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# This House Believes 0.99* Equals 1!

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 Voting Style: Open Point System: 7 Point Started: 9/22/2014 Category: Philosophy Updated: 2 years ago Status: Post Voting Period Viewed: 1,295 times Debate No: 62102
Debate Rounds (1)

32 comments have been posted on this debate. Showing 1 through 10 records.
Posted by 9spaceking 2 years ago
incredible. A purple circle ties the powerful Ajabi.
It's even a one round debate! Ajabi should have been able to insta-snipe-insta-win this thing.
Posted by BoggyDag 2 years ago
The problem with all of your alleged proofs is that theyare basically circular.

Ajabi claims that 0.99*=1 due to 1/9=0.11*.
The latter of which is not proven within this context, as it is the same problem. You use a misconception of infinity to prove itself.
In your particular case, you use Aristotle's law of non-contradiction, but we alreay know, thanks to Heisenberg, that not all matters are true or false, some are just "undetermined".
Infinity, by its definition, can not be determined in the way you do. Contradiction is not enough, as in infinity, more than the limited two alternatives you list may be possible.
First, you would have to prove that the law of non-contradiction applies to infinity.
It doesn't, as I showed in my round. Either the distance between any two numbers is larger than zero, or all numbers become the same. Yet the distance obviously cannot be measurably large, as then there would be no continuity.
Both are true. I repeat: name a number between 0.0*1 and 0!
Both are not identical. The difference is nigh zero, infinitesimally small, but NOT zero.

All you can do is quote mathematicians who were actually wrong, in making circular conclusions. Like Euler corroborating a statement about a limit using a formula that contains a limit!
What you can't for the life of you is answer a single of my questions. Which means you are just a copyist, not a thinker. Which is why you keep repeating yourself.
As long as you evade the challenge of naming the number between zero and 0.0*1, you are just pathetic.
If no other than zero infinitesimals exist and numbers are continuous, then any conceivable number must exist. Since I can imagine 0.0*1, the number must then exist, and you must back up your claim that there's room between this number and 0.
All you prove is the same neglect I have covered in my round already, which means you are biased.
Case closed, "judge".
Posted by Enji 2 years ago
Sorry if you thought my previous post was an explanation for why the two are contradictory, but the proof is roughly as follows.

First you define the real numbers using the ordered field axioms and the completeness axiom; Richard Dedekind's continuity axiom works (and tends to be used in more rigorous contexts), but the least upper bound / greatest lower bound property is similar and easier to deal with.

Next you assume the real numbers contain non-zero infinitesimals, and you attempt to define a cut above all the positive infinitesimals, or identify the supremum to the set of positive infinitesimals (positive infinitesimal numbers being numbers less than 1/m for all natural numbers m). The result is that no such Dedekind cut exists, or that there is no supremum to the set of positive infinitesimals, which contradicts how the real numbers is defined. The conclusion of this proof by contradiction, therefore, is that the real numbers do not include non-zero infinitesimals. This proof is covered in Keith Stroyan's textbook "Foundations of Infinitesimal Calculus", which, if you're interested in learning more about non-standard analysis, is an excellent resource on the topic.

This is why the real numbers can be proven to satisfy the Archimedean property, whereas number systems which contain non-zero infinitesimals (such as the hyperreals used in nonstandard analysis and your alleged 'better' real numbers which you brought up in the comments), by definition, do not.

Given this background, it's easy to see why in the real number system there is no need for neighboring numbers or for a minimum non-zero distance between two real numbers; for any distance 1/m>0, there exists a smaller distance 1/(k*m)>0 where k>1.

For future reference, the debate consists of the content presented in the debate rounds, and not any content presented in the comments. You didn't make any mention of conduct during the debate -- you first mentioned conduct in the comments section.
Posted by BoggyDag 2 years ago
@Enji: Since I specifically ASKED to award Ajabi points for conduct DURING the debate, please by all means do.
Pretentious pr*ck.
Posted by BoggyDag 2 years ago
That's a recursive false conclusion.
If - as I propose - a non-zero infinitesimal is considered infinitely small, but still adjacent, there's no PRACTICAL, only a philosophical difference. Which can be easily accepted intuitively. There needs to be a difference larger than zero between any two non-identical numbers. It just needs to be infinitely small, hence explicitly NOT Zero.
I say the exact opposite. If you accept only zero infinitesimals, you reach identity of all numbers, not a continuous set.
However, if we define what you call zero-infinitesimal AS the number 0.0*1, what's the big deal? I say it is only a philosophical one, not one that can not be mended by change of perspective.
Posted by Enji 2 years ago
The real numbers cannot be both continuous and include non-zero infinitesimals; if a number system contains non-zero infinitesimals then it is not continuous, and if it is continuous then it does not contain non-zero infinitesimals. Your attempt to merge both creates a self-contradicting number system, which is a much bigger dilemma than the problems you have with the real number system.

The facts of the debate remain as follows: (1) you specifically refer to the real number system in the debate, and you specifically refer to the three axioms which define the standard real number systems within your argument; (2) your argument relies on the existence of non-zero infinitesimals which do not exist in the real number system -- further you deal with infinite decimal places improperly, and hence you fail to satisfy your burden of proof; (3) Ajabi's second (and first) proof are correct in the context of the real numbers, thus satisfying his burden of proof.

Therefore my vote is well corroborated by the content of the debate.

The comments section is not to be considered for voting, or I would award conduct to Pro as well.
Posted by BoggyDag 2 years ago
So, what you are saying is that every old master is true, and progress should be forbidden. Arrogance, your name is Ajabi.
This is a philosophical debate. If it is not allowed to question Euler, how will we ever advance? I worship Euler, but that does not man he was not eventually bound to have a border. In this particular case, Euler needed to corroborate his idea of real numbers. It must have been difficult enough to establish the irrtional numbers. Expanding knowledge is an iterative process. One step at a time. Even Euler was only human. One of the most brilliant.
Who says I didn't already write that book? Hm?
You don't know who I am and what my merits are, and I'd prefer it to stay that way.
Posted by Ajabi 2 years ago
Boggy you should obviously write a book so you can go and collect your award at Stockholm, seeing how you basically went against everything Euler said in his Algebra. Not to mention you...urhg...im not even going to argue.
Posted by BoggyDag 2 years ago
I quote from my comment further down:
"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 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.
I really wish you would at least learn to read properly.
Posted by Enji 2 years ago
I know I said I wouldn't return, but I'm going to say this anyway: you reference three axioms of the real numbers in the debate: the field axioms (as you do arithmetic such as 1-0.9), the order axioms (e.g. neighbouring numbers), and the continuity axiom (as you state the real numbers are the completion of the rational numbers by combining them with the irrational numbers). It can be rigorously proven that any field (with a range from negative infinity to infinity but not including either) which satisfies these axioms must be isomorphic -- this means you must be talking about the same real numbers as 'mainstream' mathematicians in the debate. In non-standard analysis, which you mention solely in the comments, the continuity axiom is not satisfied; your alleged "actual" real numbers which include non-zero infinitesimals, therefore, are not the real numbers discussed in the content of the debate because this would contradict the continuity axiom.
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