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

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Voting Style: Open Point System: 7 Point
Started: 10/4/2014 Category: Philosophy
Updated: 3 years ago Status: Post Voting Period
Viewed: 984 times Debate No: 62630
Debate Rounds (2)
Comments (4)
Votes (2)




My opponent may prove me wrong in their first round, by naming a number between 0 and 0.0*1.
I will be rebutting in round 2. My opponent will - so that we may have an equal number of rounds - type "No round, as agreed upon" on their second round or forfeit the debate.
Good luck!


The resolution of this debate states two things: (1) There is no number between 0 and 0.0*1, and (2) 0 and 0.0*1 are not identical.

My opponent claims that I may prove the resolution wrong by naming a number between 0 and 0.0*1, however it is also sufficient that I prove 0 and 0.0*1 are identical. Unfortunately I will be unable to defend my argument from my opponent's rebuttals, whereas my opponent will be able to tailor his response to the claims I make. For this reason, previous statements made by my opponent on related topics will be considered indicative of the arguments my opponent intends to put forth, and the debate should be considered in its wider background.

In his debate with Ajabi, my opponent claims that since the Real numbers are continuous, the existence of neighbouring Real numbers is “obvious.” Consequently, there must exist a difference between two adjacent Real numbers, and this difference must be positive, but infinitesimally small, or otherwise all numbers would be equal. Therefore, 0.9* and 1 are not identical; instead they are adjacent Real numbers and, according to my opponent, the difference between them is 0.0*1. [1]

In this debate, I will argue the following: (1) the Real numbers are consistent and mathematically useful and should be used to judge this debate, (2) my opponent’s ‘neighbouring Real numbers’ are not a concept which can be derived from the axioms of the Real numbers and the allegedly obvious implication that the Real numbers contain non-zero infinitesimals is false, and (3) 0.0*1 and 0 are identical because there is no positive Real number between them.

(1) The Real Numbers
The Real numbers are described by mathematicians as the complete ordered field. This is because the Real numbers are defined by the algebraic field axioms, the order axioms, and the continuity axiom, and it can be proven that any ordered field which satisfies these axioms must be isomorphic (i.e. they are the same field). [2]

The consistency of the Real numbers is universally accepted by mathematicians. Gödel’s completeness theorem implies that a system is consistent if and only if it has a model (the model existence theorem), and the Real numbers can be formulated as a model, so therefore the Real numbers must be consistent. [3] The consistency of the Real numbers is so highly regarded that the Real number system is often used to prove the consistency of other mathematical systems (for example, Euclidean geometry) in proofs of relative consistency; in other words, if the Real numbers are consistent then so is a lot of other widely-used mathematics.

Perhaps a simpler basis for accepting the consistency of the Real numbers is that inconsistency is death in mathematics. If the Real numbers were truly inconsistent or flawed, then any proof involving the Real numbers or relying on their consistency would be wrong. Consequently, if any mathematician could prove that the Real numbers were inconsistent then the Real numbers would be abandoned. The Real numbers are still widely used and no mathematician has contested their consistency, so it is reasonable to accept their consistency along with every other mathematician.

Since my opponent specifically refers to the Real numbers in his debate with Ajabi, and uses them to form his basis for claiming that 0 and 0.0*1 are not identical but there is no Real number between them to conclude that 0.9* and 1 are not equal, the Real number system should be used to judge this debate.

(2) Neighbouring Numbers
My opponent suggest that because the Real numbers are continuous, there must exist neighbouring numbers. As a result of the existence of neighbouring numbers, my opponent claims that there must also exist non-zero infinitesimals, and the difference between two adjacent Real numbers (for example 1 and 0.9*) is such an infinitesimal. [1]

Curiously, the concept of neighbouring numbers actually makes more sense when considering discontinuous, incomplete number systems such as the natural numbers, {1, 2, 3,…}. For example, in the natural numbers, the number 2 neighbours the numbers 1 and 3, there is an indivisible, non-zero difference between them (the number 1.5 does not exist in the natural numbers). Clearly, neighbouring numbers is not a concept derived from continuity as my opponent claims, since it can easily be applied to discontinuous number systems. In fact, while neighbouring numbers is a concept which can be applied to discontinuous number systems, it cannot be applied to the Real number system because the Real number system is continuous.

This is because the Real numbers are infinitely divisible. As a result of the continuity axiom of the Real numbers, between any two non-identical Real numbers, there are infinitely many other Real numbers. A simple proof by contradiction (see resources) that neighbouring Real numbers do not exist goes as follows: Suppose there exists two neighbouring Real numbers m and n such that the difference m-n = k>0 where k is a Real number. Then, following the field and order axioms of the Real numbers, there must exist a smaller difference, k/2. Hence there exists a number between m and n such that n < n+k/2 < m; however, this contradicts the assumption that m and n are neighbouring numbers. However, contrary to my opponent’s claim, this does not imply that all Real numbers are equal because the difference between them is still greater than 0; there is simply no smallest difference because there is no smallest positive Real number.

