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Zero Divided by Infinity = All possibilities?
- Thread starter JustBusiness17
- Start date
JustBusiness17 said:
JustBusiness17 said:
JustBusiness17 said:
Josan said:
You seem to be confused. You see, a mathematical system has no internal semantics, only internal rules. As long as you follow the rules, the meaning of the outcome doesn't matter. Mathematical models only adopt significance when correctly applied to a phenomenon that follows the rules sufficiently closely for predictive purposes.
More abstractly, consider the field axioms.
We are guaranteed a multiplicative identity, 1
An additive identity, 0
A set of elements
A scheme for defining the model we need for both binary operations (in this case addition and multiplication)
Inverses for every element with respect to the binary operations
we are also given
commutativity (x+y) = (y+x) and (x*y) = (y*x)
distributivity x*(y+z) = (x*y+x*z)
associativity x+(y+z) = (x+y)+z
But lo, infinitely many constructs fulfill this! In fact, for every power of every prime number, there is a field of that size! They are called the integers modulo that prime to that power. Addition and multiplication are defined algorithmically.
For instance, GF(4)
0
1
x
x+1
with the characteristic function x^2+x+1 = 0
Code:
Multiplication:
| 1 | x | x+1
-------------------------
1 | 1 | x | x+1
x | x | x+1 | 1
x+1| x+1 | 1 | xJosan said:
You are correct in deducing this, however; this should indicate to you that your thoughts about how random mathematical rule sets (they might not seem random to you if you've only had exposure to calculus) might say something about reality are in fact similar in nature to astrology.
You need to realize that mathematics is inherently devoid of meaning outside of the rule sets (though you can develop many useful intuitive ways of navigating the rules so that they seem more meaningful). It seems as though you are not following a coherent rule set, but rather a very vague one that you have pieced together.
If you are trying to deduce facts about external reality from mathematical concepts, without input from external reality, then you are not going to do better than medieval science. Empiricism overrides rationalism ::necessarily:: I cannot stress this enough! There are uncountably infinitely many rational models for reality, but they are all worthless without investigation in to which one coincides with the way things actually are.
On Cantor and Infinity
You are very mistaken about Cantor and you have a very muddled conception of infinity. It is a counter intuitive subject matter and so takes a good amount of time to grasp.
Cantor was likely bipolar and delusional towards the end of his life. He suffered from mental afflictions that I can assure you did not result from his study of infinite sets. If you want proof of this, you only need to look at the multitude of (relatively) mentally stable set theorists who enjoy a much stronger grasp on the theory than Cantor did. I won't say whether Cantor's obsessive nature with regard to his work exacerbated it, but it's often highly romanticized in popular accounts of Cantor's life (few though they may be).
There are gradations of infinity. What is meant by 'infinite' in Cantor's work is probably infinitely more boring than you seem to have interpreted it as. You see, the two most common gradations of infinity are:
Countable Infinity
What is meant by this is illustrated by the natural numbers. There is no largest, but there is an obvious next number at any point in the line. If you give me 8, I can construct 8+1 and call it 9. There is no number N such that 8<N<9. It is not a defined term in the natural numbers. In other words, I can count the numbers! They are countable!
What is somewhat more remarkable is that this can be done with rational numbers! That is, the set f all fractions! The simplest way of showing this is the following:
Consider the number 133/231. Re write this as 1330231. This is a natural number, 1,330,231, so you see you can do this with any fraction, because the 0 is a place holder! So what does this mean?! It means that for every fraction, there is a unique number! It also means that I can count fractions up like natural numbers! They are countable!
and
Uncountable Infinity
This is a much stranger idea, and it applies to the real number line, which cannot be given an assignment of natural numbers. Why is this? It has to do with the fact that some of the numbers cannot be represented as fractions, but this is ancillary to the overarching issue.
The key to understanding uncountable infinity is the somewhat counter intuitive notion of non-finite representations. Consider the set of all finite strings of letters of the English alphabet. If we assign the numbers 1-26, we can encode strings in the following way:
ABC = 123
AAA = 111
ZZZ = 262626
As you can see, the set of finite strings can be encoded fully into natural numbers. But what happens when we allow strings that are not finite? We get numbers that are non-finite, yet distinct!
