Gödels incompleteness theorem

A Brief History of Time Updated

This is a Leonberger blog, but sometimes I also post reviews for books that are not about Leonbergers and when I do it is books that I love and that I want others to read. Today I am posting a review for a book that I loved, “A Brief History of Time: And Other Essays” Hardcover – September 1, 1998, updated in 2017. It was written by Stephen Hawking. A long time ago I read the original “A Brief History of Time” but now I read this updated version. It is still the original but the adjustments/corrections for more recent discoveries are described in the appendix. Contrary to what Amazon claims I don’t think this is a book “Told in language we all can understand”. In my opinion you need a little bit of a physics background or at least a serious interest in the subject. Otherwise, it will be too much abstract information at once. I should add that I bought the Hardcover.

Hardcover – Publisher : Bantam; Anniversary edition (September 1, 1998), updated 2017, ISBN-10 : 0553109537, ISBN-13 : 978-0553109535, 240 pages, Item Weight ‏ : ‎ 2.31 pounds, dimensions ‏ : ‎ 6.18 x 0.83 x 9.29 inches, it cost $16.14 on US Amazon. Click here to order it from Amazon.com.
Paperback – Publisher : RANDOM HOUSE UK; First Edition (January 1, 1990), ISBN-10 : 0553176986, ISBN-13 : 978-0553176988, 211 pages, Item Weight ‏ : ‎ 2.31 pounds, dimensions ‏ : ‎ 4.33 x 0.71 x 7.05 inches, it cost $20.85 on US Amazon. Click here to order it from Amazon.com.
Kindle – Publisher : Bantam; 10th edition (May 4, 2011), ASIN : B004WY3D0O, 242 pages. It is currently $9.99 on Amazon.com. Click here to order it from Amazon.com.
Audiobook – Publisher : Phoenix Books, Inc., Release date : January 06, 2022, ASIN : B09NLFY54Z, Listening length 5hrs 49 minutes. Click here to order it from Amazon.com.

Picture of Stephen Hawking with a galaxy in the background, plus title and author name and golden sticker saying, “Includes new material” — Front cover of A Brief History of Time: And Other Essays Hardcover by Stephen Hawking. Click here or the picture to visit the Amazon.com page for the Hardcover version of the book.

Amazon’s description of the book

Published more than two decades ago to great critical acclaim and commercial success, A Brief History of Time has become a landmark volume in science writing. Stephen Hawking, one of the great minds of our time, explores such profound questions as: How did the universe begin—and what made its start possible? Does time always flow forward? Is the universe unending—or are there boundaries? Are there other dimensions in space? What will happen when it all ends?

Told in language we all can understand, A Brief History of Time plunges into the exotic realms of black holes and quarks, of antimatter and “arrows of time,” of the big bang and a bigger God—where the possibilities are wondrous and unexpected. With exciting images and profound imagination, Stephen Hawking brings us closer to the ultimate secrets at the very heart of creation.

This is my five-star Amazon review for A Brief History of Time: And Other Essays by Stephen Hawking

Relativity, Quantum Mechanics, Black Holes, and Cosmology in just 200 Pages.

First, I should mention that even though I bought the hardcopy version of the book released in 1998, I received the updated version released in 2017. The same will probably happen to you if you buy it. I certainly did not mind. The 2017 version is identical to the 1998 version, but it contains additional material including corrections that are featured in an appendix at the end of the book. For example, in 1998 it was discovered that the expansion of the universe is accelerating, making the Friedmann models he discussed in chapter 3 almost obsolete. In addition, since 1998 Hawking’s no-boundary condition has become more certain, as well as the existence of multiverses. Therefore, it is important not to skip the appendix, and perhaps it is best to read the appendix first, so you know what to ignore in the original text of “A Brief History of Time”. I should mention that I read the original book from 1990, a very long time ago.

The book covers a lot of material. He describe past models of the universe, space and time and special and general relativity, light cones, cosmology, the expanding universe, quantum physics, Heisenberg’s uncertainty principle, the two slit experiment, the wave particle duality, anti-particles, Feynman’s sum over histories, elementary particles, particle spin, fundamental forces, entropy, black holes, event horizons, space-time singularities, the cosmic censorship hypothesis, virtual particles, the big bang, the inflationary model and the new inflationary model, the anthropic principle, imaginary time, quantum gravitational effects, the no-boundary condition, three arrows of time, Gödels incompleteness theorem, Einstein-Rosen bridges, or so called wormholes, supergravity, string theory, the unification of physics, renormalization, eternal inflation and the multiverse, etc. In summary, he covers a lot in less than 200 pages.

