.Miscellaneous snippets of information on any subject
I have enjoyed discussing infinity in earlier blogs, today I am moving to the other end of the scale. The complexity and detail that occurs in the nanoworld is stunning. Click on the link below to reveal the relative size of the fundamental objects of life on an interactive webpage created by Learn.genetics, the genetic science learning centre. Advance the slider to fall deeper into the rabbit hole!
Pigs can fly!
Another apparently ridiculous assertion! After all, did I not state in my last post that infinity is a concept that describes something unbounded? So how can anything be greater than something that is unbounded? The answer is that it depends on the way it is unbounded. Lets consider the infinite list of all the positive numbers in sequence 1,2,3 etc. You know that the list is complete because you have not missed any integer. After all, by definition you have listed them in order (2 definitely follows 1 and 3 definitely follows 2 and so on).
Now think of the infinite list of even numbers listed in sequence 2,4,6 etc. Again the infinite list of these numbers is complete since by definition you have listed them in order. You can now map each number on the first list to each number on the second (1 maps to 2, 2 maps to 4, 3 maps to 6 etc). There is no number on the first list that you cannot map to a number on the second list since any integer can be multiplied by 2 and every number that is divisible by 2 is on the second list. These lists are both unbounded but they are unbounded in the same way (because of this one to one mapping) and are known as “countable” lists.
Now try and do the same thing with the positive real numbers. You write the first positive real number down ….umm, what are you going to write? Perhaps you think of 0.1 but then you know that 0.01 is less than that. So where are you going to start? You literally cannot write down a list of all the real numbers in sequence as whatever number you start with you have missed out a number smaller than it! As a result you cannot possibly map each integer on our list of integers to a corresponding number on the list of real numbers because whatever real number you choose to map to the integer 1, there is another (actually an infinity of other) smaller real numbers that you could choose to map it to. So you see that the list of all real numbers is not even countable and therefore it is a bigger infinity than the list of all positive integers (or the infinite list of any other countable things). You think that must be it, but what is even more astonishing is that mathematicians have identified lists that are even more unbounded than the list of real numbers. Good grief, my head aches…time to stop!
Photo by Freddie Marriage on Unsplash
What about infinity plus a million? You guessed it: they are equal. Infinity + 1 = infinity plus a million = infinity! This does seem counter intuitive until you recognise that you are only lured into thinking that infinity plus something is bigger than infinity because you are unconsciously treating infinity as a number, which it is not. Infinity is an expression used to describe a concept of something that has no bound. This is not something that we are used to handling within our highly bounded 3 dimensional physical world!
Indeed “infinity plus one” is virtually the same as saying “imagination plus one” and only appears to make slightly more sense because we know that the number line is also unbounded and we are usually first exposed to infinity in that mathematical context, which understandably establishes a connection between the two in our minds. Notwithstanding, the connection is certainly not that they are both numbers!
Photo by Gabriel Manlake from Unsplash
I am fascinated by infinity, mainly because it's a wonderland of provably bizarre but true counter intuitive concepts (like Schrodingers cat or Einsteins time dilation effect) and such things are always interesting. So, I plan to embark on a short series of infinity related blogs.
I will start with a wonderfully evocative description of eternity (the "time" manifestation of infinity). Unfortunately I do not know who came up with it so I cannot give a proper credit. An internet search appears to credit different people but none with any apparent authority so I will put it down to that well known genius "anon"!
"Think of a ball of steel as large as the world, and a fly alighting on it once every million years. When the ball of steel is rubbed away by the friction, eternity will not even have begun."
This description of a minute fraction of eternity is so large that we can hardly take it in. Here's a great article titled Putting time into perspective that nicely presents some ideas about why.