∞

Think of any number between 1 and 10. If you guessed the same number that I was thinking of, I would be taken aback because I thought you have some magic power that allows you to read my mind rather than just being a pure guess out of thin air. Okay, let’s try another one. What’s the biggest number you can think of? 10, 100, 1,000, 50,000, 100,000, 1 million, 5 million, 1 billion, 1 trillion, 1 quadrillion, 1 quintillion, 1 googol, 1 googolplex, 1 googolplex factorial, 1 googolplex to the power of 1 googolplex to power of 1 googolplex…. Come on! You can go much higher than those numbers. What?! You have given up already? No matter how big your number is, you can always go 1 higher. By then we would have run out of words to describe these massive numbers and space on our calculators to fit in these numbers whilst we solve equations full with unknown variables. Normal scientific calculators can display 15 digits on the screen and 99 on scientific notation, hence the biggest number those calculators can handle is 99.999999999999 x 10^99. CAS calculators can handle much bigger numbers but they barely make it along the number line. If you had a piece of paper long enough to traverse the equatorial circumference of Earth (40,070 km, 24,900 miles) with each digit about 0.5cm wide to be legibly written from left to right, that number would be 8,014,000,000 (8 billion and 14 million) digits long. A googol is 10 to the power of 100 (10^100) and a googolplex is 10 to the power of a googol (10^(10^100)). Those 2 large numbers were named by Edward Kasner’s nephew, Milton Sirotta in Kasner and Newton’s 1940 book, Mathematics and the Imagination. The highest ever numerical value printed on a banknote was a Hungarian note for 1 sextillion pengõ (10^21, 1 milliard bilpengõ) in 1946. Behind it is the 100 trillion Zimbabwean dollar note printed in 2009. Depending on where you live, e.g. UK, Australia, USA and France, you will habituate using either short scale or long scale naming of large numbers. For example, 10^3003: In short scale Americans, Canadians and modern British call it a Millinillion. In traditional British long scale, it’s called a Thousand Quingentillion. Traditional Europeans refer to the long Peletier scale, calling it a Quingentillard. A discernible pattern emerges for every 6 zeros added to a 1,000 (10^3) in the traditional British and European long scales. Any number larger than a millinillion would have to resort to scientific notation when pronounced. E.g. 10^1382369 = 10 to the power of one million 3 hundred and 82 thousand 3 hundred and 69. To see what names are allocated for other large numbers click on the link below:
https://en.wikipedia.org/wiki/Names_of_large_numbers

The fact is our number line is endless in both the positive and negative directions. What about infinity? Unfortunately, infinity is NOT a rational number. It’s a kind of number to represent something that is endless. Countable Infinity refers to the infinite number of natural numbers going in a positive direction. Natural numbers include 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, … and so on, meaning you can verbally count them one by one. The world record for the highest number ever counted to by any person is 1 million by American Jeremy Harper in September 14, 2007 at 7.25pm EST. If every number he pronounced from 1 to 1 million lasted 1 second, just note that 1 million seconds is about 11.574 days. He recorded a live stream video of himself counting up to 1 million that went for 89 consecutive days. He got regular sleep, adequate food and water but even so, he had the courage and dedication to remain in his apartment and continue counting uninterrupted for 16 hours per day. Uncountable infinity includes every real number between the natural numbers. First number is obviously 0, then what’s the next number? 0.000000000000…1. There are infinitely many zeros in the decimals before we eventually come to a 1 at the end. That’s why it’s called uncountable. You will never finish saying the next number after 0 because you can always add another 0 to the following decimal.
Let’s divide all numbers into even and odd categories. Even numbers end in 2, 4, 6, 8 and 0 meaning when halved will give a whole integer, whilst odd numbers end in 1, 3, 5, 7 and 9 meaning when halved will give a number with a fraction or decimal. It’s obvious there are the same number of even numbers as well as odd numbers. Would you believe that the set of even or odd numbers has the same amount as the set of the natural numbers? When you divide 2 certain numbers to give out a rational number with a repeating decimal, we mark that decimal with a solitary dot on top of the decimal. e.g. 1/9 = 0.111111… = 0.1*

What about numbers with a non-repetitive sequence of decimals like π (pi) and e (Euler’s number)? They’re called transcendental numbers. They’re defined as real or complex numbers that aren’t algebraic, which excludes roots of non-zero polynomial equations with integers or rational coefficients. I’ve seen many talent shows featuring contestants given the challenge to recite the first 100 decimals of π. Can you recite even more decimals of π? Challenge yourself when you have the time at home. Almost all real and complex numbers are transcendental, since the algebraic numbers are countable while sets of real and complex numbers are both uncountable. All real transcendental numbers are irrational, since all rational numbers are algebraic. However, not all irrational numbers are transcendental. E.g. Square root of 2 gives a rational number (1.41…) but it’s not a transcendental number because it’s a solution of the polynomial equation x^2 — 2 = 0. Another example is the golden ratio, φ, since it’s a solution of the polynomial equation x^2 — x — 1 = 0.
The word “transcendental” comes from the Latin transcendĕre, meaning ‘to climb over or beyond, surmount’. It was first used in Leibniz’s 1682 paper in which he proved that sin (x) is not an algebraic function of x. In 1768, Johann Heinrich Lambert proved that Euler’s number (e) and π are irrational. In 1844, Joseph Liouville first proved the existence of transcendental numbers and invented his own constant (number) in 1851, known famously as the Liouville constant which looks like this:

The nth digit after the decimal point is 1 if n is equal to k! (! = factorial e.g. 3! = 3x2x1) for some ‘k’ values and 0 otherwise.

The first ever number proven to be transcendental was ‘e’, by Charles Hermite in 1873. If you want to see and learn the complicated proof simplified by David Hilbert, click on the link below and scroll down the web page until you see the subtitle “Sketch of a proof that e is transcendental”.
https://en.wikipedia.org/wiki/Transcendental_number
In 1874, Georg Cantor published his proof of algebraic numbers being countable and real numbers being uncountable. In 1878, he published another proof that there are as many transcendental numbers as there are real numbers. In 1882, Ferdinand von Lindemann published a proof that π is transcendental. In summary, he first showed that e^a is transcendental when a is algebraic and not zero. Then, since e^(i*π) = —1 is algebraic, i*π and therefore π must be transcendental. Karl Weierstrass generalised this approach to to the Lindemann-Weierstrass Theorem. The transcendence of π allowed the proof of the impossibility of several ancient geometric constructions involving the compass and straightedge, famously including, squaring the circle. In 1900, David Hilbert posed an influential question about transcendental numbers, known as Hilbert’s Seventh Problem:
- If a is an algebraic number, that isn’t 0 or 1, and b is an irrational algebraic number, is a^b necessarily transcendental?
In 1934, the Gelfond-Schneider Theorem answered in the affirmative. This work was extended by Alan Baker in the 1960s on lower bounds for linear forms in any number of logarithms (of algebraic numbers). Obviously the mathematical notation is too complex for my keyboard and brain to handle since I’m not a qualified mathematician. If you are curious about the the Lindemann-Weierstrass and Gelfond-Schneider Theorems, check out the links below:
https://en.wikipedia.org/wiki/Lindemann–Weierstrass_theorem
https://en.wikipedia.org/wiki/Gelfond–Schneider_theorem

