Is Cantor’s diagonalisation argument nonsense (what’s up with infinity and zero)?
In set theory, Cantor’s diagonal argument, also called the diagonalisation argument, the diagonal slash argument, the anti-diagonal argument, the diagonal method, and Cantor’s diagonalization proof, was published in 1891 by Georg Cantor as a mathematical proof that there are infinite sets which cannot be put into one-to-one correspondence with the infinite set of natural numbers.
I’m not sure how relevant or valid this proof is nowadays. I’m not a mathematician. I only learned about this ‘proof’ just a few months ago and viewed the video on the topic one more time just recently.
Again, I’m not a mathematician but I was amazed at the sheer stupidity of it. I’m sorry. And I think I have a very good and sound reason for it. So I challenge every mathematician to come with the counterargument as to why my personal arguments are wrong and my reasoning is false.
I use only pure logic and no formal mathematical notation, I simply just can’t. Well, then stick to what you do know, and let math to the ‘elite’ I can hear some say or think. But if a simple human being like me is able to disproof a well known argument just by logic, that would proof mathematicians get lost in their formal language and make things overly complex and complicated, creating all kinds of paradoxes while they do. But hey, if you’re the smart guy that can easily counter my arguments, I’m more than willing to listen and learn.
It goes like this: Cantor starts with an infinite set of natural numbers that are mapped to an infinite set of for example binary digits.
Well, here’s your first error. The notion that time is key in the concept of infinity is something that somehow escaped the attention of clever mathematicians. You see, if you are creating an infinite set, that means you’re busy with it… well for eternity, you can’t stop. At the moment you stop, you end up with a finite set. Infinite means it has no end, it goes on and on and on and on.
Infinity is not a ‘state’ or value, infinity is time related and means something goes on forever. To say ‘Here’s an infinite set of something’ is by definition not true, because at that moment it is (still) finite, you can keep adding stuff to it. We can only speak of infinity if it goes on forever, i.e. it is a concept of time, not of some value.
If there would be an infinite set of anything at this moment, there would be nothing else left, now would there? It would have consumed all space, matter and energy in the entire universe. If there was even one single neutrino left, this could be added to this set, so up tot that moment it would still be finite.
Mathematical operators are references to operations, literally actions in time. You take some amount of something, you add some amount to it and end with a bigger amount. That’s causality. You can’t have an instant infinite set of anything, it must be created first, it would otherwise violate the law of causality. But you need infinity to do so i.e. the operation will never stop.
Let’s assume we could ‘freeze’ the universe at a random moment and we could count all the particles, all the distances, all the energies, etc. It would all be finite. Only if the universe goes on, expanding forever, we can truly speak of ‘infinity’.
The term infinity in math is ill-defined and that’s why all these weird paradoxes show up every time infinity is discussed. What is 2 times infinity? What is infinity plus 1? What is infinity minus 1? What is infinity divided by infinity? All pointless and nonsensical questions.
Infinity is not a number, it is a concept. Besides that, it is self referential because it references nothing else than itself. The proof for this is that in set theory the infinite set must contain itself because it contains everything.
So when Cantor says ‘Here’s an infinite set of natural numbers, mapped to another set of…etc), that’s just plain wrong. But let’s for the argument sake see if there is something else wrong with Cantor’s argument. Let’s assume we have an infinite set of natural numbers at one moment. What happens then? Well he simply lets one set go on and be expanded even further, by playing the diagonalisation card while the other set is not allowed to do so… He tricks us intor thinking that this set is already ‘infinite’. Hey, that’s cheating! Infinite means it must expand and expand and expand… forever but one set is not allowed to expand any further, while the other set is allowed to do so? You really think you can’t add more natural numbers to any set of natural numbers. Of course you can, you even MUST if you want to make an infinite set… forever. Again, no set is infinite at a given moment in time.
