Clayton: Prove it. You just proved it for me. Clayton -
Prove it.
You just proved it for me.
Clayton -
Clayton, first, read this.
Now, please respond to my original request in an intelligent manner. You can do it, son.
You can do it, son.
Holy crap, how did you find that video of me????
:-P
Honestly, I really don't care to prove or disprove anything about objectivism. I simply find it annoying and braindead. If I had to sum up in one sentence what I don't like about it I suppose that it's because it's a kind of dogmatic nihilism which strikes me as an immediate contradiction.
I recommend you read everything about Epicurus you can get your hands on. Objectivism borrows a few good themes from Epicurean philosophy but then completely misses the whole point of life: your satisfaction.
Looks very interesting, Vive. I'll check it out.
Yeah, if you like this paradox - it deals extensively with it you may like it. Been awhile since I read it though so I cant say much about it.
"As in a kaleidoscope, the constellation of forces operating in the system as a whole is ever changing." - Ludwig Lachmann
"When A Man Dies A World Goes Out of Existence" - GLS Shackle
I don't know of a single mathematician in the world who doesn't accept .999..... = 1 I accept it, too. And we all accept it in the same way, as saying that the least upper bound of the LHS is the RHS. In addition to accepting the RHS as the least upper bound of the LHS, do you also accept that the LHS is exactly equal to the RHS, and they are in fact merely two ways of notating the same number, just as 1/2 and .5 are exactly the same number? faber est suae quisque fortunae | Post Points: 20
I don't know of a single mathematician in the world who doesn't accept .999..... = 1
.999..... = 1
I accept it, too. And we all accept it in the same way, as saying that the least upper bound of the LHS is the RHS.
In addition to accepting the RHS as the least upper bound of the LHS, do you also accept that the LHS is exactly equal to the RHS, and they are in fact merely two ways of notating the same number, just as 1/2 and .5 are exactly the same number?
faber est suae quisque fortunae
Jack,
An equality means equality, of course.
But we have to be careful with our definitions. The LHS has no other definition but the LUB of the infinite series .9+.09+....
An infinite series is not a number, and cannot equal a number. So that saying "the infinite series .9+.09+... equals 1" is false.
But the statement "the LUB of the infinite series .9+.09+... equals 1" is true.
And the equation means, by definition, the latter.
My humble blog
It's easy to refute an argument if you first misrepresent it. William Keizer
An infinite series is not a number, and cannot equal a number. So that saying "the infinite series .9+.09+... equals 1" is false. Sorry, no. An infinite series is a single number, by definition. faber est suae quisque fortunae | Post Points: 20
An infinite series ... cannot equal a number
You're using one or both of the words "equal" and "number" in a different way than mathematicians do. Please provide the cite from Hardy or anybody to back up this statement. Note that a least-upper-bound is a number and you've claimed multiple times in this thread that an infinite series equals its least-upper-bound.
a "convergent infinite sum" means, by definition, a least upper bound
I should clarify my comment. A series, by definition, is a single number. Adding infinite as a descriptor does not change that.
This isn't quite true. An infinite divergent series is not a single number.
The series itself is a number. That number may or may not be able to be calculated/defined displayed in a more pleasing manner. Regardless, the sum of the sequence is still exactly one number. It is possible that the sequence itself, preceded by a sigma, may be the only way to accurately display said number.
Numbers are not like space or time - the residue in a convergent infinite series is not an "error term" representing some kind of "unknown quantity" which we could be surprised, after enough iterations, to find differs from what it is provably equal to. When we prove that the "least upper bound" of the sequence 0.999... is 1.0, what we have proved is that "when carried out to infinity", the sum 0.999... is identically equal to 1.0. This is not some kind of speculation that we can't verify, it is actually the case. What we mean by 1.0 can also be written as 0.999... these numerals are aliases for one and the very same number.
Where analysis qualifies this is in the nature of the approach to the limit. Frequently, it matters which direction you're approaching a limit from in order to fix the limit. For example, if you approach zero in the equation 1/x, you get two different answers depending on whether you approach it from the negative side or the positve side. This gets really crazy in complex space where you can approach the limit from any direction in the complex plane.
Consider any of the closed-form equations for pi or e such as those given by Friedmanite. Those equations are not merely correct to within some epsilon-delta value that is dependent on how far you carry out the equation... the equation is exactly equal to the constant in question in the limit. That is, the proportion of a circle to its diameter is precisely equal 4 * (1 - 1/3 + 1/5 - 1/7 + ... ). There are equations that are only approximations of pi... 22 / 7; they "approach" pi in an utterly different sense than Leibniz's beautiful equation does because they do not ever, even in the limit, actually equal pi. A l.u.b. specifies the limit of a sequence of partial sums and, in the case of a converging infinite series, the l.u.b of its partial sums is identically equal to its sum both of which are a number - the very same number.
The series itself is a number. That number may or may not be able to be calculated/defined. Regardless, the sum of the sequence is still exactly one number.
It depends on what formalism you are using. Some series are particularly slippery. It is not unqualifiedly true to say that a "series is a number."
Clayton,
I've edited my previous post to clarify.
Again, you are not correct. A series diverges if its sequence of partial sums does not have a limit. Take, for instance, the harmonic series
Precisely because this series is unbounded, it is perfectly valid to say