There is no number between 0 and 0.0*1, while both are not identical

"My opponent has failed to meet this pre-set condition. I consider this debate forfeited by her not even trying to meet the conditions of a win. I will compare what my opponent did to a simple example. Had I resolved "A book can be read at night, while the lights are turned on. " she would have basically said: "Wrong. All I have to prove is that the lights are out! I win. And my opponent is an idiot (see this off-topic greater picture of us arguing in the comments of another debate). "This is nonsense of course. This debate is about the question what number lies between 0 and 0.0*1, WHILE (= "as long as", according to . http://www.merriam-webster.com... ) they are not identical. My opponent has thus evaded the actual problem, by discussing the possible case that they are identical, DELIBERATELY MISSING THE POINT in order to confuse the audience and gain an easy win. This is very bad conduct, and a forfeiture of all arguments. The same goes for her insistence that we are discussing "real numbers" following her silly little axioms here. The "broader picture" shows that I do NOT wish to discuss her idea of real numbers:' "It's long overdue to merge non-standard analysis and standard analysis to form a proper set of axioms for the actual real numbers. " I neither adere [sic] to standard analysis nor strictly to non-standard analysis. Obviously, both have strengths that I believe can be reconciled. And while I do not claim to have it all worked out, both can be merged to solve the dilemma at hand. Whereas the standard theories do not suffice to do just that. ' from . http://www.debate.org... in the comments section. Constantly and deliberately misrepresenting my intention is very bad conduct indeed. The very idea of this whole debate is to show that the axioms of the real numbers, as they stand now, contain a paradox and MUST be expanded on. My opponent is the living proof of that. The axioms claim that between any two numbers, there exist an infinite count of numbers. Yet, my opponent cannot name a single number between 0 and 0.0*1. She claims this to be de to them being identical. Then, she should not have accepted this debate, the resolution of which clearly demands the numbers to be considered distinct numbers. Now, for my rebuttals anyway. For even in her own world of paradoxes, what she writes does not make sense. Especially in her own world. My opponent concedes: "there is no positive Real number between them"She has thus already admitted that there is no real number between 0 and 0.0*1. If I can prove that both are not identical within the terms proposed by my opponent, I win even by her terms, although this debate of course is already over. What then is 0.0*1? As my opponent admits, it is a representation of a number that - written as a decimal - contains an "infinite" amount of zeros, followed by a 1. And while my opponent claims this to be "silly", if we write it as a fraction, things become much more evident, and will not confuse her so much.0.1 or "1 divided by 10" can also be written as "1/10" OR "10-1 ". 0.01 or "1 divided by 100" can also be written as "1/100", as "(1/10)2" OR "10-2 ".0.000001 or "1 divided by a million" can also be written as "1/1000000", as "(1/10)6" OR "10-6 ". All these forms depict the same thing, a number of zeros, followed by a single 1, with the exponent indicating the number of zeros left (hence the minus) to that single 1.0.0*1 or "1 divided by infinity" can also be written in that form, but since infinity is not a number that may be easily plugged into an equation, we have to write it as a LIMIT. [On why infinity (or ∞) may not be plugged in, consider this: ∞+1=∞ -> subtract ∞ on both sides of the equation, and you get 1=0, which is a contradiction. This equation would not work with an real number: 2+1=2 is just wrong. So ∞ does not work like other numbers, and we take that into account by using a LIMIT. ]This LIMIT can be written as limx->∞ 10-x. What my opponent does is show that this limit is 0, calling it an "infinitesimal", which by the faulty definition she uses means it must be zero. Remember, however, that all infinitesimals ARE LIMITS by their very definition. And now, bearing this in mind, read the following. This is from one of my opponent's own debates (source: . http://www.debate.org... ), from her own round:"Limits are useful in showing what functions tend towards, however they do not show what functions are equal to. For example, limx->0 0/x = 0, however this does not mean that 0/0 is equal to 0. XX may tend towards 1 as X approaches 0, however it is not equal to 1. "This means that, on my OPPONENT'S insistence we consider other debates as sources, she has contradicted herself. The number she calls "nonsensical" is a veritable and thoroughly examined limit of a function, the function f(x)=10-x. She claims it to be zero, yet on the other hand declares it to be not zero, as limits do not show what the value is equal to. This proves by her own terms that 0.0*1 (or 10-∞, although this is not a mathematically correct way of writing this) is NOT equal to zero. So, my opponent admits that 0.0*1=/=0 despite her claims to the contrary. She has also admitted that there are no numbers between the two. Which means she has lost this debate, furthering my idea of adjacent real numbers and the invalidity of the commonly accepted axioms due to inherent contradictions. In layman's terms: mathematicians who proclaim that the axioms of the ALLEGED "real numbers" are consistent only achieve this by arbitrarily declaring that a limit is equal to its end value, OR NOT, just as they need to serve their ends. This wibbly-wobbly kind of mathematics may be good for children's television, but it is not logically sound. Addendum: Why last decimal can be known and important regardless of its positionMy opponent claims limx->∞ 10-x to be nonsensical, while this is a corroborated theme of standard analysis. Weird enough. But nonetheless, the last decimal of this number is known for all values of x that are natural numbers {1; 2; 3; . .. ; ∞}. It is 1, as shown above. So, what does this tell us and why is it important? For quite simple reasons. We know that the number 10-100000000000 is uneven, for example, as the last decimal - without even looking at it - must be uneven, as it is 1. The same goes for 0.0*1. We must consider this, as certain rules apply for even numbers. The number 0.0*2 for example, is an even number, and can be divided by 2, resulting in 0.0*1.0.0*1 however, is not even. Attempting to divide it gives us back 0.0*1 itself, as this is the number adjacent to Zero. It's basically a property very similar to its neighbor, Zero. If you divide 0 by any other number, you get 0 as a result. We all accept this easily, and nobody cries "paradox" over this. Zero is considered a "neutral element" in this context. I say for division only, this property extends to its direct neighbour. Zero is also a neutral element in addition, its neighbor 0.0*1 is NOT. Which means: adding Zero to itself also results in Zero, and we would never get anywhere if there were NO non-zero infinitesimals. How do we ever reach any other number than Zero if my opponent is right? Starting with Zero, as we should do in any number system, as the absence of number would be the only common factor in all perspectives on mathematics, we would never reach 1, if we didn't add something larger than Zero. Since all numbers must be the sum of others, except for Zero, this is obviously an infinitesimal problem. Add 1 to Zero. Where does the 1 come from? Well, it's 1/2 + 1/2. Where do those come from? 1/4 +1/4, and s forth. While we reach always smaller numbers, we can never reach zero this way (a fraction can only be zero if the numerator equals zero), which is a good thing! Because we reach 0.0*1, the thinnest thread in the fabric of the mathematical field. And it must not be zero, or nothing would exist. But standard analysis rejects this idea and tells us everything is made from nothing, while at the same time claiming the opposite (see above, on limits being zero or NOT, whenever they please). So, we gain information from the last decimal, without knowing where it is. It may very well be infinitely far away, but the laws of mathematics still apply beyond infinity. Even numbers can be divided by two without complications, uneven ones cannot.0.0*1 thus is not nonsensical, it contains the notion of being the smallest, indivisible number adjacent to zero. One that can be used to add and subtract, as opposed to zero.