
#37
Mar1212, 09:45 PM

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PS: If we take your logic as is, then you are arguing all questions whose answer is "zero" is indeterminate. Or (worse) all finite answers are indeterminate. Neither is correct. 



#39
Mar1412, 04:17 AM

P: 75

"Sufficiently complex mathematical systems cannot be contradictionfree." via Wikipedia... "Gödel's incompleteness theorems are two theorems of mathematical logic that establish inherent limitations of all but the most trivial axiomatic systems capable of doing arithmetic." http://en.wikipedia.org/wiki/Gödel's...eness_theorems 



#40
Mar1412, 04:22 AM

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How do you "prove" that a poker hand can have 5, and only 5 cards? It is laid out as a rule of the game, and in like manner, maths is a game where we pick whichever rules we want to play with. Those rules are called "axioms" Obviously, we may construct as many maths games we want. Just like we can invent new card games, by laying down some new set of rules. 



#41
Mar1412, 04:34 AM

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#42
Mar1412, 05:40 AM

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P: 16,543

1) Our current mathematical system can never be shown to be contradictionfree. 2) A mathematical system can never be AND complete AND contradictionfree. (1) is certainly problematic, but it does not mean that there are actually contradictions in current math. We just can't prove it. So we more or less accept it on faith. 



#43
Mar1412, 06:08 AM

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And, to add to micromass:
Jus because there MIGHT be some contradiction deeply embedded,as yet unrecognized bu us, in our preferred mathematical system, we need not worry too much about it, since if we DO notice it, it might well be easily remedied, by, for example, adding some pedantic little detail in an axiom formulation that doesn't do anything else than preventing just that contradiction from happening. All previous results that (1) did NOT depend upon the flawed axiom to begin with, and (2) won't depend upon the new axiom would remain unaffected, and perhaps practically all the results which DID use the flawed axiom to begin with. The contradiction might be more lethal than that, of course, but that it should be so is no implication that follows from the fact that we do not know if our current system is free of contradictions. 



#44
Mar1412, 08:25 AM

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P: 14,433

In other words, mathematical systems of sufficient complexity cannot be both consistent and complete. Consistency is essential, but completeness is not; it's just a nice thing to have. Mathematicians have given up on completeness (Hilbert's second problem) and assume consistency. 



#45
Mar1412, 11:27 PM

P: 28

and 0/0 is just a product statement, it might have came from some complicated f(x) = h(x)/g(x) where one simply got 0/0, when computing the limit. undefined is a statement, where you get R/0, where R is any real number. 0/0 or anyother form of 0^n/0^n, when u compute limits of f(x) as a whole, which give rise to the method of L'Hopital, when computing limit of f(x), when h(x)/g(x) gives you 0/0. its 2 different things, one is the axiom & another is taking limits and tend to 0/0. Additional Reference: Steward J' Calculus 7th ed. 



#46
Mar1512, 12:31 AM

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P: 16,543

If we have numbers 0 and 0, then its division 0/0 is not defined. This has nothing to do with limits or with functions. If you have functions, then you can perhaps deal with [itex]\lim_{x\rightarrow 0}\frac{\sin(x)}{x}[/itex]. But, you do NOT divide by 0 here. The limit means that if you get close to 0, then the expression sin(x)/x gets close to 1. You NEVER evaluate the function sin(x)/x in 0. So you NEVER deal with 0/0. 



#47
Mar1512, 02:22 AM

P: 28

Mathematics are not just about the calculations. The calculations are to solve a particular situation, a problem statement, in which this mere question does not have.
This question lacks the problem statement we need to determine how we can solve it mathematically. To put it in a sense where 0 = nothing, so 0 / 0 is definitely 0. There is nothing to divide it from and with, but the statement of 0 = nothing made it possible as nothing is required to divide nothing. In short, UNDEFINED means it is undetermined UNLESS a problem statement is issued. Therefore, undefined can be anything as long as you fulfill the requisite and criteria of the conditions. 



#49
Mar1512, 03:16 PM

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P: 20,948

The argument that 0/0 = 0 arises incorrectly from the fact that division and multiplication are inverse operations. If a/b = c, then a = b * c. This is true as long as b ≠ 0. If we insist that 0/0 = 0 makes sense because 0 (the denominator) * 0 (the quotient) = 0 (the numerator), then we should also accept 0/0 = 2, because 0 * 2 = 0. Since we have gotten two different answers (and infinitely more are possible), this is a violation of the commonsense requirement that division produce a single result. The upshot is that dividing by 0 is never defined, period. 



#50
Mar1512, 09:02 PM

P: 828

I just got an idea for my dissertation: I'm going to study the patterns and cycles by which threads like this get started on these forums. I can't wait until the next .999999... thread comes along. Then perhaps a thread from an arrogant ignoramus who is convinced that there is something fundamentally wrong about the real line.




#52
Mar1512, 10:33 PM

P: 66





#53
Mar1612, 01:06 AM

P: 28

As I am typing this, I realized I have made an error, though I refuse to erase the top. I was trying to implement algebraic argument to the 0 / 0 context, which in the end I believe, would be a mathematical fallacy. Forgive me on that. I was initially planning to introduce a statement as to treat 0 as an unknown in which one could substitute for another for the cause of the current calculation, but after exercising the calculation from different aspects, I realized I was led to a spurious proof. 



#54
Mar1612, 01:27 AM

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P: 16,543

I'll explain it once and for all: 0/0 is not defined because we choose it to be undefined. We could define it if we wanted to, but we choose not to. We have very good reasons for this. First, let's define what division actually means: we say that n/m=p if and only if p is the unique number satisfying mp=n. The reason we choose not to define 0/0 is because there is no unique number p such that 0p=0. All number satisfy!! We want / to be a function: that is, every input must give a unique output. This is not satisfied, so we rather choose not to define 0/0. There is no way to prove that 0/0=0, because this would just be a definition. You can't prove definitions. There is no context whatsoever in which 0/0=0. Math works perfectly fine with not defining 0/0. So does computer science by the way: no plane ever fell from the sky because 0/0 has not been defined. Arguing about 0/0 is pointless, since you're just arguing a definition. You can agree or disagree with a definition, sure. But the fact remains that 99.999999...% of the mathematicians choose to let 0/0 be undefined. 


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