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Is the flow of my logic flawed? 
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#1
Dec2010, 11:34 PM

P: 737

If two numbers are not equal, there is infinitely many numbers between them. Therefore, numbers that are equal have a finite amount of numbers between them, 0. Therefore, numbers with a finite amount of numbers between them are equal.



#2
Dec2110, 03:53 AM

P: 91

Assuming you mean real numbers with the usual order relation (without entering into a debate about whether this needs to be said, please)... this sounds correct. It's the contrapositive of a true statement, after all.



#3
Dec2110, 04:03 AM

P: 91




#4
Dec2110, 04:05 PM

P: 737

Is the flow of my logic flawed?
Thanks. I know it sounds like a "duh!" question, but it has implications. I was considering 1 = or != .999..., and the thought about there being infinitely many [real] numbers between two numbers came to me. One can name no numbers between .999... and 1.
My understanding of numbers is that they do not exist until named, and cannot exist if they cannot be named. The argument would rely on this being true, which it is, isn't it? 


#5
Dec2110, 04:15 PM

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#6
Dec2110, 04:34 PM

P: 91

If you are not accepting formulas, then you can't "name" .999..., anywayin fact, you can't "name" most of the real numbers (we know how to write only a handful of transcendental numbers, yet we know that most real numbers are transcendentals). Yet, they're still there. EDIT: Well, really, you can "name" .999... since you'll always know the next number. But still most of the reals cannot be represented this way; there are numbers whose approximation by rationals will display no pattern at all, in fact. 


#7
Dec2110, 04:39 PM

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Hi, TylerH!
It will be useful for you to develop logacial notation skills, so that you more easily see whether your implication is valid. Let "A": Two numbers are not equal. "B": There is an infinity of numbers between two numbers. Thus, your premise is: If A, then B. This is logically equivalent to its contrapositive form: If NotB, then NotA NotB: Not infinitely many numbers netween two others (meaning there exists a finite amount of such, regarding NO numbers between them as 0) NotA: Two numbers are not not equal (meaning they are equal). Thus, your logic is fine. 


#8
Dec2110, 04:45 PM

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#9
Dec2110, 05:03 PM

P: 737




#10
Dec2110, 05:11 PM

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If you think ".999...." means "the limit of the sequence .9, .99, .999, ..." , then (since the terms in the sequence is the sum of a geometric progression) ".999...." is just another name for "1". On the other hand, if your definition is different (or if you haven't yet really nailed down WHAT your own definition is) then all bets are off untill after you have defined it. But you are on a slippery logical slope here. I suppose you probably accept that "the ratio of the circumference to the diameter of a circle" is the name of a number  otherwise usually called "pi". If so, you accept that numbers can be "named" in other ways than by a sequence of digits. Nobody knows the complete (infinite and nonrepeating) sequence of decimal digits of pi, so nobody can "name" it except by stating some property that it has, like my defintion above. So if you agree with that general idea about "naming" then what about "The smallest positive number that cannot be named". Is that the name of a number, or not? If you think not, how do you propose to define what "naming" means, so you can always tell whether "names" like that are valid or invalid? Here be (probably an uncountably infinite number of) dragons .... 


#11
Dec2110, 05:11 PM

P: 737

When I say name a number, I mean a decimal expansion. Anything that can be decimally expanded is valid. The reason for decimally expanding is to make it more clear in what interval a number lies. Like sqrt 2 or 1.41...[the ellipsis in this case means more digits, not repeating digits], so we know that it must be between 1.41 and 1.419.
I've seen the Wikipedia article, but I prefer my way for its simplicity. But sometimes simplicity leaves a residue of ambiguity. That's why I come here to consult those smarter and better educated than myself. 


#12
Dec2110, 05:37 PM

P: 91

I included the link to the Wikipedia article not to suggest a method of proof, but to show that as long as you're allowed to "name" a number by writing a series for it, there is nothing wrong with the argument. 


#13
Dec2110, 05:44 PM

P: 91




#14
Dec2110, 06:00 PM

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#15
Dec2110, 06:26 PM

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I don't see the advantage in distinguishing between an infinite number of numbers between two given numbers, and a finite number (i.e., zero) of numbers. 


#16
Dec2110, 06:55 PM

P: 737

I doesn't matter that you can't write or compute the whole decimal expansion of a number. The numbers can be named that they can be named to the point necessary. The point is to name the bounded region they exist in. You can name sqrt 2 as 1.4... and you know it is more than .999..., that's all the info I need to accomplish my ends, the rest is extraneous.
Mark44: I completely see your argument. The problem is that when I argued 1.999...=0, my friend said 1.999...=1/∞, which is right, but he refuses to admit that limits are the constant value that is approached(0, in this case), rather than the never ending sequence. He's stuck on "it will approach, but never equal" which forces me to have to try to come up with some contrived real number theory method of proving I'm right. I know that his interpretation of a limit as an infinite process of approaching is wrong, but I have no way to argue it. Suggestions? 


#17
Dec2110, 07:28 PM

P: 617

ask him what he means by .999... Mathematicians define it to be a limit. The limit by definition of limit equals 1. Once he understands that .9999… is just a short hand notation for a limit it’s hard to argue it doens't equal one. For him to say .9999… doesn’t equal one is like arguing the limit as x goes to infinity of (x+1)/(x1) isn’t equal to 1.



#18
Dec2110, 07:58 PM

P: 737

No, he knows it's a limit. He's just got a messed up understanding of limits. He says they "approach" as if approaching is an infinite process of almost equaling.
It's hard to explain his views. Here's a short excerpt: 


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