According to mathematics, infinity is a concept that describes something boundless or larger than any other natural number. It is often treated as a number (i.e. Counting or measuring things: “There are an infinite number of terms to describe this phenomenon,”) but it is NOT the same sort of number as either a natural or real number. The earliest recorded idea of infinity originated from Anaximander, a pre-Socratic Greek philosopher who lived in Miletus. He used the word “apeiron” meaning infinite or limitless. However, the earliest accounts of mathematical infinity came from Zeno of Elea (born circa 490 BCE), a pre-Socratic Greek philosopher from Southern Italy and a member of the Eleatic School founded by Parmenides. The Jain mathematical text Surya Prajnapti (circa 4th - 3rd century BCE) classifies all numbers into 3 sets: Enumerable, Innumerable and Infinite. Each of these sets was further subdivided into 3 orders:
- Enumerable: Lowest, Intermediate, Highest
- Innumerable: Nearly Innumerable, Truly Innumerable, Inumerably Innumerable
- Infinite: Nearly Infinite, Truly Infinite, Infinitely Infinite
In the 17th century, European mathematicians began using infinite numbers and expressions systematically. In 1655, John Wallis was known to have first used the notation ∞ for such a number in his work De Sectionibus Conicis and exploited it by dividing the region into infinitesimal strips of width on the order of 1/∞. But in his other work Arithmetica Infinitorum, Wallis also indicates infinite series, infinite products and infinite continued fractions by writing down a few terms or factors and then appending “&c”. e.g. “1, 6, 12, 18, 24, &c”. The infinity symbol ∞
(also called a lemniscate) is used not only in mathematics but also in modern mysticism and literary symbology.

Calculus
Gottfried Leibniz, one of the co-inventors of infinitesimal calculus, speculated that both infinitesimals and infinite quantities were ideal entities, not of the same nature as appreciable quantities, but enjoying the same properties in accordance with the Law of Continuity.
In real analysis, infinity is used to denote an unbounded limit. x —>∞
means the value of x increases without bound, and x —> -∞
means the value of x is decreasing without bound. If f(t) > 0 for every value of t, then:

Infinity is also used to describe an infinite series:

Infinity is not only used to define a limit but as a value in the extended real number system. Points labeled +∞& -∞ can be added to the topological space, producing the two-point compactification of the real numbers.

In complex analysis, ∞ denotes an unsigned infinite limit. x —> ∞ means that the magnitude of x, |x|, grows beyond any assigned value. A point labelled ∞ can be added to the complex plane as a topological space giving the one-point compactification of the complex plane. The resulting space is a one-dimensional complex manifold, or Riemann Surface, called the extended complex pane or the Riemann Sphere.

Riemann Sphere: Using stereographic projections, the complex plane can be wrapped onto a sphere, with the top point of the sphere corresponding to ∞.

In non-standard analysis, the original formulation of infinitesimal calculus by Isaac Newton and Gottfreid Leibniz used infinitesimal quantities. Infinitesimals are invertible, and their inverses are infinite numbers. The infinities are part of a hyper-real field hence there is no equivalence between them as with the the Cantorian transfinites. e.g. If H is an infinite number, then H+H = 2H & H+1 are distinct infinite numbers. This approach to non-standard calculus was fully developed in Keisler in 1986.

This hyper-real number line displays Infinitesimals (ε) and infinities (ω), [1/ε = ω/1]

https://www.youtube.com/watch?v=SrU9YDoXE88
https://en.wikipedia.org/wiki/Cardinal_number
In set theory, a different form of infinity features ordinal and cardinal infinities. Georg Cantor developed a system of transfinite numbers, in which the first transfinite cardinal is ℵ0 ,(aleph-null), the cardinality of the set of all the natural numbers. (Fun fact: aleph (ℵ) is the first letter of the Hebrew alphabet.) This modern mathematical concept of quantitive infinity was developed during the late 19th century from work by Cantor, Gottlob Frege, Richard Dedekind and others, using the idea of collections, or sets. In VSauce’s video “How to Count Past Infinity” (link above), the cardinality of a set is a measure of the number of elements of the set. For instance, set A has 3 elements {1, 2, 3}, therefore A has a cardinality of 3, which is a cardinal number. Cardinal numbers (Cardinals) generalise natural numbers when measuring the cardinality (size) of a finite set. On the other hand, transfinite cardinal numbers describe the sizes of infinite sets. In terms of bijective functions, if 2 sets have the same cardinality there will be a one-to-one correspondence (bijection) between the elements of the 2 sets. For example, if plate A has 4 eggs and plate B has 4 pieces of bacon you can correspond 1 egg from plate A with 1 piece of bacon from plate B, meaning both plates have the same cardinality. In that case, the cardinal number for both plates is 4. A fundamental theorem made by Georg Cantor states it’s possible for infinite sets to have different cardinalities meaning that the cardinality of the set of real numbers is greater than the cardinality of the set of natural numbers. A proper subset of an infinite set can have the same cardinality as the original set, something that can’t occur with proper subsets of finite sets.

A bijective function, f: X → Y, from set X to set Y demonstrates that the sets have the same cardinality, in this case equal to the cardinal number 4.

Georg Ferdinand Ludwig Philipp Cantor (1845 - 1918)

The transfinite sequence of cardinal numbers is as follows:
0, 1, 2, 3, …, n, … ℵ0, ℵ1, ℵ2, …, ℵα, …
It starts with the natural numbers from zero (finite cardinals), followed by the aleph numbers (infinite cardinals of well-ordered sets), which is indexed by ordinal numbers. According to the axiom of choice, the transfinite sequence above includes every cardinal number possible. If one rejects that axiom, the situation becomes more complicated, with additional infinite cardinals that are not alephs. The axiom of choice (AC) is an axiom of set theory referring to a Cartesian product of a collection of non-empty sets being non-empty. It states that that given any collection of bins, each containing at least 1 object, it is possible to make a selection of exactly 1 object from each bin, even if the collection is infinite. Formulated in 1904 by Ernst Zermelo, in order to prove his well-ordering theorem, he states every indexed family of non-empty sets ((Si)iCI) there exists an indexed family ((xi)iCI) of elements such that xi c Si for every i c I.

