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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:

  1. 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.
  2. 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.
  3. 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.
  4. 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

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This is actually a bit of a continuation from my earlier post on the multiverse and additional dimensions theory, and is basically supporting evidence for the multiverse (which is itself evidence for additional dimensions).

To avoid things getting messy, I’m going to break this into three subheadings.

Stephen Hawking:

You can easily do a quick wikipedia on him if you want to find out the more mundane details of his life. I’m just going to quickly mention some relevant things about him.

Hawking has a motor neurone disease related to amyotrophic lateral sclerosis (ALS) which has confined him to a wheelchair. As he is almost fully paralyzed, he communicates with facial twitches and a speech device where he enters types the words he wants to say by scrolling across a virtual keyboard with his eyes. This condition alone is proof of his genius as he rationalises complex theoretical physics in his mind without the benefit of being able to write things down. Honestly, I can’t stress enough how amazing that is. I hated 4 unit maths in high school and I had textbooks, the internet and calculators to rely on. His peers have said that Hawking works on intuition that is freakishly accurate, as if he is in tune with the universe in the same way that Einstein and Newton were.

Hawking’s black hole equation combined separate major fields of physics into one elegant formula, the first time (and only time to my knowledge) that separate fields of physics have been unified. Those of you who know a bit about physics will know why this is a big deal – there are many types of physics and they have never been unified under one single model before (which would be known as the theory of everything). The Hawking-Bekenstein entropy equation is:

Where S is entropy from thermodynamics, c is the constant for the speed of light from Einstein’s work, k is the Boltzmann constant, G is Newton’s constant for gravity, h is the Planck constant from quantum physics, and A is the area of the black hole.

Not only is this a combination of different fields of physics, it is a simple equation (which is considered mathematically beautiful, like Einstein’s e=mc^2).

The last thing I want to mention are a few of his theories that are relevant to this post. First, he provided mathematical proof for the beginning of the universe (the big bang), he not only described the mechanism of black holes mathematically, but defined many of the laws governing them (such as the event horizon, which is the region around the black hole which if you enter, there is no escape), and he also predicted that black holes would evaporate over time.

The Information Paradox:

The Information Paradox was something Hawking came to based on his work on black holes. Black holes break everything down into subatomic particles and suck them into the core of its gravity – a singularity (defined as a point that is infinitely small, infinitely dense, with an infinite gravity). The gravitational force of a black hole is so strong that not even light can escape. The implication of this is that whatever is sucked into a black hole is lost forever (or rather, it is stuck in the black hole forever).

However, Hawking later proved that black holes would evaporate over time. This is related to two facts: first, black holes emit radiation (a form of energy) and second, E=mc^2. Einstein’s equation means that energy and mass are essentially the same thing, just in different forms (you can mathematically represent energy/mass as a function of the other). What this means is that if black holes emit energy, they need to burn mass to do so; thus if a black hole runs out of mass (given that it runs out of stuff to suck in and burns out its core), it will evaporate.

Why was this huge news to the science world? Because the laws of physics (conservation of mass and energy) state that you cannot destroy mass/energy, only change its form. Information is “coded” into particles, and can never be lost. A visual representation of this would be if I tear a piece of paper to shreds. If I have all the pieces still, and a knowledge of how they fit together, I could theoretically recreate the original paper. The same applies to everything in the universe – if I burn a tree inside a containment unit, I would have everything that tree (and the fire) was made out of inside that containment unit. Theoretically, I could use those ingredients to remake the tree. However, Hawking’s proof of black hole evaporation violated this most fundamental law. If a black hole disappeared, what happened to all the information it absorbed? It would disappear with the black hole, a clear violation of the conservation of mass/energy. In essence, Hawking described black holes as huge cosmic machines that went around erasing parts of the universe and proclaimed that parts of the universe were missing as a result. Physicists were mind boggled and needed to disprove this theory. Why? Because the implications were that if black holes could violate this law, then the law was no longer a law of the universe. If it was no longer a law, that means that information anywhere in the universe could potentially be erased, and not just inside black holes. Further, at this time more black holes were discovered – there were supermassive black holes and micro black holes. There could even be micro black holes existing in your room as you read this. If black holes have the power to erase information, how can you say anything you know or see or feel or believe is real? Nothing is certain if everything is impermanent. This caused a huge fuss and was known as the Information Paradox.

