Does Counting Exist? - An Incomplete Mathematical Argument Using Cantor's Methods

[RJL738]

Try this concept. Yoctation, Zeptation, Autation, Femptation, Pication, Nanation, Micration, Millation, Centation, Negative Decation, Noventation, Negative Octation, Septation, Sexation, Quintation, Quadation, Negative Exponentiation, Division, Subtraction, Zeration, Addition, Multiplication, Exponentiation, Tetration, Pentation, Hexation, Heptation, Octation, Nonation, Decation, Hectation, Kilation, Megation, Gigation, Terration, Petation, Exation, Zettation and Yattation.

[theboombody]

@RJL738 Sounds awesome.

[Tikeslar]

First of all, the definition of a countable set is a bijective to the naturals. Claiming that a set isn't bijective to itself isn't possible in even the most minimal axiom systems.

Second of all, Cantor's diagonal is a general one about cardinality, the reals-naturals example is a classic result, but still an example. True, it's useful to functional analysis (and frankly, to all fields of math), but it's true home is set theory.

[theboombody]

@Tikeslar I made a follow up to this video that clears some things up. It introduces the concept of beginningless numbers to show that the whole numbers are not countable if you classify beginningless numbers as whole numbers. The naturals probably wouldn't include these numbers.

I also made a video where I show that a two-to-one bijection exists between two particular sets where a one-to-one bijection also exists. This creates a problem for me.

[claymullis]

@theboombody

The problem with your argument is that you say "beginningless numbers" can be classified as whole numbers. By doing this you have redefined whole numbers. The important thing to note here is that while whole numbers can counted infinitely, they by definition must have a least significant term. This is not the case with the Real numbers, which can by definition have no end.

Your argument only applies to your modified version of whole numbers.

[theboombody]

@claymullis Well said. I have to make a distinction between natural numbers and whole numbers for my argument to work. For a while I actually took my own video seriously.

[JamesTR4]

ok, so what if i take the other diagonal. And then construct a number with different digits from the top-right bottom-left diagonal. We still get a different number that isn't counted. Cantor's diagonal argument still holds, because you haven't removed any elements from the set of real numbers.

[theboombody]

@JamesTR4 Yeah, I made an error in this video, that's why I made a follow up one. But the follow up one is interesting in that it shows that the set of whole numbers is uncountable only if you include whole numbers with infinite digits, like the ever-increasing sum of a diverging infinite series. I call these "beginningless numbers."

[mynameisradar]

if you are right. does this mean that theorys involving numbers like string theory isn't real either? physicist use math alot to solve the greatest mysteries of the cosmos are you saying there wrong? if so can we ever figure out how old the universe is or what happen before the begning? string theory and m theory provide mathmatical answers to questions such as this.

[theboombody]

@mynameisradar Is it better to lie and have an answer or to tell the truth and have no answer?

I'm not against exploring to find the truth. My own video goes into an exploration of unfamiliar beginningless numbers territory. So people can look into string theory all they want, but you should be careful about instantly accepting what someone says. I don't care how good their reputation is. Nothing is above questioning. Math may be useful, but it may still have errors.

[mynameisradar]

@theboombody i see.

[00wassup00]

Of course counting exists. 1,2,3,4,5,6,7,8,9,10.

[theboombody]

@00wassup00 That's not real counting though. That's just "accepted" counting.

[theboombody]

@00wassup00 What about all the beginningless numbers and endless numbers inbetween?

[mateo3470]

See: Logic of Actual Infinity and G. Cantor's Diagonal Proof of the Uncountability of the Continuum by AA Zenkin

Source: Rev. Mod. Log. Volume 9, Number 3-4 (2004), 27-82.

"'Real"Analysis is a Degenerate Case of Discrete Analysis" ~ Doron Zeilberger

Look at P-Adic numbers. The mirror argument is interesting, but not rigorous.

[theboombody]

@mateo3470 (Not Rigorous = Sloppy), and the statement of equality I just put in parenthesis isn't rigorous either.

I like that Zeilberger quote. I want my quote to be truth is more inconsistent than lies. Does someone have dibs on it already?

[theboombody]

@mateo3470 Wow, those P-Adic numbers are interesting. Who would have known I'd be exposed to such wild ideas by simply posting an idiotic video?

[mateo3470]

@theboombody

This doesn't invalidate Cantor though because they aren't considered the "proper" definition of Natural number. But Zenkin shows the actual fallacies of Cantor in a more rigorous way.

[theboombody]

@mateo3470 I made a follow up video to specify that I don't consider his conclusions invalid. What fallacies did Zenkin find?

[mateo3470]

@theboombody

Well, don't rush to make that one. The mathematical community isn't to honest about this. David Hilbert said the following: "No one shall expel us from the Paradise that Cantor has created." That means they weren't going to care if contradiction came up. Zenkin showed that the ordering of the list actually does fatally affect the result. Furthermore, there is an arbitrary prohibition of reindexing. Read his paper!