Consider that we might have a rule describing a string that is not finite:
Four Z's followed by an A and repeated. Then we get 262626261......262626261.........
Now consider that we might have a bunch of strings that are similar to the above string, but at the beginning they are different! Maybe it starts with AAB before it gets in to the ZZZZAZZZZA.... pattern! So it begins with 112.
If we count all of these sort of strings to be the same as (ZZZZA)...... proper, we have defined what is called an equivalence class. Equivalence classes of this kind are the part of the set of all strings, finite or infinite, that make it uncountable. It is not the ones with regular patterns, however, these are enumerable. What if some of the strings had no finite repeating pattern? They would not have a finite description!
How does this play in to a proof of uncountability? The following is Cantor's diagonal argument and you might find it very counter intuitive, though I would be glad to assist you in understanding it.
Consider that you have the set of every single binary string of 0's and 1's of infinite length and you lay them down, one under each other and place a number next to each number (viz A):
Code:
<i>
</i>1. 1010001001000101001.......
2.0100101110111011110........
3. 0100100100101000101.......
.
.
.
.Consider that the numbers and the order you choose them is is arbitrary, it doesn't matter which is first or second etc.
Now, consider taking the first string, and getting a string different than it by changing the first slot. Now we have:
0010001001000101001.....
Now what if we just took that first digit '0' and started making a new string with it? What if we went down to the second line and picked up the second number, a 1, and flipped that to a 0? Then we went down to three, and took the third number in the string, a '0', and flipped it to a '1'.
We have so far:
001
A string that is different from each of the strings in at least one of the first three spots.
So what are we doing here? We are looking down the diagonal, and picking up the numbers and flipping them. If we do this for infinitely many numbers, we have defined a new string that is different in the first slot from the first string, different in the second slot from the second string, etc. So it is different from every string in at least one spot.
What does this mean? IT means that no matter what countable collection of infinitely long strings you pick, it will not include all of the strings. You can always construct new strings that are not in the set. What does this mean? It means that there is no countable set that you can fit all of the infinite strings in to. So then there is no way to say that any infinite string is the first, second third etc. without missing some strings on the way. If this doesn't make sense at first, keep thinking about it for a day or two and see if it makes sense then. If not, I can attempt an alternative explanation if time permits.
Tying in the Real Numbers
How does this relate to the real numbers? Quite directly! Consider that a real number is a natural number plus a decimal number. For instance, Pi. 3.14159265358...... Think about Pi as a string; 31414926....... Now consider taking all of the numbers that are made up of an infinite number of integers placed one after another ( an alternative way of thinking about the structure of irrational numbers).
If we use the diagonal argument, we can change each digit at one place in each string to make a new string not in the list.
E.g.:
Code:
<i>
</i>1234567898789.......
2345345656777.......
13242354656488.......
12233445566778......
1234356667878999..
3265789000989898...Taking these first few strings, we generate the partial string 233449. If we hit a 9, we simply subtract 1 instead of adding 1. We can continue this to define a string that is not in the list, coming to the same problem at before with the 1's and 0's.
I hope this clarifies the nature of mathematical infinity to some degree.
Your Conjecture
You seem to have been very muddled in your thinking. You see, the mathematical concept of infinity is well defined relative to the real number system as far as conventional use in calculus goes. 0/Infinity is 0 by the limit 0/n as n goes to infinity. This extends generally.
1/n = 0 as n gets large
2/n = 0 as n gets large
What is really meant? What is really being said is that given a small number D, as small as you want, I can provide a value for n that makes 2/n of size D. That is, I can make the term as close to 0 as I want.
What is meant by 0/infinity is generally 0/n as n gets large. Any other conception of infinity has no meaning unless you explicitly define it. If you have vague ideas, then you aren't doing mathematics. You have to deal with closed definitions that take in to account ever aspect of what you are considering. You have to take a group of basic assumptions that allow you to infer all of the properties of the symbols involved.
So you should see that your claim that "taking into consideration the limitless nature of infinity, zero divided by infinity creates limitless possibilities for something within nothing." has no well defined mathematical or physical meaning unless you will explicitly provide us with the axioms, why you chose them, and how you derived your proof of the above conclusion, and furthermore, what you mean by limitless in the given context.
These issues are not easy, and take time and patience to grasp.
