Technically, zero is a number. A variable is a symbol that stands in for a number that isn’t necessarily fixed. Don’t know if that makes a difference.
There’s a difference between “natural numbers” (or “counting numbers”) and “whole numbers”, that’s not too important to what Magnus was arguing. Magnus was arguing that NULL is not a mathematical variable to attempt to “prove” that all infinite series are not the same size!
I’d like to see another argument from him to this regard.
I’d like to see the first argument. (No, I won’t scroll up and read).
Hmm, doesn’t sound like an argument I made.
Are you referring to this?
EDIT: Checked the link you posted. So no, this is not it.
Honestly Magnus !!!???
You’re going to play dumb here?
Remember your:
1.) 1
2.) -
3.) 3
4.) -
5.) 5
6.) -
Argument !!! This argument was used by you to “prove” that some infinite sets are larger than others!!!
Null is still a fucking number Magnus … it’s still in 1:1 correspondence
Decimal notation. Whole numbers over one, and fractions of 1 over a whole number.
We all know 1/2 of 1 is one half, 1/2 of 1/2 is 1/4, and 1/2 of 1/4 is 1/8… 1/2 is larger than 1/4 which is larger than 1/8, 1/8 is larger than 1/16 which is larger that 1/32, which is larger than 1/64, 1/64 is larger than 1/128. 1.5 is larger than 3/4 but it is half of 1.5.
1/9 is larger than 1/99 which is larger that 1/999.
[attachment=0]999999.png[/attachment]
I may live in my own little world but decimal notation says there are no 1’s in .9 recurring, some of you may be forgetting that 0.9 recurring is another way of saying 10/9 recurring. But I could be wrong…
Is 1.0 = 10/9 recurring?
I propose that any of the digits 0 - 9 in a base-ten system repeated infinitely isn’t a number. But I would venture the thought that any digit recurring over itself recurring is 1.
I would also agree with Magnus that 1/3 is not equal to .3 recurring. It is unresolved in a recurrence, at best, it is as close as a base-10 system allows you to get without being equal. When a recurrence shows up it is telling you it is not equal.
Null is not a member of the set.
1 → -
2 → 2
3 → -
4 → 4
…
What you call “null”, which is really a dash “-” that indicates an absence of element, is not a member of the set of even natural numbers.
Following your logic, all finite sets are equal in size just as well. Take a look:
1 → 2
2 → 3
3 → -
4 → -
That was supposed to show that there is something wrong with the method that you, Cantor and many others use.
The question is: given two infinite sets, how do you determine whether they are of equal size or not?
N = {1, 2, 3, …}
2N = {2, 4, 6, …}
Are N and 2N of equal size or not?
How do you decide?
Sure, you can put them in one-to-one correspondence but you can also put them in any other correspondence you want.
So what makes you think that just because you can put them in one-to-one correspondence that they are equal?
It either equals null or zero, take your pick. At this point, you’re arguing for the sake of arguing, you stopped using mathematical arguments. Your argument is psychosis, no bearing on reality. You want so desperately for there to be orders of infinity, that you have ceased rational discussion.
and yes, N and “2N” are equal in value, what so many people have tried to explain to you, is that you CANNOT use operators on infinity!!
At what point does the sequence terminate? Sure N and 2N have different values, but ONLY in termination, infinity does not terminate though, and so many people in this thread keep trying to explain that to you.
This invites a kind of meta-discussion – a discussion about discussion.
The following isn’t specifically directed at you, so you can safely ignore it, but if you want, you’re welcome to take it in and bake a response for it.
You can interact with people in a large number of ways – indeed, an infinite number of ways, a pretty large infinite number of ways (: – and each one of these ways have certain consequences; and each set of consequences can be compared to every other in order to determine the most preferrable one.
If, for example, you want to be respectful, I believe you absolutely must abide by the rule that says “Make sure that the other person wants to hear what you have to say”. Just in case, I will repeat, you don’t have to be respectful to other people, you act as you see fit, but if you want to be respectful, I believe that’s the way to go.
If you think it’s fruitless to have a discussion with me, that’s fine, and you’re absolutely free to act in accordance with that belief, by say, not trying to explain stuff to me. But by telling me things I don’t want to hear – e.g. that you think that I’m arguing for the sake of arguing, that I stopped making valid arguments, etc – you are being disrespectful. The question is: do you really want to be disrespectful?
And then there’s the general question of the extent to which it is useful to talk to people when they don’t want to listen to you or when they don’t want to listen to what you have to say.
If you think you can educate people that way, make them more intelligent, more capable in life, I don’t think that’s the way to go. Forcing people to do things against their will can certainly make them do things you want them to do but at the cost of becoming confused.
Let that be the end of this meta-discussion.
Let us return to the subject.
I don’t know what it means that N and 2N are equal in value. N and 2N are symbols representing the set of natural numbers and the set of even natural numbers, respectively. You are surely not saying that N and 2N are equal sets i.e. that they have the same elements?
What sequence?
Magnus… so basically “whoa”
You don’t even understand this discussion. That’s not on you, but on me.
Like I say “everyone is a genius, if you don’t understand something, it either wasn’t explained simply enough or there’s nothing to understand in the first place”
Slide Rule.
I played with Adobe Illustrator 88. You could crash the program by folding a segment of a line parallel to itself. The stroke cast was infinite. Sent the computer into a recurring loop, attempting to calculate where the parallel lines of the stroke width would converge. Never, and that caused some problems for a graphics program. So how many radial degrees of one will a computer program resolve too… a ridiculously more number of decimal points then were required to fly a man to the moon. But back then they had just 8 bits to work with.
en.wikipedia.org/wiki/Pierre_B%C3%A9zier
The founder of the mathematics upon which 2 and 3 D surfaces are defined within computer systems.
npr.org/sections/ed/2014/10 … n-the-moon
Just three points in space relative to an origin.
X over 1.1 over X
For example, 9/1 and 1/9 ?
And what are the decimal representations of those fractions?
Posting the same stuff over and over is not helpful.
I am still waiting for Silhouette to define “undefined”.
His argument so far has been that the word “undefined” cannot be defined. This implies the word is meaningless given that only meaningless words cannot be defined. But why would someone use meaningless words when discussing mathematics?
I have absolutely no idea what it means for a number of anything (not only 3s) to be undefined.
My best guess is that what he’s saying is that the number of 3s is unspecified.
But the number of 3s in (0.333\dots) is certainly not unspecified. The ellipsis tells us that the number of 3s is infinite. That’s not the same as unspecified. Unspecified means we didn’t specify (i.e. say) how many 3s there are.
Even if the number of 3s in (0.333\dotso) is unspecified, which it isn’t, how does it follow that (0.333\dotso) equals (\frac{1}{3})? If the number of 3s is unspecified, what follows is that (0.333\dotso) is indeterminate because it can be any of the following: (0.3), (0.33), (0.333) and so on. It means that (0.333\dotso) is a CATEGORY of numbers, not a SINGLE number.
On the other hand, phyllo made an argument that (\infty) is not a number because (\infty \div 2) cannot be expressed more simply. I responded by saying that (i \div 2), which is widely accepted as a complex number, cannot be expressed more simply either. His response was to conveniently ignore this point.
i isn’t a number. But at least it’s well defined.
Feel free to research it.