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• Mayur Pawar

# The argument: Doesn't infinity exist in physics?

Infinity, the concept of something that is unlimited, endless, without bound. Since the time of the ancient Greeks, the nature of infinity was the subject of many discussions among philosophers. In the 17th century, with the introduction of the infinity symbol and the infinitesimal calculus, mathematicians began to work with infinite series and what some mathematicians (including l'Hôpital and Bernoulli) regarded as infinitely small quantities, but infinity continued to be associated with endless processes. There are many mathematical theories and proofs of describing infinity in mathematical along with finite values or sets.

Zeno's Paradox comes to mind. and this might be a good example of how math is an imperfect representation of the world. the distance from one point to another can be separated into pieces, each 1/2 the length of the last (.5D, .25D, .125D, etc...). using summation notation, you could technically add up all the parts to equal the distance, D. but also, theoretically, I think (as Zeno thought), one can infinitely divide D into an infinite number of pieces. thus, you would have to add an infinite number of increasingly infinitesimal distances. applying this to the real world, if one crossed one section of the distance at a time, then you could never get to the other point because you would cross and increasingly infinitesimal distance, thus crossing and the infinite number of distances. and the infinite number of real numbers exist, both outside of and inside of 1. so if one actually applied this to the real world, because the matter has volume and thus a dimension of length, there could theoretically be an infinite number of pieces of matter, increasingly infinitesimal. so you could make the conclusion, perhaps, that infinity exists in the matter because space around (and of) us is infinitely "deep" (is filled with an infinite number of infinitesimal particles of matter). which also brings to mind the oddity that since one does appear to go from one point and reach another, one can traverse infinity, just as one traverses infinity in math by counting from 1 to 2.

Another one, Mathematically, infinity must exist because there must exist the infinitesimal. For example, there is an infinite number of real numbers between 1 and 2 (technically the more appropriate term might be uncountable, but by its definition, still infinite).

One might argue that any practical number, such as the

√ 2

which represents the diagonal of a square, might trail off with all zeros eventually if one were to calculate out the decimal places far enough. But if that were true, it would be possible to represent

√ 2

as a ratio of integers, albeit very, very large integers. But here is rough proof that this is not possible. Suppose

√ 2

can be represented by a ratio of integers, M/N (where M and N might be very large, yet still finite).

Suppose that we reduce M/N such that M and N are the smallest units possible. Now there are 3 possibilities regarding M and N.

(a) M and N are both odd.

(b) M is odd and N is even.

(c) M is even and N is odd.

Note that M and N cannot both be even, because then we could simply divide both by 2, and retain the same ratio.

Given the above, we could say,

√ 2 = M /N

Squaring both sides gives us

2 = M^2/ N^2

which is,

2 N^2 = M^2

Here we can be confident that M is even because 2N is even: The the square of an odd integer gives an odd number. 2 times any integer is even thus 2N is even. Thus if M2 is even, M must be even. That also implies that N must be odd (because M and N cannot both be even).

Since M is even, let's make a substitution that 2L = M.

2 N^2 = 2 L^2

2 N^2 = 4 L^2

N^2 = 2 L^2

But this means that N must also be even since multiplying any integer by 2 creates an even number. making both M and N even, which contradicts the original proposition. Thus

√ 2

cannot be represented by a ratio of integers.

√ 2

belongs in some different class of numbers, that cannot be represented by a ratio of finite integers (namely irrational numbers).

One way to interpret this is

√ 2

has a truly infinite number of decimal places to represent it - they do not trail off to all zeros eventually. Another way to interpret this is that

√ 2

cannot be represented by a ratio of integers -- at least not finite integers -- at least not unless both M and N are infinite. So if one believes that squares exist, and if one believes that squares have diagonals (i.e. one believes that isosceles right triangles exist), arguably one must believe that infinity exists.

But talking about physical infinity, many scientists believe that physical infinity doesn't exist in real-world; even though the mathematical proofs are available. For example, a gravitational singularity which is found at the centre of every black hole. We know that there is no degenerate matter that can maintain equilibrium with the force of gravity because it is too strong. There is no known force in the universe that can stop this incredible warping of space. The result is an astronomically large quantity of matter being crushed into a point in space with zero volume and infinite density.

P = M / V

= M / 0

The problem is there can be no infinite. so there can be no infinite acceleration of gravity. if the singularity is the sum of all things that exist today (layman's interpretation) then the sum of all things today must equal the singularity. So if there is no infinite gravity today then there wasn't in the singularity. So in my most humble opinion, the singularity cannot be modelled by the math or there was no singularity because it is an impossible state.

But there are certain solid proofs of the existence of a singularity. In general relativity, a singularity is a place that objects or light rays can reach in a finite time where the curvature becomes infinite, or space-time stops being a manifold. Singularities can be found in all the black-hole spacetimes, the Schwarzschild metric, the Reissner–Nordström metric, the Kerr metric and the Kerr–Newman metric and in all cosmological solutions that do not have scalar field energy or a cosmological constant.

One cannot predict what might come "out" of a big-bang singularity in our past, or what happens to an observer that falls "in" to a black-hole singularity in the future, so they require modification of physical law. Before Penrose, it was conceivable that singularities only form in contrived situations. For example, in the collapse of a star to form a black hole, if the star is spinning and thus possesses some angular momentum, maybe the centrifugal force partly counteracts gravity and keeps a singularity from forming. The singularity theorems prove that this cannot happen and that a singularity will always form once an event horizon forms.

In the collapsing star example, since all matter and energy is a source of gravitational attraction in general relativity, the additional angular momentum only pulls the star together more strongly as it contracts: the part outside the event horizon eventually settles down to a Kerr black hole (see No-hair theorem). The part inside the event horizon necessarily has a singularity somewhere. The proof is somewhat constructive – it shows that the singularity can be found by following light-rays from a surface just inside the horizon. But the proof does not say what type of singularity occurs, spacelike, timelike, orbifold, jump discontinuity in the metric. It only guarantees that if one follows the time-like geodesics into the future, it is impossible for the boundary of the region they form to be generated by the null geodesics from the surface. This means that the boundary must either come from nowhere or the whole future ends at some finite extension.