This diagram illustrates each Si and xi represented as a jar and a coloured marble, respectively.

In this diagram, Si is a family of sets indexed over the real numbers (R). There is a set Si for each real number (i), with a small sample shown above. Each set contains at least 1 and possible infinitely many elements. The axiom of choice allows us to arbitrarily select a single element from each set, forming a corresponding family of elements (xi) also indexed over the real numbers, with xi drawn from Si. Generally speaking, the collections may be indexed over any set I, not just R.
https://en.wikipedia.org/wiki/Axiom_of_choice

For a finite number of sets, selections can be made without invoking the axiom of choice. For example, here are 3 sets of natural numbers: X {1,2,3}, Y {5, 10, 32}, Z {58, 2498, 49823}. The choice function here is “select the smallest number from each set”. In that case those numbers are 1 (from X), 5 (from Y) and 58 (from Z). Even if we have infinitely many sets, as long as it’s from the natural numbers, we can still choose the smallest number from each set to produce a new set. Thus, the choice function is defined to be the set of elements we handpicked. However, no choice function is known to have collected real numbers from non-empty subsets (if there are non-constructible reals). In this case, we invoke the axiom of choice.
Russell coined this analogy:
Let’s say you have an infinite collection of pairs of shoes, choose the left shoe from each pair to obtain an appropriate selection. That makes it possible to directly define a choice function. Next you have an infinite collection of pairs of socks (assuming each pair of socks have no distinguishing features), there is no obvious way to make a function that selects 1 sock from each pair, without invoking the axiom of choice. An axiom, in mathematics, is defined as for any set X of non-empty sets, there exists a choice function (f) defined on X. It’s expressed mathematically as this:

Thus, the negation of the axiom of choice states that there exists a collection of non-empty sets that has no choice function.
The general definition of an axiom or postulate is a statement that is taken to be true, serving as a premise or starting point for further reasoning and arguments. It comes from the Greek axíōma meaning “that which is thought worthy or fit”, “that which comments itself as evident”.

https://en.wikipedia.org/wiki/Zermelo–Fraenkel_set_theory
The axiom of choice is one of many axioms in axiomatic set theory proposed by mathematicians Ernst Zermelo and Abraham Fraenkel in the early 20th century, known as the Zermelo-Fraenkel Set Theory with Axiom of Choice (ZFC). It was written in order to formulate a theory of sets free of paradoxes such as Russell’s Paradox. In 1908, Zermelo proposed the first axiomatic set theory, Zermelo Set Theory. However in a letter to Zermelo sent by Fraenkel in 1921, this theory was incapable of proving the existence of certain sets and cardinal numbers whose existence was taken for granted by most set theorists of the time, notably the cardinal number ℵω and the set {Z0, ℘(Z0), ℘(℘(Z0)),…}, where Z0 is any infinite set and ℘ is the power set operation.
The following axioms per se are expressed in symbolic terms in first order logic. All formulations of ZFC imply the existence of at least one set.

(1) Axiom of Extensionality
= If x and y have the same elements, then they belong to the same sets.

(2) Axiom of Regularity (Axiom of Foundation)
= Every non-empty set ‘x’ contains a member ‘y’ such that x and y are disjoint sets, which implies that no set is an element of itself and that every set has an ordinal rank.

(3) Axiom Schema of Specification (Axiom Schema of Separation or of Restricted Comprehension)
= The subset of set z obeying a formula φ(x) with 1 free variable x written as,
{x c z : φ(x)}, always exists. (Axiom Schema means there is 1 axiom for each φ). Let φ be any formula in the ZFC language with all free variables among x, z, w1, …, wn (y is NOT free in φ). Then:

Note that this axiom can only construct subsets, and does not allow the construction of sets with general forms: {x : φ(x)}.
This restriction is important in order to avoid Russell’s Paradox and its variants that accompany naïve set theory with unrestricted comprehension.
(4) Axiom of Pairing
If x and y are sets, then there exists a set which contains x and y as elements:

(5) Axiom of Union
The union over the elements of a set exists. For instance the union over the elements of the set {{1,2}, {2,3}} is {1,2,3}. The axiom of union states that for any set of sets F there is a set A containing every element that is a member of some member of F:

(6) Axiom Schema of Replacement
It asserts that the image of a set under any definable function will also fall inside a set. Let φ be any formula in the ZFC language with all free variables among x, y, A, w1, …, wn so that in particular B is not free in φ.

If the relation φ represents a definable function (f), A represents its domain, and f(x) is a set for every x c A, then the range of f is a subset of some set B.
(7) Axiom of Infinity
Let S(w) be the w U {w}, where w is some set. {w} is a valid set by applying the Axiom of Pairing with x = y = w so that the set z is {w}. There will exist a set X such that the empty set O is a member of X and, whenever a set y is a member of X, then S(y) is also a member of X.

A set X has infinitely many members but it requires an establishment that these members are all different from one another. If 2 elements are the same, the sequence will loop around in a finite cycle of sets. So the axiom of regularity prevents this from happening.

(8) Axiom of Power Set
For any set ‘x', there is a set ‘y’ that contains every subset of x.

The Axiom of Specification is then used to define the power set P(x) as the subset of such a set y containing the subsets of x exactly:

(9) Well-Ordering Theorem
For any set X, there is a binary relation R which well-orders X. This means R is a linear order on X such that every non-empty subset of X has a member which is minimal under R.

Axioms 1 to 8 define ZF without the Axiom of Choice, hence Axiom 9 is equivalent to the Axiom of Choice.
https://en.wikipedia.org/wiki/Axiom
Other Axioms not included in ZFC:
(a) Axiom of the Empty Set
= An axiom of Kripke-Platek set theory and a variant of general set theory that Burgess calls “ST” in 2005. Using the axiom of extensionality, it states in natural language that essentially an empty set exists.

(b) Axiom Schema of Collection
= Closely related and often mixed up with the axiom schema of replacement. While replacement states that the image itself is a set, collection, on the other hand, states that the superclass of the image is a set.

(c) Axiom of Constructability
= A possible axiom of set theory asserting that every set is constructible. It is usually written as V = L, where V denotes the Von Neumann Universe & L denotes the Constructible Universe.

