I'm watching the first video now, and I can say this dude almost creeps me out... if I were able to feel creeped out. Maybe that's not the best way to describe it, but he looks like he's prone to... uh... cracking? Going crazy? Doing something wild and unexpected? It's funny, I've never really gotten this feeling from more than a handful of people before, lol. It's definitely his smile and how bugged out his eyes are.
Anyway, now that I've finished the video, I'm not necessarily sure how I feel about the information presented. A lot of it mirrored what I already posted, but some of it I need to look up because the video left me asking questions that were never answered. Unless my questions are already answered (which I assume they indeed are, and the dissonance here is caused my lack of understanding... or rather lack of knowledge, leading to a lack of understanding) else where, I'm not sure I can buy into the theory that there are some infinities that are larger than others. I feel like the arguments presented and the examples given were more a result of the semantics our language used to describe the math clashes with the actual math.
For instance, what he presented as the "bigger infinity" is only came about as a conclusion due to the inadequacy of viewing any given number from a single perspective. That is to say, 0.5 can be written as 1/2, 0.5, in base 5 it is written as 0.2, etc. You can write a number however you need to write it, and you can assign units to any number so as fix certain "impossibility" errors you would get from his diagonal number he changes based on a rule that wouldn't be listed, and it would allow for that number to be listed when previously it could not have been based on the limitations assigning language to mathematical concepts imposes. Using a single number system to describe or write a number also poses limitations, using a single base number system as and its inherent inability to list all fractions in a given range due to the inherent nature of infinity is flawed. Expanding the given range allows you to reach all real numbers. You can impose finite limitations on the number line, and you will never reach the beginning, nor the end no matter which end you start counting from. His logic is faulty two fold, as I see it. A) you could never reach any new number using his logic when trying to meet either end of a finite range. How can you go from 0.1 to 0.2 when there are an infinite number of decimals between them? 0.100000000000000001, 0.0001, 0.111111111111111, 0.1212343, etc. There is never a point you could change numbers at all based on that type of thinking. The reason I find for this "paradox" is one that actually explains that it is no paradox at all. The premises behind the argument are incorrect, which is what leads to this mind boggling issue of seemingly being unable to ever reach a different "number".
The reason here is because infinity is actually seen as a preventing factor in ever reaching the beginning or end of a finite range. By the very nature and definition of the concept, the beginning and ending numbers must be included in the range you have chosen. Just because you could endlessly be writing out decimals between 1 and 2 doesn't mean that "infinity" does not include 1 and 2. To suggest otherwise means a fundamental misunderstanding of what infinity truly "is" must be occurring somewhere. The other idea is that any a given range is actually finite in itself. Functionally it makes sense to call writing all the possible numbers 1 and 2 "finite", but really it isn't. What are 1 and 2 over 10 and 20, 0.1 and 0.2, or any different set of numbers you can apply a given rule to that you apply to all numbers in between? You have an infinite set of choices before you in making different numbers to chose between. Say you square all numbers in your range, you wind up with the decimals and numbers between 1 and 4. You might say that introducing the rule means introducing new numbers to the list, but truth be told I don't see why the results of any possible rules applied to the numbers in a given range can't be included part of that range. When you run the rule backwards you can return to what you would call as the "smaller infinity" that was originally generated by the numbers and decimals between the original range. Their values are all equal, yet given the possibility of an infinite number of rules for changing the numbers, every number you could possibly encounter can be found in that list. Trying to call one infinity larger than another infinity because what our intuition says about the different examples we can generate that seemingly allow for a greater size of infinity to exist is flawed. That conclusion can only come about if you willingly leave out numbers from the list that should be included in the list in the first place.
I'm foreseeing arguments that my argument basically states that no given numerical value can be greater or less than another, but that simply isn't true. 10 is still greater than 5, 1 still less than 3. I mean, what's the difference between saying that and trying to list all the numbers between 1 and 10, and 1 and 5? Shouldn't there be more between 1 and 10 than between 1 and 5? While it's true that there are more numbers between them, if say, we count by 1. You have 1, 2, 3, 4, 5, 6, 7, 8, 9, and 10 in the range of 1 and 10, and 1, 2, 3, 4, and 5 between 1 and 5, and we can easily count that there are 5 less numbers between the two ranges. However, my argument doesn't make this observation false. If you noticed what happened when I listed those whole numbers, I was purposefully excluding numbers that should be included in these lists (when trying to infinitely list numbers in between) because they did not fit my "rules" for being written. I am not dealing with infinity, I am dealing with whole numbers. When it comes to whole numbers, there are twice as many between 110 than 15. The guy in the video said it himself. Infinity is not a number, it is a concept. In order for the defining criteria of infinity to be met, or in other words for him to use the word infinity accurately in his examples, he must include every number possible in his range, no matter your start point or end point. The starting point and ending point do not affect the numbers in a list when you are truly making an infinite list. Believing some numbers are missing is merely an oversight caused by not thinking of what you are truly doing all the way through. When it comes to infinity, because it is not a finite value represented by a symbol of our choosing (it is a symbol representing an infinite value, after all) it is not bound to the rules and logic that follows for symbols representing finite values (like 1, 5, 0.67, 2/3, 7 to the 8th power, etc.). Any finite starting and ending points in a range where we list all the infinite numbers contained between are as good as illusions. No matter how you slice looking at infinity, it's always going to be infinite in every aspect you look at, at any perspective you view it from. It will have infinite decimals (divisions) in between, infinite powers, infinite multiples, infinite differences, infinite additions, infinite anything you can think of. You can make any combination of those functional operations that you want, there are and infinite number of them, and there are an infinite number of answers for those infinite operations.
The second you talk about counting infinitely between something, you are talking about all numbers possible. Or more precisely, a value that never ends (and even never begins in a way, lol). No matter what, you will find the same numbers in one of these supposed "smaller" infinities that you find in a larger infinity.
edit: a few of my arguments are poorly written and some are invalid, after rereading what I wrote. I'll try to compile a more concise, and valid argument here in a while.


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