I also urge you to reconsider making grand assumptions like the idea that the universe came from nothing. We do not fully understand the nature of the origins of the universe, and the theoretical attempts to understand it require years of work to comprehend. Making such assertions necessarily will yield no resemblance to reality as you do not have the means to test your hypothesis. The answers nature gives us are often counter-intuitive, so you should not be surprised if physics does not at all line up with how you think it should. There is no alternative to evidence based reasoning in an attempt to understand our universe.
Zetetic said:
I am well aware of this, did I indicate otherwise? Perhaps I did, in that case I apologize, I wrote my comment in haste. I was just trying to clearify that 0 is a number, just like any other number. This might seem obvious to anyone who knows math, but I have on many occasions had tireing discussion only to find out that it is the misconception that 0 is a concept, not a number that is the problem. All I was trying to specify was that 0 is a number, to be exact the only real number that is neither negative, nor positive. I realize now tthat I phrasphed myself as if that is the sole definition of the number 0, which I never meant to imply, I was simply pointing out one of it's properties as a juxtaposition to the statement that 0 is solely a representation of nothing.
Zetetic said:
Okey now, this is not what I said at all. Again, I was simply trying to point out, that you can't make arguments based on reality when talking about math. I have never stated that the rules are random, nor do I consider them to be. And for your information, I have taken several courses in calculus, thanks for assuming otherwise.
So in summary. My comments were made in haste and were intended to add a slight clarity to his argument, as well as an attempt at some humor (which obviously failed). However, I think you misrepresent my comments entirely.
Other than that. Awesome post.
Master_Ghost_Knight
New Member
No! You have made the mistake to try and do math with apples, there is no such thing as apples in math. Zero is by definition an number a such that for any number b: a+b=b.That is it. The fact that you can establish a paralel between the proprety of addition to counting apples and zero as having no apples has absolutly no effect in math. There is an ent that represets nothing in math, a.k.a. Void, it is not a number but it has mindfucking propreties in set theory.lrkun said:
Master_Ghost_Knight said:
Zero divided by any number is equal to zero. That is math. Now with respect to your point, it does not make sense. I am not in anyway saying otherswise.
0/1 = 0 0/(1,2,...,n) = 0.
f(x) = 0/(n)
where n= (1,2,...,n)
= 0
to illustrate
If there is an apple = 1 apple.
if there is no apple = 0 apple.
if you divide no apples with an apple. you can't. Therefore, I was right when I said you can't divide nothing with something. Consequently it was divided zero times.
Hence 0/ any number except zero equals 0.
Master_Ghost_Knight
New Member
What do you mean?lrkun said:
Re-reading your post, I can atribute a second interpretation where you might wanted to introduce a graphical ilustration, I can see how you can object to that.
Anyways a graphical ilustration is not exactly how we show 0/n=0 with n=/=0 even tough is not a bad way to go an intuition of it.
Just on a side note, the proof goes like this: a division in fact doesn't exist in math as a full operand 1/b in fact means a number c such that for any b not zero c*b=1 and we call c = 1/b. So all we have to prove that 0*c=0 for any real c.
And now it is easy using the axioms:
1. The sum identity a+0=a, from here 0+0=0
2. The distributive proprety a*(b+c)=a*b+a*c
3. And finaly the existance of the simetric, for any "a" there is "b" such that a+b=0, and we call b = -a
c*0=c*(0+0)=c*0+c*0 <=>c*0+c*0-c*0=c*0-c*0 <=> c*0=0
Master_Ghost_Knight said:
My illustration did not miss the mathematical equation. It means what it says. 0 / (1,2,...,n) = 0
The topic is zero divided by infinity = all possibilities. I say the answer is zero not all possibilities.
-oOo-
Zero divided by infinity. I need to make corrections to my previous posts. Reviewing my answers, one can't really divided zero by a infinity, because in math infinity is not a number.
What to do?
Use limits - where one divides zero by a number. Next, divide zero by a bigger number. Then keep the divisor bigger as much as one desires.
Ex. 0 divided by 5 = 0. 0 divided by 50 = 0. 0 divided by 500 = 0. So on and so forth. The answer will always be zero. >.<
Please correct me if I'm wrong.