If you are curious to learn more about axioms, click on the link below.
https://en.wikipedia.org/wiki/List_of_axioms

https://en.wikipedia.org/wiki/Ordinal_number
Another type of number mentioned in Cantor’s set theory is an ordinal number. Introduced in 1883, they accommodate infinite sequences and classify derived sets Ordinal numbers describe the way a collection of objects are arranged in a particular order, one after another. Any finite collection of objects can be put in order just by counting: labelling the objects with distinct whole numbers. Hence, ordinal numbers are the “labels” that help arrange collections of objects in a certain order. For instance:
1st = First
2nd = Second
3rd = Third
4th = Fourth
5th = Fifth
6th = Sixth
.
.
.
Question: Why do we say “first”, “second” &”third” instead of oneth, twoth or threeth? Unfortunately, no one really knows the answer. What is known in etymology, is that “first" was derived from the Old English “fyrst”, which originated from the superlative (-est) form of words related to the Proto-Indo-European word “per”, meaning “forward”. Fyrst as an Old English adjective means foremost, going before all others; chief, principal; from the Proto-Germanic *furista meaning foremost.
“Second” is derived from the Latin secundus, which means “following”, “next in series”, “subordinate”, “inferior". As an adjective, “second” comes from the 1300s Old French word “second, secont.

In mathematics, ordinal numbers describe the order type of a well-ordered set, which is a set with relation such that:
- Trichotomy: For any elements x & y, exactly 1 of these statements is true
(1) x > y, (2) y = x, (3) y > x
- Transitivity: For any elements x, y & z, if x > y & y > z then x > z
- Well-foundedness: Every non-empty subset has an element x (a smallest element) such that there is no other element “y” in the subset where x > y.
When dealing with infinite sets we have to distinguish between the notion of size, described by cardinal numbers, and the notion of position, described by ordinal numbers. While any set has only 1 size (cardinality), there are many non-isomorphic well-orderings of any infinite set.
Well-ordered sets are totally ordered sets (given any 2 elements one defines a smaller and a larger one coherently) in which there is no infinite decreasing sequence, hence infinite increasing sequences. Ordinals are defined by the set of ordinals that preceded it. e.g. Ordinal 42 is the order type of the ordinals less than it i.e. {0, 1, 2, 3, 4, … 41}.
The smallest infinite ordinal is ω (omega - Greek letter for v), which is the order type of the natural numbers. Does anything come after ω? Yes, there is ω+1, ω+2, ω+3, … and so on. After all of these we approach ω*2 (ω+ω), then ω*2 +1, ω*2 +2, ω*2 +3, … and so on, then ω*3 (ω+ω+ω) and then ω*4, ω*5, ω*6, … and so on. We can formulate a pattern using this equation ω*m + n, where m and n are natural numbers). After this equation, we come to ω^2 (ω*ω), then ω^3, ω^4, ω^5, … and so on. Then we come to ω ^ω, then ω^(ω^ω), then later ω^((ω^ω)^ω).
After an infinite number of powers of ω, we come to ε0 (Epsilon nought). We can continue this sequence indefinitely far and repeat everything we have labelled above until we reach the first uncountable ordinal, which expresses the set of all countable ordinals, ω1 . Then after that is ω2 , then ω3 , ω4, ω5, … and so on, until we approach ωω, then ωωω, ωωωω, … and so on.

This diagram represents a graphical “matchstick” of the ordinal ω2 . Each stick corresponds to an ordinal of the form ω*(m+n) where m and n are natural numbers.

https://en.wikipedia.org/wiki/Continuum_hypothesis
By then we’ve run out of one-word superlatives to describe even larger numbers, and we resort to scientific notation instead. e.g. 1.3 x 10^23344 . One of Cantor’s most famous theories in mathematics is the Continuum Hypothesis (CH) proposed in 1878 that hypothesises about the possible size of infinite sets. It states that there is no set whose cardinality is strictly between that of the integers and the real numbers. In 1900, David Hilbert presented it as one of his 23 problems and answering it is independent of ZFC set theory. So either the CH or its negation can be added as an axiom to ZFC, with the resulting theory being consistent if ZFC is also consistent. Complementing earlier work by Kurt Gödel in 1940, this independence was proven by Paul Cohen in 1963.
Cantor provides 2 proofs that the cardinality of the set of integers is strictly smaller than that of the set of real numbers, which are the uncountability proof and the diagonal argument.
https://en.wikipedia.org/wiki/Georg_Cantor%27s_first_set_theory_article
(a) Uncountability — Georg Cantor’s first set theory article
Published in 1874, the first theorems of transfinite set theory concerns infinite sets and their properties. One of these theorems states that the set of all real numbers is uncountably, rather than countably, infinite. Cantor proved this using his first uncountability proof on the set of real algebraic numbers. To prove they’re countable, he defined the height of a polynomial of degree ’n’ with integer coefficients as: n — 1 + |a0| + |a1| + … + |an|, where a0, a1,… an are coefficients of the polynomial. Then he ordered the polynomials by their height, and the real roots of polynomials of the same height by numeric order. Since there are only a finite number of roots of polynomials of a given height, these orderings put the real algebraic numbers into a sequence. Cantor then produced a sequence in which each real algebraic number appears just once by only using polynomials that are irreducible over the integers. The table contains the beginning of Cantor’s enumeration:
Cantor’s Enumeration of the Real Algebraic Numbers

Cantor’s second theorem states that given any sequence of real numbers (x1, x2, x3,…) and any interval [a,b], there is a number in [a,b] that is not contained in the given sequence. To prove this theorem, we use open intervals (a,b) where the set of real numbers are larger than a (> a) and smaller than b (< b). To find a number in [a,b] that is not contained in the given sequence, let’s construct 2 sequences of real numbers as follows: - Find the first 2 numbers of the given sequence that are in (a,b). - Label the smaller of these 2 numbers by a1 and the larger by b1. - Similarly find the first 2 numbers of the given sequence that are in (a1,b1) - Likewise, label the smaller by a2 and the larger by b2. - Continue this procedure until you generate an extremely long sequence of intervals: (a1,b1), (a2,b2), (a3,b3), … such that each interval in the sequence contains all the succeeding intervals. This implies that the sequence a1, a2, a3, … continues to increase whilst the sequence b1, b2, b3, … continues to decrease. These intervals generated can be either finite or infinite. If they are finite, let (an, bn) be the final interval. If they are infinite, take the limits a∞ = limn —> ∞, an and b∞ = limn —> ∞, bn. Since an < bn for all values of n, either a∞ = b∞ or a∞ < b∞. Thus there are 3 cases to consider: (1) Since at most one xn can be in this interval, every y-value in this interval except xn (if it exists) is not contained in the given sequence. The last interval is labelled as (an, bn)

(2) a∞ = b∞. Then a∞ is not contained in the given sequence since for all n: a∞ belongs to (an, bn) but xn does not. Cantor states that without proof xn ∉ (an, bn).

