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I remember as a kid we’d settle arguments by whoever could say the highest number (or multiply by it; such as “I hate you more times 100, or I’m 50 times better than you). At some point we all learned of the existence of infinity, and that would settle it. Whenever someone said “infinity plus one”, the other would object by saying “but infinity plus one is still infinity!” implying that infinity was as big as you could get, and that all infinities were the same size. Turns out they’re not.

This may sound counter-intuitive to some of you. After all, infinity alludes to an endless size right? How can two things that are endless be bigger or smaller than the other?

Well the properties of infinite sets have been studied for quite some time. I’m not sure if he was the first, but Galileo Galilei identified a conceptual problem with infinite sets in his final scientific work *Two New Sciences*. I’m not interested in giving a history lesson so I’ll keep this post short and let you do you own research if you don’t believe me (and by now, you should have much more faith in me than that!).

His work identified a property of infinite sets that is now known as Galileo’s Paradox. The gist of it is this:

- Some numbers are squares (2^2 = 4; 3^2=9; etc.) whereas some numbers are not (prime numbers and other non-square integers, etc.). Therefore all numbers must consist of both squares and non-squares, and thus be more numerous than just the squares alone.
- However, every square has exactly one positive number that is its square root (basically just reversing the square operation). Therefore, there are as many squares as there are square roots.
- For every number, there is exactly one squared value (you can square any number by multiplying it by itself). Therefore, there are as many square roots as there are numbers (because every number can be squared so every number has a square root). Since there are as many squares as there are square roots, and there are as many square roots as there are numbers, then there must be as many squares as there are numbers.
- However, our premise was that there are more numbers than there are squares. Therefore it is a “paradox”, the solution to which is that some infinities are larger than others.

Galileo simplified this in his work with a very apt analogy that I think will help people visualise this concept better. Imagine two lines, one longer than the other. You can say that both lines are made up of an infinite amount of infinitesimally small points, however, you can also clearly see that one is longer than the other.

Later on, the mathematician Georg Cantor provided mathematical proof and definitions for sets and trans-infinite numbers, to which a great deal of resistance was put up by the other mathematicians of the time. The existence of larger infinities was and is of great philosophical importance. You can Google more of Georg Cantor if you wish; this post is merely for me to show you that not all infinities are the same.

Also, here’s an 8 minute video explaining this concept with some different examples by Numberphile. The guy presenting has a creepy smile and is way too excited about maths, but it’s still quite informative: http://www.youtube.com/watch?v=elvOZm0d4H0

And a much faster explanation by MinutePhysics (about 2 minutes): http://www.youtube.com/watch?v=A-QoutHCu4o

Life is a learning process and as a child (before I entered senior year of high school), I learned some things that really blew my mind. I’m sure some of these were experienced by us all whereas others some of you still don’t know (hopefully, or this won’t be that interesting).

**The refrigerator light:**There’s a button between the door and the inside of the fridge that is pressed when the door is closed. I remember trying to close the door slowly to see when the light turned off. Finding out how it worked sort of ruined the fun for me, as I could then manually turn the light on and off. This button also stops the beeping that occurs when you leave the fridge door open for too long (because pressing the button makes the fridge think it’s closed).**E=mc^2:**The implications of Einstein’s famous equation probably escapes most people. Everyone knows the equation but not many know what it means. Since my mother is a nuclear physicist, I learned this one pretty early on. We can tell, from basic mathematics, that an increase in energy (E) means an increase in mass (M), because C is a constant (speed of light) and does not change. The implication is that the closer you travel to the speed of light, the more mass you get (or heavier you get). Further, the equation unites two concepts: the conservation of mass and conservation of energy, wherein mass and energy cannot be created or destroyed, but only changed in form. Further, the equation describes a mass-energy ratio where, in a closed system, energy comes from the mass of an object (for example, a flashlight emitting light will become slightly lighter than when it is not emitting light).**Heavier objects fall at the same speed as lighter objects:**Galileo’s discovery was mind boggling to me as a child because I couldn’t come to grips with the idea that a super heavy object like a tank would fall at the same speed as a sheet of paper. Even now, it sounds a bit weird, but what determines the speed an object falls at is the air resistance. A fun experiment to do is dropping a book and a leaf at the same time. The leaf falls slower because of its shape which increases air resistance. But what if you put the leaf on top of the book? They fall together because the book is taking all the resistance on itself.**Microwave ovens do not produce heat:**This is a cool one that I think most people still don’t understand. When you reach into a microwave oven to get your food you don’t feel that hot air you get from opening a conventional oven door. Why? Microwave ovens don’t produce heat. They produce an electromagnetic field. Because water molecules are dipoles (one side is positively charged and the other side is negatively charged) and because electromagnetic waves are made up of an alternating electric and magnetic field, the microwave basically pushes the water molecule in a direction depending on the field. Since the field alternates, the water molecule is basically spinning and releasing heat to nearby molecules. The reason why you shouldn’t put metal in the microwave is because it can accumulate a high voltage that will cause a dielectric breakdown of the air inside the microwave which can release harmful gases.**Spaghettification:**This is a cool feature of black holes. Once you enter the event horizon of a black hole, your body is affected by gravity at different levels. Assuming you go feet first, your head experiences less force than your feet, so at some point you simply split in half. These two new portions of your body will experience the same effect and split again into quarters. This process will continue until you are broken down into a molecular level and essentially become a stream of subatomic particles that gets sucked into the core of the black hole. This process is known as spaghettification.

Edit: I’m in a lecture right now and very bored so I thought of a sixth thing.

6. **Time is the fourth dimension:** I’m sure many people have heard of this but not many fully understand why. Think of it as a mathematical graph. I’m sure everyone has encountered simple graphs in their lifetime. A 2D graph has two axes, the x and y coordinates. You can identify the location of a point based on their coordinates (x,y). Now, a 3D graph (which you’ll see in higher levels of mathematics). The principle is the same. There are now three axes, and if you want to identify the location of a point, you need the three coordinates: (x,y,z). Now, adding a fourth dimension doesn’t make sense right? Because our world is 3D, comprised of objects that have three dimensions that we can touch, right? Now we introduce the universe. As we know, the universe is expanding from a single point at which the big bang occurred. That means as you travel towards the “edge” (there is no edge, I’m just making a point) of the universe, you are essentially going back in time (seeing things that have existed longer than Earth, which is only 4.5 billion years old, whereas the universe is 14.6 billion years old). Since the universe loops on itself (not necessarily a sphere but similar in that you won’t ever reach an “edge”), that means we can’t rely on only three coordinates to locate a point in the universe. We need to know how old that point is, hence time is the fourth coordinate. Thus, the universe is actually 4D. We only consider things to be 3D because Earth is so tiny that the fourth dimension makes no difference on Earth (we just say the whole Earth is 4.5 billion years old, rather than saying the core is older than the crust).