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Define Infinity

Dazz
Posts: 1,163
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10/28/2013 3:27:25 AM
Posted: 3 years ago
Q1. What is infinity? And when there are infinite series of numbers between two real numbers, what is a real number?

Q2. How can we define distance_while moving (if really moving) from a real number (if it is started from a real number, or infinity otherwise) towards the next real number (to cover the distance). Hence, if the distance is covered by changing the position from 1 point to another, how can the infinity be "infinity" any more, between these two points?

*Request to all:
1. Please state precise and comprehensive answer.
2. Please always comment for the original post question. Avoid to start replying each other. It's good to state your personal logic (if you have to support your view/answer) rather than going for terminating others logic. As if reader is capable, he/she can differentiate between logical and illogical points, by self.
3. I'm the exception, for request # 2, as I have to check and verify the presented logic.
4. Both of the questions are subject to have logical error (if you find any, share!).

Thanks for cooperation.
Remove the "I want", remainder is the "peace". ~Al-Ghazali~
"This time will also pass", a dose to cure both; the excitement & the grievance. ~Ayaz~
themohawkninja
Posts: 816
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10/28/2013 4:35:35 PM
Posted: 3 years ago
At 10/28/2013 3:27:25 AM, Dazz wrote:
Q1. What is infinity? And when there are infinite series of numbers between two real numbers, what is a real number?


A real number a value that represents a quantity along a continuous line [1]. An easier way to think of this is that a real number is any number that can be represented by a part, parts, whole, or multiple objects. You can't have sqrt(-1) of an apple, but you can (physical limits of how small you can divide something aside) have pi apples.

As far as what infinity is, there are two infinities that I know of. Those that are listable/countable, and those that are not. If I recall correctly, a listable infinity would be like Hilbert's Hotel [2]. While you can't count to infinity, you can count the individual members that make up the infinity. An uncountable infinity might be (I am not certain) something like the surface area of the universe, whereby it's ever-expanding (hence infinite), but there is no defined way of measuring it (hence, uncountable).

Q2. How can we define distance_while moving (if really moving) from a real number (if it is started from a real number, or infinity otherwise) towards the next real number (to cover the distance). Hence, if the distance is covered by changing the position from 1 point to another, how can the infinity be "infinity" any more, between these two points?


If I understand you correctly, you are asserting what is known as Zeno's paradox, which states (as an example) if you have your hands one meter apart, and you go to clap them, you can look at your hands clapping from the perspective of halving the distance between your hands per unit time, which means that you will never get to the full meter (1/2 + 1/4 + 1/8 + 1/16... is less than 1) [3]. However, you can clap your hands, so why does this work? The paradox breaks if you write the equation as an infinite series, whereby the answer is what the sum of all sums tends towards, which in the case of Zeno's paradox, is 1 [4].

1. http://en.wikipedia.org...
2. http://en.wikipedia.org...
3. youtube.com/watch?v=u7Z9UnWOJNY
4. http://www.skeptic.com...
"Morals are simply a limit to man's potential."~Myself

Political correctness is like saying you can't have a steak, because a baby can't eat one ~Unknown
themohawkninja
Posts: 816
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10/28/2013 4:46:45 PM
Posted: 3 years ago
Also, there are always an infinite number of real numbers in between two real numbers, unless (I suppose) one of those numbers is irrational, as then you wouldn't be able to determine what value is 1/inf greater than it, as you can never discern the last digit.

To pose the above as a question, in case you don't understand: While 3.2 is greater than pi, at what SPECIFIC point on a number line is something greater than pi? I believe that the answer is simply (pi + (1/inf)).
"Morals are simply a limit to man's potential."~Myself

Political correctness is like saying you can't have a steak, because a baby can't eat one ~Unknown
drafterman
Posts: 18,870
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10/28/2013 8:39:57 PM
Posted: 3 years ago
At 10/28/2013 3:27:25 AM, Dazz wrote:
Q1. What is infinity?

Infinity is the description for something without end.

And when there are infinite series of numbers between two real numbers, what is a real number?

There are many ways to answer this.
http://en.wikipedia.org...

I guess the simplest one would be to say that the infinite decimal expansion of a number is a real number.

Q2. How can we define distance_while moving (if really moving) from a real number (if it is started from a real number, or infinity otherwise) towards the next real number (to cover the distance). Hence, if the distance is covered by changing the position from 1 point to another, how can the infinity be "infinity" any more, between these two points?

I'm not sure what you're asking. If you're asking how can there be an infinite number of something between two points between which is a finite distance the answer is that those points have no dimension and, therefore, take up no "space."