I have a degree in physics (I became an engineer) and I have an interest in these kind of topics, and therefore I understood most of the content in this book, at least at some level. However, I can see how people without a background in physics would have a hard time with this book. Hawking is for the most part doing a great job explaining these topics, but many of the topics are very abstract and the book covers a lot of them. I think it might be too much for some people, but I don’t see that as necessarily a fault of the book. It is just an acknowledgement that this is not an easy subject. If you want to understand what we know about the universe you have a lot of work ahead of you, no matter how great your teachers or authors are.

One potentially controversial item is the implications of the no-boundary condition derived from the combination of general relativity and quantum mechanics (quantum gravity). It is natural to think that the universe has either existed for an infinite time or that it had a beginning. The no-boundary condition offers up a third option. Space-time is finite and yet there is no singularity. In addition, the universe is self-contained meaning it does not have a beginning or an end. Just like in a universe that has existed for an infinite time there is no moment of creation.

In chapter 10 he discusses wormholes and time travel. Kurt Gödel, the guy with the incompleteness theorem, showed that under certain circumstances General Relativity allowed for time travel. Also, when you travel faster than the speed of light you are traveling backwards in time, something most science fiction authors depicting spaceships traveling faster than the speed of light conveniently ignore. However, the conclusion of the discussion that followed was basically, in practice you can probably not time travel. Just imagine that you could travel back in time and kill your mother. That way you would never be born so now you could not travel back in time and kill your mother, and poff, now you exist again, but now you can travel back in time and kill your mother. Time travel comes with various logical problems. By the way where are all the time travelers from the future?

One thing I disagree with was that he on page 156 says that intelligent beings can only exist in the expanding phase of the universe. His explanation for this statement is not convincing and I don’t believe it. However, since we now know that the universe is likely to expand forever it is a moot point. He also keeps calling entropy “disorder”. This is very common, but “disorder” in common vernacular is a vague term that does not exactly correspond to the mathematical definition of entropy, and this should at least be pointed out. These are very minor and unimportant complaints, but I wanted to mention them. The Amazon description states: “Told in language we all can understand”, which as I mentioned is not really true. But that is the Amazon description of the book, not a problem with the book.

In summary, I think this is a very interesting and informative book and I think it is well written and well organized. The fact that it is difficult reading for many people is because of the subject matter and is not the fault of the author. I highly recommend the book to anyone with a background in physics and anyone else who is really interested in the subject and doesn’t mind looking up concepts a bit more in depth. I give it five stars.

Endorsements for the book plus the text of the Amazon description of the book as well as an introduction to the author. — Back cover of A Brief History of Time: And Other Essays Hardcover by Stephen Hawking. Click here or the picture to visit the Amazon.com page for the Hardcover version of the book.

About the Author

Stephen Hawking’s ability to make science understandable and compelling to a lay audience was established with the publication of his first book, A Brief History of Time, which has sold nearly 10 million copies in 40 languages.

Hawking has authored or participated in the creation of numerous other popular science books, including The Universe in a Nutshell, A Briefer History of Time, On the Shoulders of Giants, The Illustrated On the Shoulders of Giants, and George’s Secret Key to the Universe.

I can add that I also read The Grand Design by Stephen Hawking, which I also recommend but with the same caution as for this book. It’s a bit abstract.

Some Things are not Meant to be Known

As I mentioned in a previous blog post I’ve been thinking about launching a second blog. The topic would be facts, or insights that are widely disputed or misunderstood amongst the public, yet important and known to be true. However, I am also thinking about adding another aspect to the blog and that is facts, or insights that are highly surprising to people, yet important and known to be true. Curious or strange facts if you will. Not strange trivia but important stuff. People wouldn’t necessarily dispute these facts, well they might, but they may appear unintuitive to a lot of people. So, the true facts that people are disputing would be “cluster-A” and the facts that just would seem strange, ponderous, or counter intuitive would be “cluster-B”. I will post about both.

This particular blog post gives an example of a cluster B fact. To express myself a little bit simplistic, science does not know everything (otherwise it would stop), but it knows a lot. By studying the light from a distant star, we can determine what elements it is composed of. The star may be composed of 71% hydrogen, 27% helium, 1% Lithium, and 1% other elements, and we can know that just from its light. We can determine the distance to the star, how it is moving compared to us, its temperature, roughly its age and longevity, and more. 150 years ago, we could not have dreamed of this capability.

Bright white star with a planet and a moon. — We can know so much about a star from its light. Shutter Stock Illustration ID: 566774353 by Nostalgia for Infinity.

Yet we know that there are things we can never know, no matter how advanced science becomes. Infinite experimentation, super intelligence, a quintillion super genius, infinite time, cannot breach some knowledge. The universe itself forbids some knowledge. It also means that the statement “nothing is impossible” is false. My natural reaction to such a claim is, “come on you can’t say that with certainty”, and I expect many others will feel the same. However, the reason some knowledge will never be attainable is that physical laws as well as mathematics and logic forbid some knowledge. Some things are not meant to be known. I will explain in the four sections below: the event horizon, the Heisenberg uncertainty principle, beyond the observable Universe, and Gödels incompleteness theorem.