My opponent attempts to get around this problem by proposing the existence of non-zero infinitesimals. However, the existence of non-zero infinitesimals contradicts the continuity axiom of the Real numbers. Using the continuity axiom, it is proven that the Real numbers are Archimedean which means they do not contain non-zero infinitesimals. [4][5] Zero is the only Real infinitesimal, however this poses a problem to my opponent’s argument: if neighbouring Real numbers exist and there is an infinitesimal difference between them, then the difference between any two Real numbers must be zero; alternatively if the difference between any two non-identical Real numbers is non-zero, then neighbouring Real numbers cannot exist.

Consequently, my opponent’s concept of neighbouring Real numbers and Real, non-zero infinitesimals is false and self-defeating.

(3) 0.0*1 and 0 are identical
The notation 0.0*1 means that following the decimal point there are infinitely many zeros, and at the end of the zeros there is a one. This is a bit of a nonsensical notation, since if there are infinitely many zeros then there is no final zero onto which to append the one, but regardless this entails that 0.0*1 < 1/10k for every natural number k. In other words, 0.0*1 is an infinitesimal. As established above, in the Real numbers there are no non-zero infinitesimals; therefore 0.0*1 must be 0. The resolution is negated.

[4] (pages 9-10)

Additional Resources
A mathematically rigorous consideration of the consistency and completeness of RCF (which includes the Reals):
An introduction to the use of proof by contradiction in mathematics:

Debate Round No. 1


The condition for a win in this debate was clearly set: "My opponent may prove me wrong in their first round, by naming a number between 0 and 0.0*1."
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 ) 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 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: ), 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 position
My 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.


No round as agreed.
Debate Round No. 2
4 comments have been posted on this debate. Showing 1 through 4 records.
Posted by BoggyDag 3 years ago
@ Domr: I forgot to explain it to a layman.
We can never reach the center of the universe, as it is too far away. What you say is that it doesn't EXIST because we can't get there.
You are also saying that since you cannot see the difference, it is not there. But reality doesn't care for your perception. It doesn't matter how far you remove something, it doesn't stop existing.
Same goes for that 1 at the end of infinity. Yes, it is seemingly a paradox.

But so is the alternative. I will show you this:

If 0.9*=1, which is what my opponent claims and where this debate took its origin, then 0.1*+0.9*=1.1*.
This, however, cannot be true.
Repeat thisto infinity, but you do NOT get a 1 repeating. 1+9=10, so there's ALWAYS a ZERO in there, not an ever-repeating 1!
Posted by BoggyDag 3 years ago
You are the very reason I think this site is filled with laymen and hence totally devoid of any real function.
I'll try to explain.
You accept 0.9* as an infinite number, and you accept that hence its last decimal is 9, whether it exists or not.
Now look:

See this? There is always one LESS zero behind the decimal mark. This means that YOU are the one with a double standard, accepting infinite decimal places with 0.9 repeating, but denying 0.0*1 the same number of decimal places. Why is it that you people never understand that even in infinity, te laws of mathmatics still work? You cannot operate with a different number of decimal places on two sides of an equation! For every 9 you add on the one side, the 1 on the other moves back one decimal place. That does not mean it ceases to exist. Feel free to explain how you believe moving this to infinity REMOVES the 1.
And as I clearly explained, the last decimal can be of importance without ever being reached.
What I'm trying to tell you is that school maths, to which you cling in your layman perspective is INCOMPLETE. But you are - like many others - so deperate to prove that what you learned in school is important and that you must prove that you understood maths after years of mental torture that you would never admit that it was all just mumbo-jumbo. Which school maths IS.

@ Enji: Irrlevant, it's OF COURSE an infinite number of zeros.
Posted by Enji 3 years ago
I think even if you did assume 0.0*1 was supposed to indicate a finite number of 0's (so 0.0*1>0), my claim in section (2) that "between any two non-identical Real numbers, there are infinitely many other Real numbers," so "there exists a number between m and n such that n < n+k/2 < m" would satisfy my burden of proof regardless.
Posted by Domr 3 years ago
There is really only ONE way to vote on this debate.

0.0*1 I can only assume * is to represent infinite or repeating Zero's. If it represents infinite or repeating Zero's, then there can NOT be a One at the end. Because there is no end to the Zero's.

Either the * is infinite, or finite.

If it is truly infinite (or repeating) then the One does not exist meaning Con wins the debate because 0.0*1 (since the One cannot exist, the number is zero.)

If the * is FINTIE, then the debate was misleading to assume seemingly infinite/repeating 0's.

Since Pro put the ONE at the end, this number in its entirety does not exist, nullifying the debate because the premise itself is flawed.

I choose to NOT vote on a debate with a flawed premise, and urge others NOT to vote as well.
2 votes have been placed for this debate. Showing 1 through 2 records.
Vote Placed by iamanatheistandthisiswhy 3 years ago
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Reasons for voting decision: This is one of those debates that will go around forever until the ideas of infinity are fully comprehended. I don't want to award points as I think both sides made interesting arguments, and in fact both sides can learn something from each other as to why the other side rejects their opinion. The other major reason not to award points is that I don't like one round debates. I want to see rebuttals :)
Vote Placed by RainbowDash52 3 years ago
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Reasons for voting decision: Both sides made bad arguments by confusing numbers with real numbers.