(3) a∞ < b∞. Then every y-value in [a∞, b∞] is not contained in the given sequence since for all n: y belongs to (an, bn) but xn does not.

(b) Cantor’s Diagonal Argument
Published in 1891, George Cantor’s mathematical proof demonstrates infinite sets cannot be put into a one-to-one correspondence with the infinite set of natural numbers. He considered the set T of all infinite sequences of binary digits (i.e. each digit is 0 or 1). He begins with a constructive proof of the following theorem:
If s1, s2, … sn … is any enumeration of elements from T, then there is always an element ’s’ of T which doesn’t correspond to any sn in the enumeration.
e.g.
s1 = (0, 0, 0, 0, 0, 0, 0, …)
s2 = (1, 1, 1, 1, 1, 1, 1, …)
s3 = (0, 1, 0, 1, 0, 1, 0, …)
s4 = (1, 0, 1, 0, 1, 0, 1, …)
s5 = (1, 1, 0, 1, 0, 1, 1, …)
s6 = (0, 0, 1, 0, 1, 0, 0, …)
s7 = (1, 0, 0, 0, 0, 1, 0, …)
…
Next, a sequence ’s’ is constructed by choosing the 1st digit of s1, 2nd digit of s2, 3rd digit of s3, and so on. Generally for every ’n’, the nth digit as complementary to the nth digit of sn will have its digits swapped from 0 to 1 or vice versa.

s1 = (0, 0, 0, 0, 0, 0, 0, …)
s2 = (1, 1, 1, 1, 1, 1, 1, …)
s3 = (0, 1, 0, 1, 0, 1, 0, …)
s4 = (1, 0, 1, 0, 1, 0, 1, …)
s5 = (1, 1, 0, 1, 0, 1, 1, …)
s6 = (0, 0, 1, 0, 1, 0, 0, …)
s7 = (1, 0, 0, 0, 0, 1, 0, …)
…
s = (1, 0, 1, 1, 1, 1, 1, …)
By construction, our new sequence ’s' differs from each sn since their nth digits differ. Hence ’s’ cannot occur in the enumeration. Cantor then uses a proof by contradiction to show that the set T is uncountable. It begins by assuming that T is countable. Then all of its elements can be written as an enumeration s1, s2, … sn … After applying the previous theorem to this enumeration produces a sequence ’n’ that doesn’t belong to the enumeration. However this contradicts ’s’ being an element of T and therefore belonging to the enumeration, which implies that the original assumption is false. Therefore, T is uncountable.

In further detail, the CH states that the set of real numbers has minimal possible cardinality which is greater than the cardinality of the set of integers. That means every set of real numbers (S) can either be mapped one-to-one into the integers or the real numbers can be mapped one-to-one into S. The power set of any set is the set of all subsets of S (or family of sets over S), including the empty set and S itself, denoted as P(S). For instance:
If S is the set {1,2}, then the subsets of S are:
- {} = Empty / Null Set
- {1}, {2}
- {1,2}
Then the P(S) or Power set of S is {}, {1}, {2}, {1,2}. 2 numbers in a set gives me 4 subsets in the power set. Here’s another example:
If S is the set {1, 2, 3}, then the subsets of S are:
- {} = Empty Set or Null Set
- {1}, {2}, {3}
- {1,2}, {2,3}, {1,3}
- {1,2,3}
Then the P(S) is {}, {1}, {2}, {3}, {1,2}, {2,3}, {1,3}, {1,2,3}.
This time, 3 numbers in a set gives me 8 subsets in the power set. Notice a pattern here?
If S is a finite set with |S| = n elements, then the number of subsets of S is |P(S)| = 2^n, Cantor’s diagonal argument shows that the power set of a set (infinite or not) always has strictly higher cardinality then the set itself. His theorem demonstrates that the power set of a countably infinite set is uncountable infinite i.e. P(ℵ0). So the fact that real numbers are equinumerous with the power set of the integers, the CH says that there is no set (S) which: ℵ0 < |S| < 2^ℵ0 Assuming the axiom of choice, there is a smallest cardinal number ℵ1 greater than ℵ0. and the Continuum Hypothesis, in turn, is equivalent to the following formula: - 2^ℵ0 = ℵ1 = c
A Generalised Continuum Hypothesis (GCH) says that for all ordinals α
- 2^ℵα = ℵα+1
Asserting that the cardinality of the power set of any infinite set is the smallest cardinality greater than than of the set.
c denotes the cardinality of the continuum, which is shown to be greater than that of the natural numbers, ℵ0. Generally speaking, there are more real numbers (R) than natural numbers (N). The GCH states that if an infinite set’s cardinality lies between that of an infinite set (S) and that of the power set of S, P(S), then it either has the same cardinality as the set S or the same cardinality as P(S). For any infinite cardinal (λ) there is no cardinal (κ) such that λ < κ < 2λ .

https://en.wikipedia.org/wiki/Hilbert%27s_paradox_of_the_Grand_Hotel
Hilbert’s Paradox of the Grand Hotel (Hilbert’s Hotel)
In 1924 lecture entitled “Über das Unendliche”, David Hilbert launched a thought experiment illustrating a counterintuitive property of infinite sets. Imagine a hotel with a countably infinite number of rooms and every room is already occupied / booked. Suppose a new guest arrives and wishes to accommodate in this hotel. You would think that the hotel won’t have enough rooms to accommodate any newly arrived guests. No need to worry! The receptionist can (simultaneously) shift the guest in room 1 to room 2, the guest currently in room 2 to room 3, and so on. You begin to see the pattern by moving any guest from their current room number ’n’ to their new room number ‘n+1’. After all this shifting, room 1 is vacated and the new guest can move into room 1. By repeating this procedure, you can make room for any finite number of new guests although your current guests may not like you shifting them from room to room.
It’s also possible to accommodate a countably infinite number of new guests too. You can move the guest currently in room 1 to room 2, the guest currently in room 2 to room 4, and so on. You’ll notice a pattern that the guest occupying room number ’n’ with move to room number ‘2n’, and all the odd-numbered rooms (which are countably infinite) will be vacant for our new guests.