I should say in my future blog I will explore each of the four examples below more in depth, and put them in their own blog posts, and I might add or remove examples as I learn more.

The Event Horizon of a Black Hole

A black hole is a region of spacetime where gravity is so strong that nothing, including light, can escape it. The boundary of no escape is called the event horizon. If you pass the event horizon you cannot come back out no matter how much energy, you expand. Nothing can escape, no matter, no radiation, not light or other electromagnetic radiation, and no information. Nothing at all can escape. The curvature of time and space itself forbids it. I should add that right at the event horizon, there is so called Hawking radiation, but without complicating things it is not the same thing as escaping a black hole.

Black hole devouring a planet. — Black Hole Stock Photo ID: 2024419973 by Elena11

Some black holes are formed when large stars die and collapse. These black holes are estimated to have a mass of five to several tens of solar masses. However, there are also super massive black holes that reside in the center of galaxies. The super massive black hole at the center of our galaxy, the Milky Way, is called Sagittarius A* and is estimated to have a mass of four million times the mass of our sun. The largest known supermassive black hole TON 618 is 66 billion times more massive than our sun. There are an estimated 100 million black holes in our own galaxy, the Milky Way. One interesting fact is that celestial objects can orbit a black hole, just like planets orbit the sun, but as you get too close you will rush, at the speed of light, into the depths of the black hole You are “poff and gone”.

The fact that nothing, including information, can escape a black hole means that we can never observe what is on the inside. You can venture inside and be lost. You can extrapolate from physical laws what might be inside, but you can never observe and report what is inside to planet Earth.

The Heisenberg uncertainty principle

The Heisenberg uncertainty principle states that it’s not possible to know the position and momentum of an object with perfect accuracy at the same time. Another way of saying that is that we cannot know both the position and speed of a particle, such as a photon or electron, with perfect accuracy. The formula is: dX * dP >= h/4pi , uncertainty in position (dX) times uncertainty in momentum (dP) is larger than half of Planck’s constant, which is very small. There is also an energy and time precision : dT * dE >= h/4pi. It basically means that there are no perfectly exact measurements or knowledge. Everything is a bit fuzzy. Planck’s constant is very small, so Heisenberg uncertainty principle does not matter for everyday objects, but it matters when sizes are very small (positions, energies, etc.) Note, Heisenberg uncertainty principle is not regarding a limitation of our equipment, but a limit set by a law of physics. It is a limitation set by the Universe.

Beyond the observable Universe

The observable universe is a ball-shaped region of the universe consisting of all matter that currently can be observed from Earth or its space-based telescopes. The radius of the observable universe is 46.6 billion light-years. The size of the observable universe is growing. Unfortunately, at those distances, space itself is stretching/expanding faster than the speed of light. Since no signal or information can travel faster than the speed of light we are losing, not gaining, celestial objects from the observable universe. Further, in the past we’ve lost many galaxies this way. I can add that the universe may be infinite. Since the expansion is accelerating, we will keep losing more galaxies beyond the boundary of the universe and some galaxies were always lost (with respect to observation).

A picture of a galaxy full of stars. — A view of a galaxy full of stars. Photo by Pixabay on Pexels.com

If we are wrong about the fact that the expansion of the universe is accelerating, then perhaps we can observe more galaxies in the future. But if not, then there are galaxies that we have never observed, galaxies that we can never observe, and there are galaxies that will become unobservable in the future. Depending on the size of the universe we may never be able to observe more than an infinitesimally small portion of the universe. Again, the universe is stopping us from knowing something.

Gödels incompleteness theorem(s)

The theorem states that in any reasonable mathematical system there will always be true statements that cannot be proved. In other words, to find a complete and consistent set of axioms for all mathematics is impossible. There are forever hidden truths in mathematics. For the case of natural numbers this means that there will always be statements about natural numbers that are true, but that are unprovable. I can add that there is also a second incompleteness theorem that states that a formal system cannot prove that the system itself is consistent. Basically, there are limits to mathematics set by logic.

There are forever hidden truths in mathematics in the form of unprovable truths. Photo by Andrea Piacquadio on Pexels.com

I have a book on Gödels incompleteness theorem, which I have not read, but I will read it before I make a post about it.

Some other topics for strange yet important facts

Light speed in vacuum is a Universal Constant.
Time is relative (deeper explanation).
Non-simultaneity and simultaneity are relative.
The strange double slit experiment.
Quantum Tunneling.
Quantum entanglement.
The Monty-game-door trick, 3-doors.
The butterfly effect.
Mandela effect.

What do you think about mixing in some very strange but important facts in my future blog about facts people dispute even though they are known to be true?