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January 13th, 2016, 07:07 AM   #21
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Originally Posted by Karma Peny View Post
You cannot have a definition that includes the summation from 0 to infinity and then claim it is not a process. The notion of going from 0 to infinity describes a process.
You don't understand the notation. I have already explained it to you twice. The notation does not describe a process, it describes the terms. All the terms can be thought of as existing at once. But there is no infinity in the definition anyway. So your argument makes no sense.
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January 13th, 2016, 07:11 AM   #22
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Originally Posted by Karma Peny View Post
The fraction 1/3 can be expressed in entirety as a fraction, or in some bases, like base 12, but not in base 10. It cannot exist in base 10 because it cannot be actually constructed or even shown how it could possibly be constructed in base 10.
Of course it can. A third is represented by a decimal point followed by an unending string of 3s. There is no other number that is thus constructed, so there is no ambiguity.

But the important point is that the representation doesn't matter. A third is a third, however you write it down.

You might as well claim that some emotions do not exist in English because the language does not have the ability to express it exactly. It's clearly nonsense. We get around the problem by using other ways to express ourselves, or by using combinations of words.

Last edited by v8archie; January 13th, 2016 at 07:17 AM.
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January 13th, 2016, 07:13 AM   #23
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You don't understand the notation. I have already explained it to you twice. The notation does not describe a process, it describes the terms. All the terms can be thought of as existing at once. But there is no infinity in the definition anyway. So your argument makes no sense.
My argument is one of construction, which I pointed out right below the part you quoted.

People who have a problem with the notion of infinity cannot think of the terms all existing at once. This is the bit that makes no sense to us. There are so many problems with this idea it is difficult to know where to start.

But for this 'infinitely many' idea to be a valid notion in mathematics it needs to be well understood. If it was well understood and properly defined, then it would be possible to describe a process to convert 1/3 into base 10.

No such description is possible.

Last edited by Karma Peny; January 13th, 2016 at 07:24 AM.
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January 13th, 2016, 07:20 AM   #24
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So what if you can't write the number down. It doesn't mean that the construction doesn't exist. But again, it doesn't matter if it doesn't exist. It wouldn't stop the number existing.
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January 13th, 2016, 08:50 AM   #25
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Again, the definition of the notation $\sum \limits_{n=0}^\infty a_n$ is that it means $\lim \limits_{m \to \infty} \sum \limits_{n=0}^m a_n$. This limit in turn is defined to be the number $L$, if such a number exists, such that for every $\epsilon \gt 0$ there exists an integer $M$ such that $\left|L - \sum \limits_{n=0}^m a_n\right| < \epsilon$ for all $m \gt M$.

There. Not an "infinity" in sight. No "infinite" sums either. All perfectly sweet for an "extreme finitist" to agree with.
"For all m>M" is just another way to express infinity. Are you saying we can dispense with the problem of infinity by giving it a different name? That's a novel approach.

Last edited by zylo; January 13th, 2016 at 09:03 AM.
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January 13th, 2016, 09:04 AM   #26
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No it's not. The $\displaystyle m \gt M$ are all finite. Infinity isn't even a number.
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January 13th, 2016, 09:10 AM   #27
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No it's not. The $\displaystyle m \gt M$ are all finite. Infinity isn't even a number.
A finite set of natural numbers has a greatest member. Could you please give us the number?
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January 13th, 2016, 09:21 AM   #28
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A finite set of natural numbers has a greatest member. Could you please give us the number?
Sure. It's m. Now take your limit as m goes to infinity.

I really don't understand the point of this thread. The decimal representation of a real number is just that....a representation. Two points: 1) Nobody said you can actually write out the decimal "expansion" of a real number. 2) Nobody said the decimal representation of a real number is 1 to 1 with the real numbers. Apparently 0.9999.... is equivalent to 1.

What's the problem here?

-Dan
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January 13th, 2016, 09:27 AM   #29
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Do you not understand the difference between a set and a number? $m$ is a number drawn from the set of all integers greater than the number $M$. The definition states that it doesn't matter which number $m$ you draw from the infinite set of numbers greater than $M$, the inequality will continue to hold.
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January 13th, 2016, 09:30 AM   #30
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My argument is one of construction, which I pointed out right below the part you quoted.

People who have a problem with the notion of infinity cannot think of the terms all existing at once. This is the bit that makes no sense to us. There are so many problems with this idea it is difficult to know where to start.
Why can't they exist all at once?

Consider the line leading from you to your nearest McDonald's.

That line may be broken down into indefinitely smaller sections, as a theoretical process, but they must definitely all exist for you to traverse it to obtain your BigMac and Fries.

Interestingly, most mathematicians are hung up on analysis to the extent they find synthesis difficult.
Your problem seems to be the reverse one.

Last edited by skipjack; February 29th, 2016 at 02:09 AM.
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