Later, a theoretical physicist, Leonard Susskind came up with an alternative theory to solve the Information Paradox. The science world breathed a sigh of relief, but Hawking was determined to prove Susskind was wrong.

Unfortunately, at this point Hawking’s ALS got even worse. He was hospitalised but miraculously, he survived and went back to work. By now, he was so paralyzed that he had to get a student to help him work. Hawking would feed him ideas and the student would do the calculations and try to prove the concepts. As Hawking’s ALS got worse, his work became frustratingly slow. Now, his student tries to anticipate what Hawking wants to say (Hawking types the first few letters of a word and he guesses what word Hawking means).

Anyway, after getting out of hospital, Hawking went to a renown physics conference and made a public statement. He admitted that he had been wrong – information was not erased. However, he also declared Susskind wrong and claimed to have solved the paradox himself. I haven’t read this paper (it’s quite recent and he hasn’t provided mathematical proof yet), but from what I’ve gathered, his solution is as followed: information is not erased because it is transmitted to an alternate universe. The multiverse theory predicts an infinite number of universes, and inevitably, some of these universes will have no black holes. If there are no black holes, there is no way for information to be lost (this is a logic loop similar to time travel – if the cause of the problem doesn’t exist, you can’t have a problem in the first place). Basically, information will be transmitted through universes until it reaches a universe with no black hole, and since there is no black hole, the information can’t be lost.

Progress is ridiculously slow now – Hawking can only put out a few words a minute. Personally, I think it would be an astounding tragedy (especially in Hawking’s mind) if he becomes fully paralyzed and unable to spread his knowledge to the world. What would be more horrible was if he proved the existence of other universes but was unable to tell us. Imagine being trapped in your own mind with a universe shaking idea, fully proven, but unable to tell anybody around you.

Implications and the Multiverse Theory:

I really hope Hawking survives long enough to fully prove his new theory. It would be a tragedy for him to die with this work uncompleted as it would be definitive proof of a multiverse (because a multiverse would be necessary for the laws of physics to remain absolute).

How does this relate to my previous post? Well, as a quick refresher, I recently thought of the idea (which other scientists have also supported) that every singularity contains a universe. The reason for this is because our universe originated from a singularity that caused the big bang. Logically, all singularities have the potential to big bang and spew out its contents (a universe). There are singularities at the core of every black hole, meaning that there are hundreds of millions of universes inside our own universe, and that our own universe could just be the core of a black hole of an even greater universe (which would, by necessity, have more dimensions than us).

By the way, a quick note on the dimensions; further support for my suggestion that even more dimensions exist (and that our universe belongs to a universe with more dimensions) can be found in quantum mechanics and string theory. These two branches of physics study predict, by necessity, the existence of at least 11 dimensions. I think the fact that we’re working on string theory and alternate dimensions can be likened to the “fourth wall” in theatre – the characters of the play should not be aware of the audience’s existence but sometimes they “break the fourth wall” and hint that they do acknowledge an audience’s existence.

Back to the implications though: I believe in my previous post I suggested that the matter (or information) sucked in by a black hole is used to create a new (and smaller) universe. I made this post without thinking of Hawking’s multiverse, but the two concepts coincide well. Hawking states that by necessity, these other universes must exist to contain information that is taken from our universe. That is tantamount to what I said, that these alternate universes contain matter (information) from our current universe.

Essentially, I was beaten to goal again – this time Hawking came up with the idea before me. This stuff happens inevitably, and I admit, much of my knowledge is inspired by Hawking, but I can’t help but feel like I’m travelling in a rut because I’m arriving at the same conclusions as others. Breakthroughs need radical thinking that forges a whole new path or the thinker will inevitably run into the same dead-end as someone before them. This is way out of my depth already (I have long since lost any mathematical reasoning and have been relying on theoretical physics to rationalise my conclusions). I eagerly await Hawking’s work and am filled with admiration at the thought that even now, while I write this blog post, Hawking is painstakingly putting out a couple of words a minute to his student who is so close and yet so far from proving a multiverse.

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