David Hilbert (1862 - 1943)

Now an infinite number of bus coaches carrying infinitely many passengers arrive at your hotel doorstep. How are you supposed to accommodate all those guests in your hotel? There are different methods to solve this problem depending on the seats in those coaches being already numbered (or use the axiom of countable choice). For each of the following methods, consider a passenger’s seat number on a coach to be ’n’, and their coach number to be ‘c’, and the numbers ’n’ and ‘c’ are then fed into the 2 arguments of the pairing function.
(1) Prime Powers Method
Vacate the odd numbered rooms (i.e. 1, 3, 5, 7, 9, …) by sending the guest currently in room number ‘i’ to room number 2i , then put out the first coach’s load in room numbers 3n, then the second coach’s load in room numbers 5n, and so on. You will a notice a pattern that for a coach number ‘c’ we use the room numbers pn, where p is the ‘c’th odd prime number. The solution vacates certain rooms, however rooms with odd non-prime numbers may or may not be useful to the hotel such as 15 or 33, because they will be unoccupied.
(2) Prime Factorisation Method
Each passenger is on a particular coach (c) and sitting on a numbered seat (s). Put each person into room number 2s x 3c (presume 0 for the number of people already in the hotel, 1 for the first coach etc.) Every number has a unique prime factorisation which makes it easy to see every person getting a room, whilst no two people will end up in the same room. e.g. If you were sitting on seat 1 on the first coach, you would end up in room number 6. This solution will obviously leave certain rooms empty.
(3) Interleaving Method
For each passenger, compare the lengths of ’n’ and ‘c’ as written in any positional numeral system, such as decimal. i.e. Allocate each hotel resident as being in coach number 0. Whichever number is shorter, add leading 0s to it until both values have the same number of digits. Then interleave the digits to produce a room number producing the following digits in order [1st digit of coach number], [1st digit of seat number]-[2nd digit of seat number]- etc.
e.g. A hotel guest (coach 0) in room number 3143 will move to room 03010403 (i.e. room number 1,070,209). A passenger on seat 2389 of coach 479 will go to room 2437899.
This method will fill the hotel completely, hence we can extrapolate a guest’s original coach and seat by reversing the interleaving process. First add a 0 at the front if the room has an odd number of digits e.g. 2, 145, 39495. Then de-interleave the number into 2 numbers: the seat number consists of the odd-numbered digits and coach number is the even-numbered digits.
(4) Triangular Number Method
Those already in the hotel will be moved to room number (n2 + n)/2, or ’n’th triangular number. Those in a coach will be in room number (((c+n—1)2 + c + n — 1)/ 2) + n, or (c + n —1) triangular number plus n. This will fill all the rooms by only 1 guest.
(5) Arbitrary Enumeration Method
Let S := {(a,b) | a,b c N}. S is countable since N is countable, hence we may enumerate its elements s1 , s2 , … and so on. Let sn = (a,b), assign the ‘b’th guest of the ‘a’th coach to the ’n’th room (consider the guests already in the hotel as guests as 0th coach (a = 0). Thus we have a formula to assign each person to a room without skipping any rooms.

https://en.wikipedia.org/wiki/Extended_real_number_line
Do you remember solving all those algebraic problems and drawing graphs of different shapes and scales on a 2D plane in high school mathematics? As far as I can remember, there are logarithmic, linear, quadratic, trigonometric (sin, cosine, tan, and its inverses), exponential, powers, fractions, roots, differential and many others. My teachers taught me that in order maximise credit in my final mathematics exams, I have to put arrows on the ends of the x- & y-axis to indicate that this graph is continuously infinite in both positive and negative directions. You’ll notice that every question you are asked to solve or draw is accompanied by the mathematical statement (x c R). This statement means that x includes all real numbers on the affinity extended real number system. The limits of our real number system are capped by positive infinite and negative infinity, even though they are not real numbers. There are handy in describing various limiting behaviours in calculus and mathematical analysis e.g. Theory of measure and integration. It is often denoted by

or [–∞, +∞] or ℝ ∪ {–∞, +∞}.
Mathematicians often describe the behaviour of a function f(x), as either the argument “x” or the function value f(x) gets quite large in a sense. e.g. Consider the function:
f(x) = 1/x

As the value of x gets infinitely bigger, the graph would appear to plummet then level out as f(x) approaches the y-axis (y = 0). As the value of x gets closer to 0, the graph would continue to rise astronomically but f(x) would never reach 0. This type of behaviour is marked by a dotted line called an asymptote. For this function of x, we have a horizontal asymptote at y = 0 and a vertical asymptote at x = 0. By definition an asymptote of any curve is a line such that the distance between the curve and line approaches zero as one or both of the x or y coordinates tends to infinity. It’s derived from the Greek word asymptõtos, meaning “not falling together”, first introduced by Apollonius of Perga in his work on conic sections. This limiting behaviour is similar to the limit of a function at a real number, except that there isn’t any real number x approaches. So adjoining + ∞ & -∞ to R allows the formulation of a "limit at infinity” with topological properties similar to those for R.
https://en.wikipedia.org/wiki/Asymptote
There are 3 kinds of asymptotes: Horizontal, Vertical and Oblique asymptotes. For curves given by the graph of a function y = f(x), horizontal asymptotes are horizontal lines that the graph of a function approaches as x tends to +∞ or -∞. Vertical asymptotes, on the other hand, are vertical lines near which the function increases without bound. Oblique asymptotes are sloped lines that equal any finite value but zero, such that the graph of a function approaches it as x tends to +∞ or -∞. A curve intersecting an asymptote infinite many times is called a curvilinear asymptote if the distance between the 2 curves tends to zero as they tend to infinity, shown below.

If you’re currently studying calculus, you may encounter asymptotes when you learn to compute them using limits.
(a) Vertical Asymptotes:
If the line x = a is a ‘vertical asymptote’ of a graph of the function y = f(x) then at least one of the following statements below is true:

The limit as x approaches the value “a” from the left (from smaller values)
The limit as x approaches the value “a” from the right (from larger values)
e.g. (1) If f(x) = x / (x—1), the numerator approaches 1 and the denominator 0 as x approaches 1, so

and the curve has a vertical asymptote x = 1

Sometimes the function f(x) may or may not be defined at ‘a’, and its precise value at the point x = a doesn’t affect the asymptote.
e.g. (2)

This function has a limit of +∞ as x approaches 0+ (from the left), f(x) has the vertical asymptote x = 0 but f(0) = 5. Thus the graph of this function does intersect the vertical asymptote once, at (0,5). It is impossible for the graph of a continuous function to intersect a vertical asymptote more than once.

In trigonometry, tangent graphs have an infinite number of vertical asymptotes if the graph is continuous, for example:
f(x) = tan (x), where x is in radians.

