You must temper your confidence with reason. You are so caught up in trying to avoid any show of weakness that you are, no offense intended, making a fool of yourself.
You have, so far, demonstrated many of the characteristics of people we call
crackpots.
(1) You are extremely insistent on your correctness.
(2) You base your position on "common sense" -- and the assumption that everybody else's common sense must agree with yours, and those who disagree with you are simply refusing to listen to their common sense.
(3) You base your position on your inability to imagine the alternative -- and the assumption that everybody else must also be unable to imagine the alternatives.
The typical crackpot also tends to ignore any evidence that they may be incorrect -- you've one-upped them, though: you've
acknowledged evidence you may be incorrect (you know full well how to construct a line segment of irrational length), and have come to the realization that there's a problem... yet you still maintain your position with absolute confidence.
I hadn't noticed these tendancies in your earlier posts -- I suppose you're just responding to a perceived attack on your person?
Of course I'm not saying you're stupid: everybody has their own capabilities.
However, I am explicitly saying that you're being blinded by your egotism. You refuse to accept that everybody does have their own capabilities, and that someone else might be able to do something you cannot at the moment, such as being able to "fathom" infinity.
In my estimation, based on watching many people assert this point, the main reason many people have trouble "fathoming" infinity is simply because they're convinced that it's "unfathomable" -- I could tell them about all sorts of things like cardinality, the extended reals, and non-archmedian fields. I could tell them about how using the adjective "infinite" is usually more appropriate than the noun "infinity", and so on. However, these people will invariably respond (roughly) that I cannot possibly be speaking about infinity, simply because I understand what I'm talking about.
This is another example of the same problem. You hold this belief that your ability to think and reason is absolutely perfect -- anyone who disagrees with your conclusions must be mistaken or insane or something.
You were responding to this quote of HoI:
It is true that if you try to write out an irrational number in decimal form, you can't do it but that has nothing to do with "plotting a length". Every real number corresponds to an exact point on the number line (that is, basically, the definition of "number line".).
Where he makes a point of separating the notions of the "decimal form" and a "real number".
One of the hangups some people have, yourself included it seems, is separating these two notions. In their mind, a decimal number is the "only way" to write a real number. When they see other ways of denoting a real number, such as:
\sqrt{8} is the unique number
x such that
x² = 8 and
x > 0
or
\sqrt{8} is the length of the hypotenuse of a (Euclidean) right triangle whose sides have length two
they think "These denotations are simply different ways of saying 2.828...".
Another hangup people have, again yourself included it seems, is that the notation 2.828... "really means" that you're supposed to start with 2, and then you continue on to 2.8, and then to 2.82, and then to 2.828, and so forth. And since this algorithm never finishes, they think that the decimal notation cannot represent an actual number. Since they believe that this is the only way you're ever allowed to actually handle a real number, they have problems.
Incidentally, they aren't
too far from the mark -- but they refuse to budge at all, and thus never see the light.
In the rigorous sense, a decimal number
is simply a function that allows you to compute something called the "n-th digit", whatever that means. When I write something like:
0.454545...
this is shorthand for "the n-th digit of this number is 4 if n is negative and odd, 5 if n is negative and even, and 0 otherwise". (The places are numbered ...(3)(2)(1)(0).(-1)(-2)(-3)...)
This
is something we can manipulate in its entirety -- we're not doomed to forever add digits one at a time, never getting anywhere. I can, for instance, add it to 0.545454... and prove (in finite time!) that the result is equal to 0.999999..., which is known to equal 1.
I was hoping it would make you aware that you are projecting yourself onto others -- that maybe you would come to realize on your own that not everybody will agree on what you maintain is "common sense".
Another characteristic of logical debates is
burden of proof. When we say to you that there does exist a point on the number line for every real number, you are (generally) justified in asking us to prove our assertion, and it would be unfair for us to ask you to either accept it or prove us wrong.
Conversely, when you tell us that there cannot exist a point for each irrational number, we are justified in asking you to prove your assertion, and it is unfair for you to ask us to accept it or disprove you.
To state it more succinctly:
When someone makes an assertion, the burden is on
that person to justify their assertion. There is
no burden on everybody else to disprove that person.
You're wrong.
If some whacko on the internet defines "the number line" to consist of all barnyard animals and nothing else, then in that context, "the number line"
really does mean the collection of all barnyard animals and nothing else.
However, all mathematicians have essentially agreed on the default meaning of "number line". Furthermore, it is exactly this "number line" that is taught in schools.
So, when no alternative is specified, the term "number line" refers precisely to what is written in that book from the PhD from CIT.
And finally
If you want to make this about strength, instead of mathematics, then I'll win: I wield the bigger stick.
) -- I was trying to point out some of the behaviors associated with crackpotism so you can avoid them! (Yes, I'm aware it sounds an awful lot like calling you a crackpot, but I felt it important to say, and it's difficult to say it without sounding as such)