Master_Ghost_Knight
New Member
Well we simply say right of the startthat 0/inf=0 no matter how you get there, for all aplications it is a none issue. An informal demonstration can be done by majoration of the modulus that for any real number a, b and c with abs(c)>=abs(b) and b and c different from zero then abs(a/b)>=abs(a/c), since inf>c with c being a real number then abs(a/inf)<=abs(a/c). Now given a=0 and c=1 we have that abs(0/inf)<=abs(0/1) <=> abs(0/inf)<=0 and since abs is a function with conter domain R+0 then abs(0/inf)=0 <=> 0/inf=0.
Josan said:
I found your post confusing and hard to follow. I saw the specific phrase speaking about infinite possibilities and infinity being limitless, and deduced that it was likley that you were using a layman's definition of infinity and conflating it with mathematical definitions of infinity and drawing confusing conclusions. It appears that it is not the concept that confuses you or frustrates you, but rather trying to teach it to people who have varied layman's accounts of various concepts that they are not willing to suspend for the sake of understanding you. This might be partially because you are inexperienced at teaching, and likely partly because of general refusal to learn. Many people are more interested in appearing competent than in understanding what is being said. I have trouble with this, we all do, and our confused notions tht we cling to often get in the way of communication.
I would suggest explaining 0 historically, perhaps by bringing up Fibbonacci, and characterizing 0 as 'mathematical', rather than 'colloquial' or 'lay' 'zero'. I suggest not defining colloquial zero, as it represents what ever confused notion that is held by the person happens to be. Once you have established that mathematical 0 is not what they have encountered outside of mathematics, they might lose their biases. Then you must illustrate how zero is used and what rules it has. Make it clear that 0 is part of an abstract game, and that their existential considerations (if these are the problem you generally run into) are unrelated to it. They will likely admit that this is not what they were talking about and stop using the metaphor of 0 representing 'nothingness'.
You might make it clear that the formal mathematical use diverges from what they are driving at, and that their thoughts are strictly non-mathematical. 0 as a mathematical entity is clearly defined. When applied to reality, it is only defined with respect to an object that has to be specified prior to it's application. I only use it in front of a noun. There is zero charge, there are zero eggs, there are zero molecules. They are making a categorical mistake. They apply zero globally when only local applications make sense. Their problem is not realizing that zero requires a noun. Once they can specify a noun to which they are applying zero to, then they will no longer be confused.
Josan said:
If you noticed, I suggested that you have only had exposure to calculus, but not too much higher mathematics. I never indicated that you have not taken calculus. That aside, I actually should not have quoted your post, I thought it was something written by JustBuisness17 because I had been up all night (for unrelated reasons) and was writing an exhaustively long post with lots of detail and lost track of who I was quoting.. Most of what came after your post was not directed at you so much as it was directed at justBuisiness17. I, mistaking the quote for another one of his, used it to parlay in to my further points about mathematical infinity.
I apologize for confusing your joke post with JustBuisness's. That was stupid.
Josan said:
As you now know, I was confused! Thank you for the compliment! Unless you are being sarcastic! In which case, refer again to my above explanation if you still feel sour! I will offer no further consolation! :lol:
Zetetic said:
My bad. I have an easier time explaining challening concepts to people in person, when I can see their response on a word-by-word basis. Also, english is not my native language, so I find discussing physics and math difficult in english, as I tend to innocently mix up phrases that sound similar, but are completely different!
Zetetic said:
This. This is exactly what I was trying to get at. I was actually hoping Irkun would respond with either a confirmation that 0 was a number, or why he found that confusion, that way I could go into a similar explanation as you just posted. Mine would never, ever be this well formulated or clear-cut, so thank you again for a great post! =)
Zetetic said:
Again, my bad. I shouldn't have taken offense, you were just trying to clearify, which should always be welcome! =)
Zetetic said:
No, I wasn't being sarcastic at all =) It was a really good post, enjoyed reading it from start to finish.
And thank you for pointing out why what I posted was confusing, I realize now (even more than before), why what I said was quite stupid =)
Josan said:
Actually, I should clarify. I did not realize my mistake until after having written that part of my previous post as well and meant to delete it. It is JustBuisness's writing that I find to be confusing and hard to follow.
I suspected that English might not have been your first language due to unusual idiosyncrasies in your writing, but you have a better command of the English language than most, so I was hesitant to ask as it might be interpreted as an insult.