Is the Axiom of Choice Controversial??

As I think more and more about the essence of cardinality, my opinion on the subject shifts, so what I believe one day may be different than what I believe the next. Currently though, I am completely convinced that a line with a length of two inches has more points than a line with a length of one inch, despite the ability to create a bijection between the two lines. The longer line is bigger, just not UNMATCHABLY bigger. But a line of any length is certainly unmatchably bigger than a collection of separate points. Countable infinity is unmatchably bigger than any finite number, and uncountable infinity is unmatchably bigger than countable infinity. Still, that doesn't mean one countably infinite set can't be bigger than another countably infinite set. The integers are bigger than the naturals, for instance.

However, I do not disagree with all of the findings of set theory. I believe the irrationals are far more numerous than the rationals even though you can find an irrational between any two rationals and vice versa. And even if a good chunk of the ideas in set theory are false, it's interesting to explore the things their falseness might yield.

I'm toying with the idea of rates being used as a measure of the size of infinite sets. Here's an example: Let's say you have an infinite number of points evenly spaced along a number line, and let's say each point is a vertex of an equilateral triangle with sides less than the distance between points on the number line. The vertices of the triangles will always be three times the points on the number line, because both are increasing at constant RATES.

A few other things to note:

-Obviously I had a mic problem when I recorded the word "area" at the beginning, but I didn't feel like recording everything all over again for just one word, so I left the problem uncorrected. The rest of the sound quality is fine.

-This video applies some ideas explored in a previous video I published. The previous video is available here: http://www.youtube.com/watch?v=URAvLTbBcHY

-I take some liberties with the word surjective at the end of the video, but I think under the circumstances such liberties should be permitted.

Category:

Education

Tags:

• axiom
• of
• choice
• set
• theory
• mathematics
• dispute
• controversy
• indeterminate
• Georg
• Cantor
• countability
• cardinality
• bijection
• surjective
• objective
• one-to-one
• onto
• two-to-one
• infinite
• size
• element
• choose
• true
• false
• proof
• philosophy
• logic
• reality
• fantasy
• Hilbert's
• Paradox
• the
• Grand
• Hotel

[Rokker815]

@theboombody can I just remind you that anyone can post anything on wikipedia and that wikipedia articles are not usually mathematically rigorous. You are interpreting this statement (which I have read, along with the rest of the article) incorrectly. I have explained to you many times now how you should be interpreting statements like this. I have run out of patience with you; I am sorry. Continue thinking whatever you like, but I assure you: you are completely unequivocally wrong.

[theboombody]

@Rokker815 To be honest, I didn't understand most of the words in the hyperlink. But as bad as my memory is, I do remember things people have told me in the past.

Wikipedia cardinality article - "Cardinal arithmetic can be used to show not only that the number of points in a real number line is equal to the number of points in any segment of that line, but that this is equal to the number of points on a plane and, indeed, in any finite-dimensional space. "

[Rokker815]

@theboombody it doesn't matter what equinumerous sounds like it means. Point is, you didn't even bother clicking the hyperlink to check what it really meant. And at no point has anyone EVER told you "For sure, these two lines of different length have the same number of points exactly." This is what YOU, with your limited understanding, have inferred from what you have read. Ever even questionned how a set with an infinite number of elements can possibly have a "number" of elements?

[theboombody]

@Rokker815 Equinumerous is an extremely misleading word in this case, you have to admit. It would be like me defining a tree as "water falling from the sky." Mathematically, I can do that, but it sure leads people to think I'm talking about rain.

I admit a slight obsession working with things like infinity which cannot logically be grasped. I just didn't like someone telling me, "For sure, these two lines of different length have the same number of points exactly."

[Rokker815]

@theboombody [cont] (0,1) and (0,2) have the same cardinality, right? Then by DEFINITION, they are equinumerous. At no point has Dedekind or any other mathematician formally said anything about the same "number" of elements since such a statement when dealing with infinity is MEANINGLESS!!!!!! I still don't get your problem! You seem to have FUNDAMENTALLY MISUNDERSTOOD the article to which you referred me. How can you disagree about a DEFINITION??

[Rokker815]

@theboombody [cont] subsets of infinity equalling infinity. You need to read the article again. There is nothing in there that should be causing you concern. It is not saying that the subset (0,1) is EQUAL to (0,2), but since (0,1) is a proper subset of (0,2) and can be mapped bijectively onto it, (0,1) and (0,2) are equinumerous, which if you bothered to read the article on what that means, it doesn't mean same number of elements, it means same cardinality. You agree that [cont]

[Rokker815]

@theboombody [cont] contains the infinite subset {x: 0<x<2, x is rational}, but this does not map bijectively onto (0,2). It is a different cardinality: countable. This means there is a way of listing the elements such that it takes a finite time to count to any given element (sure, an infinite amount of time to count them all, but given an element of the set, I can be sure I'll eventually reach that particular element). And nowhere in the article does it say anything about [cont]

[Rokker815]

@theboombody [cont] IF a set contains a proper subset which CAN be mapped bijectively back on to the original set, (so we don't always HAVE to choose that map, but if one exists, we're in business for proceeding with what the definition has to say), then we say that original set is Dedekind-infinite. What is the problem here? (0,2) has a proper subset (0,1) which can be mapped bijectively onto (0,2) does it not? So by definition, (0,2) is Dedekind-infinite. (0,2) also [cont]

[Rokker815]

@theboombody if two sets are infinite, and both are at the same "level" of infinity (i.e. "size class", cardinality) then indeed both sets contain the "same number" of elements. The problem you continue to have is to try to give meaning to this number. There isn't really a "number" of elements, since they are infinite. They are the same level of infinite though, and that's good enough. Finally, if you read that wikipedia entry, almost everything on there is just DEFINITIONS. [cont]

[theboombody]

@Rokker815 I called the "level" of infinity a "size class" earlier. And I agree with all you're saying here, but many do not. Go to Wikipedia and see what they say about Dedekind-infinite set. It says a bijection is the same as being equinumerous. It says a subset of infinity is equal to infinity. Do I agree? Heck no, but it seems there are people out there that do. That's why I have to make videos like these.