[theboombody]

@mateo3470 That Zenkin guy is hard to find on the internet. I remember that Hilbert quote. I lost a lot of motivation to continue formal study of mathematics when I found an interval could be twice as big as itself. I was like, well, even sense doesn't make any sense, so I might as well find something else to do besides making sense.

[mateo3470]

@theboombody

The article is at Project Euclid.

I see exactly what you mean. It's the result of left-hemispheric criminality of those boring people who are too nihilistic to care if they are actually being intelligible. It's a great weeding out mechanism and initiation process to find those who will blindly accept such vulgar irrationality and claim superior understanding. Frauds flock together.

[theboombody]

@mateo3470 Yeah, to be truly open minded you have to consider two options when you don't initially agree with a popularly accepted point of view. The first option is to accept the point of view, even though you don't feel right accepting it, just because you know you are ignorant or may be ignorant of the subject matter. The second is to deny the point of view, feeling that those that have accepted the view are misled. Blindly accepting everything presented to you is not being open minded.

[mateo3470]

@theboombody

But the group has nothing to do with it at some level. The truth can be discovered outside their domain if the subject isn't completely socially constructed. Math is a type of map, so it is trying to map reality in some rough but general way. I think too many people define their views in terms of groups. That is nihilism even with a pragmatic criterion for truth.

"I love people, I hate groups. People are smart, groups are stupid." ~ George Carlin

[theboombody]

@mateo3470 Do you believe mathematical axioms are socially constructed? The axiom of choice seems to be, but what about really basic stuff, like: If A or B, and not A, then B?

[mateo3470]

@theboombody

Axioms can be socially agreed upon, but it's not always that direct. It depends on what kinds of maps these assumptions spit out. The axiom of choice seems reasonable enough at first glance until you find out that it assumes selecting elements from an infinite number of bins. It's not required for the finite cases.

[mateo3470]

@theboombody

In the end, logic is very useful. I haven't found many places where logic has steered me wrong. Of course, it is very hollow on it's own. The causation and atomic propositions and assumptions don't come with the toolkit. Concerning mathematics, it certainly shouldn't be created in formal systems, but ridding the excess should involve logic. Otherwise, the structure can become a Cantorian Tower of Babel.

[theboombody]

@mateo3470 I probably bash logic more than anyone I've ever seen. It's not because I don't like logic. Like you, I agree that it's extremely useful. But I don't think that it's flawless, and from what I've seen most everyone else thinks it is. I have to pretend I'm against it more than I actually am just to be heard. It's sort of like my tendency to bash sex a lot only because I think it should be the 16th priority in life, not the first.

[mateo3470]

@theboombody

What logic do you bash and with what do you bash it?

[theboombody]

@mateo3470 I'll bash any form of logic whenever I feel like it. I'm inconsistent though. Sometimes I feel like bashing it, and other times I feel like supporting it. I just enjoy being illogical sometimes. Quite often really.

[mateo3470]

@theboombody

As long as you aren't one of those real analysis creeps, I'll accept that.

[mateo3470]

@theboombody

Watch these funny videos:

watch?v=A6it_kQeOnU

watch?v=Y0Z0raWIHXk

[VeritySeeker]

I am sorry that I haven't seen this video until now. I've been very busy lately, but and it annoys me that so many videos pass me by these days here on youtube. However, the list that you construct in the middle of the video is actually countable, not uncountable. Hence so is the last sequence that you construct. They are all countable.

:)

[theboombody]

@VeritySeeker Shoot, don't feel obligated to watch my videos man. I just value your opinion. So you're saying all sets are countable?

[VeritySeeker]

@theboombody Obligated? No, no, I haven't watched videos on youtube for quite a while, and I was browsing through the videos in the automatic list, and this one caught my eye. I like your videos a lot.

No, I say that the sets you construct are countable and that the diagonal argument does not work on the sets that you construct. There exist uncountable sets, but the ones you construct in this video are all countable.

[theboombody]

@VeritySeeker It appears to me that the diagonalization argument works with any set of infinite numbers where each number has an infinite sequence of digits. How was the set I contructed in violation of this?

[theboombody]

@theboombody Constructed. Geez. It's too easy to make typos on youtube.

[VeritySeeker]

@theboombody There is a logical fallacy here. You are assuming that the number you construct from the list is inside the set you are working with. But it is not. You are actually proving that the set you construct IS countable by actually listing it. For example, take ANY element n from your list. Then n has the property that sooner or later there will only be zeros after some digit. Then the number 0.111111..... will never occur, and so hence it is not a part of the set you are constructing.

[VeritySeeker]

@VeritySeeker In fact, with this construction, you are not even reaching all the rationals.

[theboombody]

@VeritySeeker I guess what I'm really trying to say is, why can't the diagonalization argument be used on whole numbers as well as decimals?

[VeritySeeker]

@theboombody I thought about it for a while now, and hopefully I can explain it in the following way. Cantor made an assumption, and he assumed that he listed all real numbers. Then he showed that this list did NOT list all real numbers, hence a contradiction. The question you must ask to use the diagonalization argument is: What is your initial assumption? In the list you construct, there is no violation of the initial assumption. I'm not sure if I manage to explain this good enough. :)

[VeritySeeker]

@VeritySeeker Maybe I should make a video to explain this. I feel that my words and comments are not enough. It is difficult to explain without going deeper into the subject. In fact, you are posing a very interesting question here.

