when one imagines that they have been orbiting in this fashion for an actually infinite amount of time. If we assume this, it appears to follow from “x = 2y” that the gap between “x” and “y” will become astronomically large. In reality, the equation becomes subject to logical absurdities and self-contradictory answers on both a mathematical and conceptual level. This is because the variables “x” and “y” in “x = 2y” each become “∞” because both Neptune and Uranus have orbited the sun an infinite amount of times. If one defines “x” and “y” as “∞” in “x = 2y”, then one ends up with the equation “∞ = 2∞” which simplifies to “∞ = ∞” because infinity times anything is infinity. This means that if Neptune and Uranus orbited around the sun for an actually infinite amount of time, their orbits would be the exact same! This is absurd, because the difference between “x” and “y” got bigger as the numbers plugged in got bigger, yet when plugging in infinity itself (a number much higher than any finite number) the difference became the same! Because of this absurdity, one cannot assume that Uranus and Neptune could have orbited an actually infinite number of times. If we agree that this is indeed impossible, then “x = 2y” is true in every circumstance, and actual infinities cannot exist.

Another demonstration of this is the “Hilbert’s Hotel” illustration by mathematician Dave Hilbert, where one imagines every object in some set “Sx” as a floor in a hotel. If “Sx” is comprised of a finite number of objects, the hotel will function normally. If we assume that “Sx” is comprised of an actually infinite number of objects, then logical absurdities follow necessarily. Imagine a hotel that had a finite number of floors, in which every floor had an occupant on it. If someone came into the hotel and asked to rent a floor for the night, the hotel owner would have to turn them away because all the rooms are full. Now imagine instead that this hotel had an infinite number of floors, each with an occupant on it. If someone asked to rent a floor in this case, the hotel owner could simply ask everyone to move up one floor. Person 1 would move to floor 2, person 2 would move to floor 3, and so on. This would be akin to moving every object in “Sx” up one position. After this, manager could give the customer floor 1. In the same way, every occupant could move up 100 floors and the manager could accommodate 100 new people, even though the hotel was full previously! There are other things the manager could do. If the manager needed to accommodate an infinity of new guests, he could ask the occupants to move up to a floor which was double their previous number. After this, the infinity of previously occupied floors would be empty, and the infinity of new guests could use all the rooms which the original infinity moved out of! Mathematically, this would be akin to transforming every defined “Sx” into “S2x”, which is simply nonsensical when applied to real world examples like floors and hotels. Hilbert’s hotel is therefore another reason that actual infinities cannot exist.

Liked the article? Feel free to share on it social media to support my blog! You can share with one of the buttons below, or you can share by copying and pasting the article URL from the address bar into your Facebook statuses or Twitter tweets. Thanks!

Liked the article? Feel free to share on it social media to support my blog! You can share with one of the buttons below, or you can share by copying and pasting the article URL from the address bar into your Facebook statuses or Twitter tweets. Thanks!