So here’s the issue with infinity, no set is EVER complete if you let it run for infinity because you can always add one more. So if you would order a monkey — where have we heard this one before — with an indestructible typewriter with infinite lint, infinite paper and a infinite source of banana’s (you see the sheer nonsense here?) to type out both the sets it would never run out of natural numbers, nor out of whatever binary digits or fractions or … whatever. But if it eventually quits at some moment because it gets bored or something like that what will remain are two finite sets, both equal in length.
So the issue here in my opinion is that we just can’t use infinity to do calculations, we can only use it to show there’s an error in our formula’s and we have to fix something. If you write a computer program, and you create an endless loop, you’ve done something wrong. If you do some math and hit infinity you know something is wrong. There are no infinities in this universe, not even inside a black hole. Don’t try do do clever things with infinity, it will blow up in your face. Again, it’s a concept not a number, not even something that you can describe for then it would not be infinite anymore.
To put it all into a bigger perspective, the one and only ‘thing’ that can ever be infinite is space (or spacetime?) itself because it holds everything else and will expand forever. And if this is a ‘set’, it is the only set that also contains itself because space is not expanding into something — according to scientists.
And the idea that in an infinite universe anything not only can happen but will happen an infinite amount of times? I once heard someone state this. Some silly thought experiment about a monkey on a typewriter, hitting random keys that would eventually not only type the complete Bible, but type it an infinite amount of times. I’m sure that is exactly NOT what will happen. This statement is closer to the truth:
In an infinite universe anything will happen exactly only once. Because in an infinite universe there are infinite possibilities and the chance that something happens ever again is one to infinity exactly.
My suggestion would be to replace the notion of infinity with some other concept, something like indefinite. In other words, something we can’t define because it will run for an indefinite amount of time.
What about zero?
When Cantor postulated set theory, the problem with it was pointed out by Bertrand Russel that there is a problem with sets referencing themselves (using the barber analogy). And so, the revised set theory (by sir Mellow et all) excluded sets that reference themselves. (Like the set containing all sets and the collection of sets not containing themselves). Nice.
But the empty set…. is a self-reference too, for it can only reference itself for there is nothing in it, so nothing else to reference. When we speak about this set, we speak about the set itself for it holds nothing. So here is the paradox: It is a set containing nothing are therefor it must contain itself because it is nothing… but then it contains something…. which is …nothing…etc.
So, the empty set is not a set to begin with, according to the revised set theory for no set that references itself is a valid set.
And so, zero is ultimately a self-reference because it cannot reference anything else, no value, no quantity because zero is… well zero.
To state there are 0 cows in a field is as silly as pointless, just as silly as stating there are 0 Ferrari ‘s there. It explains why dividing by zero is nonsense. Zero is a concept and not a number and is self-referencing by nature. A (true) number is always a reference to something, some value, some quantity. Zero is not.
The use of 0 in a decimal system relies on convention, in the same way bits in boolean Algebra have a specific value based on their position and is therefore very different from the meaning of ‘zero’.
The reason zero works in most cases in math is because we learned to avoid the pitfalls carefully.
But something that has a correct grammar or syntax according to some language, can be conceptually nonsense.
Take the statement : “I have zero cows”. That seems like a true and clear statement, but conceptually it is nonsense. You don’t have them, so conceptually you should say: “I don’t have any cows”.
Is this relevant or important? Well, hell yeah? The goal of the language of math is to be precise and consise. Take for example these statements:
12 * 0 = 0
and
42 * 0 = 0
so according to the rules and grammar of math we can state that:
12* 0 = 42 * 0
Now we can eliminate the zero’s at both sides and keep the statement:
12 = 42
Do we ever do this? No it’s conceptually wrong. But the syntax and the grammar is correct! With any real or natural number this would work an give a valid out come, which proofs that zero is not a number but a concept. So the language of math is flawed because both zero and infinity are ill-defined. Does it have consequences? Well, that’s something I leave up to your own judgement. Are there more paradoxes and weird things in math that relates to zero and infinity? Go find them! Personally, when I listen to theoretical physicists I hear a lot of BS these days, relating to life, the universe and everything.