[theboombody]

@VeritySeeker So, if I prove a set is countable, and I can still use the diagonalization argument on it, doesn't that sort of take away the diagonalization argument's ability to prove uncountability?

[VeritySeeker]

@theboombody Say you define a set S as in the video. Then you try to list them. Your assumption is: "I can list all the elements in the set S." Then the diagonalization argument tries to constructs a number n that is inside S. Then the argument shows that the number n is NOT in the list. Hence a contradiction to the assumption. The argument in the video fails. It doesn't fail because you cannot list the number n. It fails because n is not in the set S to begin with.

[theboombody]

@VeritySeeker I take some mathematical liberties in the video, so I do accept your point. But I do think we need to separate how we define whole numbers and natural numbers. I would say a whole number is anything that has no non-zero digits to the right of a decimal point, including a number with infinite digits, which obviously would not be a natural number. So natural numbers are countable, and whole numbers are not.

[theboombody]

@VeritySeeker Darn, I always forget about that. Thanks for this post. I included the idea behind it in my follow up video.

[tenchoosethree]

On the other hand, finding a system of representing all natural numbers by infinite strings (which is what you did by appending infinitely many zeros on the left) doesn't make them uncountable. The diagonalization will fail as soon as you try to construct a new natural number. I wonder why you didn't give it a try at the end of the video. The new "number" would have had infinitely many nonzero digits, which doesn't qualify as a real natural number.

[theboombody]

@tenchoosethree You're arguing that the type of number I get as a result of switching numbers on the diagonal must match the type of numbers I originally used in the set. That makes sense, but I don't think I'm saying natural numbers are uncountable as much as I'm saying whole numbers are uncountable. "The way I normally count" can be classified as whole numbers instead of natural numbers can it not?

[tenchoosethree]

@theboombody Well, the way you normally count is for you to decide. :)

But yes, for diagonalization to work the newly constructed element must be an element of the set. The point of diagonalization is that you make the assumption that some set is countable, i.e. you can write down an infinite list of all its elements. If you can then show that at least one of its elements will always be missing you arrive at a contradiction and can conclude that your assumption was wrong.

[tenchoosethree]

@theboombody When you write down a list of integers, and then find something which is not in the list, but also not an integer you don't have that contradiction because you haven't shown that your list of integers was incomplete.

[theboombody]

@tenchoosethree Well, to me nothing is as contradicting as something being twice as big as itself. I made another video addressing that subject.

[tenchoosethree]

In fact, there will always be at least infinitely many real numbers that get paired with a representation of infinite length, no matter how clever a system you choose. It is this very property that makes the diagonalization argument applicable and consequently the real numbers uncountable.

[tenchoosethree]

The crucial difference between natural and real numbers is that every natural number can be represented by a string of finite length over a finite alphabet (most common choice would be {0,...,9}), whereas this is not the case for real numbers.

[tenchoosethree]

When you flip the whole list around, you conclude that mathematically nothing has changed at all, and you are dead right. Unfortunately, you do not realize how much right you are. You miss the point that the set you are arguing about has not changed, too. It is still the real numbers.

[theboombody]

@tenchoosethree The set I started with wasn't the real numbers. It was just a subset of the real numbers. And when the set was flipped, it didn't include one decimal, so that's just a subset of the real numbers as well. Everything is pretty much a subset of the real numbers except imaginary stuff to my knowledge. Even the natural numbers are a subset of the reals.

[tenchoosethree]

@theboombody Yes, everything you say is true of course.

I wasn't really referring to the intuitive meaning of these symbols (whether they represent integers, fractions or whatever). I wanted to point out that you were still arguing about a set which was composed of all (and not just some) infinite strings over some finite alphabet.

[theboombody]

@tenchoosethree Interesting.  I never thought of the concept of an infinite alphabet. Although I imagine the written characters in the Chinese language come close to approaching one.

[tenchoosethree]

@theboombody Well, it might seem that the chinese alphabet is closer to an infinite one, but it really isn't. ;)

From our theoretical viewpoint it is just as powerful as any alphabet containing at least one symbol. Although words in such a small alphabet would quickly become very big, it is still the same amount of words you can build in both of them: Infinitely many, but still countable :P

[tenchoosethree]

@theboombody Hmm, I just noticed that when you allow infinite-length words, the one-symbol alphabet becomes indeed less powerful than the bigger ones. Having at least 2 symbols, you can build an uncountable amount of infinite words (you can prove that by diagonalization). On the other hand, having only one single symbol, there is only one infinite word you can build.

[theboombody]

@tenchoosethree Interesting thought. I didn't expect to get such good comments.

[theboombody]

This video's giving me an idea for something I'd call transcendental whole numbers. Rather than being represented by converging infinite series, they'd be represented by diverging infinite series. No one ever seems to count those as actual numbers though. They never get any attention.