Using the equation tan (x) = sin (x) / cos (x), since cos (x) equals to 0 (zero) when x = …, -π/2, π/2, 3π/2, 5π/2,… Then the curve has discontinuities (“holes” for these values of x, since anything divided by 0 gives an undefined result. That means there’s an asymptote at those values of x. To convert radians into degrees, replace π with 180 and evaluate it e.g. π/2 = 180 / 2 = 90o . Fun fact: 1 radian is about 57.3o.

(b) Horizontal Asymptotes
If a horizontal line y = c is a 'horizontal asymptote’ of the function y = f(x), then

when x tends to +∞ respectively.
For instance, the inverse of tangent function, also known as arctangent function (arctan(x), tan-1 (x), 1 / tan(x)) has 2 horizontal asymptotes.

The following equations above describes the line y =-π/2 is a horizontal tangent for the arctangent when x approaches -∞, and y = π/2 is a horizontal tangent for the arctangent when x approaches +∞.

(c) Oblique Asymptotes
If a linear asymptote is not parallel to the x- and y-axis, then a function f(x) is asymptotic to the straight line y = mx + c, (m ≠ 0) are x approaches +∞:

e.g. f(x) = x + 1/x

The graph of this function of x has an oblique asymptote, y = x (m = 1, c = 0) and a horizontal asymptote at y = 0. The limits below prove the presence of this oblique asymptote:

What about rational functions? At most, there have 1 horizontal asymptote or oblique (slant) asymptote, and infinitely many vertical asymptotes. The degrees of the numerator (Num.) and denominator (Den.) determine whether or not there are any horizontal or oblique asymptotes.
(a) If degree (Num.) — degree (Den.) < 0. Then there is a horizontal asymptote at y = 0.
e.g. f(x) = 1 / (x^2 + 1)
(b) If degree (Num.) — degree (Den.) = 0. Then there is a horizontal asymptote at y = Ratio of leading coefficients
e.g. f(x) = (4x^2 + 7) / (9x^2 + 3x + 12). Asymptote at y = 4/9
(c) If degree (Num.) — degree (Den.) = 1. Then there is a horizontal asymptote at y = Quotient of the Euclidean division of the numerator by the denominator.
e.g. f(x) = (x^2 + x + 1) / x. Asymptote at y = x + 1
(d) If degree (Num.) — degree (Den.) > 1. Then there is no linear asymptote but a curvilinear asymptote still exists
e.g. f(x) = 8x^9 / (3x^2+ 4)

Vertical asymptotes occur only when the denominator is equal to zero. If both the numerator and denominator are zero, the multiplicities of zero are compared.
e.g. f(x) = (x^2 — 5x + 6) / (x^3 — 3x^2 + 2x) = [(x — 2) (x — 3)] / [(x(x — 2) (x — 1))]
There are asymptotes at x = 0 and x = 1, but not x = 2, because the (x—2)s in the numerator and denominator get cancelled out.

(d) Curvilinear Asymptotes
Here are some bonus maths questions for you to try at home. Apply all the appropriate methods I’ve mentioned above to help solve and graph the following equation:
f(x) = (x^3 + 2x^2 + 3x + 4) / x
Using Euclidean division, now f(x) = (x^2 + 2x + 3) + 4/x. Therefore there will be a curvilinear asymptote y = x^2 + 2x + 3, as known as a parabolic asymptote because this graph is a parabola rather than a straight line. Furthermore there will be a vertical asymptote at x = 0.

(x^2 / a^2) — (y^2 / b^2) = 1
To work out the asymptotes, we substitute 1 for 0 and then rearrange the equation to isolate y on one side. You would end up with:
y = +(b/a)*x
Therefore the graph of this hyperbola has 2 oblique asymptotes, shown below.

(x^2 / a^2) — (y^2 / b^2) — (z^2 / c^2) = 1
The graph of this equation will be a hyperboloid, with its asymptotes forming a shape called an asymptotic cone.

https://www.youtube.com/watch?v=ffUnNaQTfZE
Ever heard of a supertask? The above link contains a Vsauce video that uses animations to illustrate famous super tasks which I found quite interesting. A supertask is a philosophical concept that describes a countably infinite sequence of operations that occur sequentially within a finite amount of time. The term was coined by James F. Thomson, known for his philosophical lamp, which I’ll delve into later on. Below are a list of supertasks that I challenge you to work out and see if you have a solution for them.

A) Achilles and the Tortoise

Zeno of Elea once claimed that super tasks were impossible to accomplish. He imagined Achilles running a race against a tortoise like the bunny and the tortoise race. Achilles is running at 1 m/s whilst the tortoise is striding along at 0.1 m/s. However the tortoise is 0.9 m ahead of Achilles. You would think that Achilles will catch up with the tortoise after 1 second of the race, but Zeno argues against this. Instead he believes that Achilles would inevitable approach the tortoise’s starting point, but by the time he has reached it, the tortoise will have already moved on to another point. This race continues like this for an infinite amount of steps before one reaches the finish line. Beginning at 0.9 m for Achilles has to catch up, then 0.09m, then 0.009 m and so on, forever. While these catch-up distances get infinitesimally small, they will remain finite, while Achilles’ catching up to the tortoise will become an unending super task.
Here’s another scenario on Achilles’ race:
Imagine Achilles running the same race but he’s holding 2 flags, a blue flag and a red flag. He will hold up a blue flag after an odd number of steps, and a red flag after an even number of steps. Achilles traverses half the distance from A to B to its midpoint AB, consequently holds up the blue flag. From AB to B, he then traverses half that distance and holds up the red flag, then half of that and blue flag is held up again, half of that and red flag is held up again, and so on. By the time Achilles reaches the finish line at B, which flag will he be holding up?

B) Thomson’s Lamp

Philosophers like James F. Thomson reject Zeno’s argument that super tasks were impossible hence motion is impossible and instead took it as a proof by contradiction where the possibility of motion is taken for granted. They rather accepted the possibility of motion and applied modus tollens (contrapositive) to Zeno’s argument to conclude that either motion is not a supertask or not all supertasks are impossible.
Imagine a lamp that turns on and off according a timer. At the start the lamp is off. After 1 min the lamp is turned on. After 0.5 min the lamp is turned off. After 0.25 min the lamp is turned back on again. After 0.125 min the lamp is turned off again so on and so on. After 2 mins, is the lamp on or off?
It seems that isn’t any non-arbitrary way to answer this question. The lamp can’t be off because it’s immediately turned back on again and the lamp can’t be on because then immediately it’s turned off again. So the answer is neither on or off and this is a contradiction hence Thomson embarrassingly admits that supertasks are impossible.

C) Gabriel’s Cake
Imagine baking a delicious cake in the shape of a square cross-section like 1m by 1m by 0.1m. That means the volume of your cake is 0.1m^3 and the surface area is 2.4 m^2. You cut the cake in 1/2 and the volume hasn’t changed but the total surface area has increased to 2.6 m^2. Next you cut one of the half-slices in 1/2 to make a quarter slice and the surface area has increased again to to 2.8 m^2. Next you cut one of the quarter slices in 1/2, which increases the surface area to 3.0m^2. You continue to cut the cake in half, in half and so on. By the time you’ve finished cutting, you will end up with an infinite number of slices of cake despite the fact that the volume has not changed. If you stack all those slices of cake on top of one another, you will need an endless amount of frosting to cover the whole cake. If Gabriel were to slice his birthday cake in this way forever, he would need to invite an infinite number of guests to his party in order to finish the entire cake, am I right?

D) Ross-Littlewood Paradox

Imagine a jar capable of holding an infinite amount of marbles all labelled 1, 2, 3, and so on. After 1 min, add marbles 1 to 10 into the jar and remove marble 1. Wait 0.5 mins, then add marbles 11 to 20 into the jar and remove marble 2. Wait 0.25 mins, then add marbles 21 to 30 into the jar and remove marble 3. Can you see a pattern here? In mathematical terms, at time ’t’ = 1 — 0.5n, I would add marbles labelled ‘10n + 1’ through to ‘10n + 10’ and remove ‘n+1’ marbles. How many marbles would be in the jar after 2 mins?
It seems I’m adding a net of 9 marbles after half of the remaining time left so I should end up with an infinite number of marbles inside the jar right? Consider these equations: [∞ + a = ∞], [∞ — a = ∞], [∞ * a = ∞] & [∞ / a = ∞]. No matter what finite number you add to, subtract from, multiply with or divide from infinity, the answer will still be infinity. So if you continue to remove an infinite number of marbles after adding an infinite number of marbles, then the jar would end up empty, i.e. no marbles will be in the jar. This paradox is called as such because we have 2 seemingly perfectly good arguments with completely contradicting opposite conclusions.
Allis and Koetsier shared their variation on this thought experiment. At the start, place marbles 1 to 9 into the jar. Instead of removing a marble, write a 0 (zero) after the 1 labelled on marble 1 so it’ll be labelled 10. After 1 min, add marbles 11 to 19 into the jar and write a 0 after the 2 on marble 2, labelled it as 20. This process is repeated ad infinitum. Since no marbles were taken out of the jar at any time, you will end up with a infinite number of marbles in the jar at the end of 2 mins. This dismisses any argument about the end result dependent on which marbles are taken out along the way.

On Wednesday November 29th 2014, Voyager 1, NASA’s farthest and fastest human-made spacecraft currently flying in interstellar space has fired up its thrusters for the first time to orient itself so it can communicate and feedback data to NASA headquarters on Earth through its antenna points. The first bytes of data transmitted via electromagnetic waves takes about 19 hours 35 mins to reach the closest receiving antenna in Goldstone, California, which is part of NASA’s Deep Space Network. That’s the last public update on Voyager 1 I could find as it flies further and further away from Earth and into the unknown stretches of space as you read this.

https://en.wikipedia.org/wiki/Observable_universe
Do you think the universe is infinite, boundless or endless? The image above is the best visualisation of our observable universe. You would think it’s described as ‘observable’ because of our modern technology that detects light or other useful information from distant stars, galaxies, planets and other cosmological entities across the cosmos. That’s called the visible universe. But it actually refers to the physical limit created by the speed of light itself. Because nothing can travel faster than light, any object further away from us than light could travel in the age of the universe simply can’t be detected, as they haven’t reached us yet. The speed of light is 299,792,458 m/s. But I always wondered why nothing can travel faster then light in our universe. I’ll look into the physics and limits of light in another post. Here are some astronomical numbers to get your minds blown. According to astrophysicists, the Virgo Supercluster (home of the Milky Way Galaxy) is at the centre of the observable universe. Then the radius of our observable universe in any direction you desire to space travel is about 4.4 x 10^26 m (46.5 billion light years, 14.26 billion parsec). Assuming our observable universe is a perfect sphere, then the volume is, using the formula (4/3)*π*r^3, it’s about 4 x 10^80 m^3. It’s predicted that the age of our observable universe is about 13.799 +/- 0.021 billion years. If you summate every star, planet, galaxy, nebula and asteroid / comet in the observable universe, it would weigh about 1 x 10^53 kg, about as dense as 6 protons per cubic metre of space which is equivalent to 9.9 x 10^(-30) g/cm3. Because there’s not much matter in space that we could see and generate heat with around 100 particles per cubic metre, the average temperature would be 2.72548 K, which is barely above absolute zero (0 Kelvin). However, there is optimism among astrophysicists and cosmologists that the universe is older than 13.799 billion years. Based on recent observations of distant stars and galaxies, predictions of a receding and decelerating expansion hence collapse of the universe were dismissed as they discovered that space itself is expanding and accelerating at an increasing rate by an force exerted by a mysterious thing that astrophysicists call, dark energy. Assuming that dark energy remains constant, so the universe’s expansion rate continues to accelerate, there will be a ‘future visibility limit’ beyond which cosmological objects may never enter our observable universe at any time in the infinite future. Outside that limit. the light it emits may never reach Earth for us to see and detect. The current future visibility limit is calculated at a comoving distance of 19 billion parsecs or 62 billion light years, if the Universe continues to expand forever, but this figure is subject to change as time ticks by. The detection of light from objects that were once close to Earth are now up to 45.7 billion light years away. Before the recombination epoch, about 380,000 years after the Big Bang, the Universe was filled with plaque opaque to light, and photons reabsorbed by other particles preventing astronomists from see these objects visibly from before that time using light or other electromagnetic radiation. What do you think we will find in the unobservable universe? Will we see more stars, galaxies, planets, solar systems, black holes etc. that we have to name? Would Voyager 1’s signal ever be detected by any life on other planets that have the suitable technology to decode the gold disk stored in that probe? Would there be Earth-like creatures on these planets? Are we all alone on his insignificant, lonely, minute planet called Earth currently orbiting around a star called the Sun, set to die within the next millennia? These are some of the many unanswered questions scientists alike simply don’t have information nor knowledge about. We are all in the dark when it comes to picturing the real story of our universe because there’s so much yet to be discovered in the darkness of space. I will delve into the details of the many mesmerising spectacles of our universe and the theories of how our universe operates and began in another post.