## The Nature of Infinity in Mathematics and Reality

Hello all!

In the past few months I've stumbled upon an issue that has played games with my mind. I feel I need some help to solve this, as I've tried various other sources and remain without answers.

Firstly, I was confronted with a mathematical proof which states that 0.9~ (to infinity) = 1. The proof is simply:
x = 0.9~
10x = 9.9~
9x = 9
x = 1
This bugs me, for from a purely logical standpoint it seems ludicrous to claim that two different numerical representations are representative of the same quantitative value.

As a result of studying this issue I have also run into a problem with my whole perception of reality. All my life, I have considered mathematics to be an ultimate truth which has certain affinity with reality. Thus, what is proven in mathematics, has for me been enough proof for such a truth in reality. In mathematics, infinity exists and is essential to the functionality of the number system. In reality, however, I struggle to find an example of infinity which can be tied directly to its existence in mathematics.

To further this conundrum, I have realized that, should there be infinite distance between any two given points in reality, there can be no points to begin with, for a point which is endlessly broken down ad infinitum has no end. My entire perception of time and space has been confused, and I'd like some help to understand why.

 Quote by upgrade This bugs me, for from a purely logical standpoint it seems ludicrous to claim that two different numerical representations are representative of the same quantitative value.
1/2 = 2/4

Be aware that the whole point of the equals sign is this.

 All my life, I have considered mathematics to be an ultimate truth which has certain affinity with reality.
We can use mathematical models to study the real world, but mathematics itself is only applying logic to axioms. It is independent of reality. You can take something which is impossible in real life and study it mathematically. Banach-Tarski theorem is a good example.

 In reality, however, I struggle to find an example of infinity which can be tied directly to its existence in mathematics.

Are you part of reality? If you are, and if you have a mental intuitive category called "infinity" somewhere in your in mind, then that philosophical construct is also part of reality. We can study that using mathematics.

 To further this conundrum, I have realized that, should there be infinite distance between any two given points
In mathematics, the concept of distance is usually codified by what we call a metric.
One property of a metric is that between any two points, the distance is always finite.

The idea that points can be infinitely far away is called "extended metric", but every extended metric can be converted into a normal metric which retains the important concepts of convergence and continuity, so its not studied that often.

 in reality, there can be no points to begin with, for a point which is endlessly broken down ad infinitum has no end. My entire perception of time and space has been confused, and I'd like some help to understand why.
The continuum is just a model of space. We don't know what RL space actually looks like, and most scientists don't care.
 Blog Entries: 8 Recognitions: Gold Member Science Advisor Staff Emeritus See our FAQ on 0.999... and infinity: http://www.physicsforums.com/forumdisplay.php?f=207

## The Nature of Infinity in Mathematics and Reality

 1/2 = 2/4 Be aware that the whole point of the equals sign is this.
I apologize for I misspoke. I should have said that no two decimal representations are representative of the same value. For example, there is no other decimal representation of ∏ than 3.14159... correct? If not, what other equivalent decimal representation is there?

 We can use mathematical models to study the real world, but mathematics itself is only applying logic to axioms. It is independent of reality. You can take something which is impossible in real life and study it mathematically. Banach-Tarski theorem is a good example.
 Are you part of reality? If you are, and if you have a mental intuitive category called "infinity" somewhere in your in mind, then that philosophical construct is also part of reality. We can study that using mathematics.
Is this not a contradiction? To assert, on the one hand, that mathematics is independent of reality, but on the other hand that anything which can be conceived of is part of reality, seems illogical. For is not mathematics a concept? I think it is important in this situation to acknowledge a certain mystery known as mind-matter dualism. Could this not be the mystery which hinders our ability to answer this very basic question?

Apart from this, though, I find it incredibly hard to believe that mathematics is independent of reality, for to claim such would be to brush aside known affinity between the two as mere coincidence. For instance, if one adds 1 rock to pile already containing 1 rock, he now has 2 rocks, which remains a truth in and of itself, apart from how we choose to address it with language.

 The idea that points can be infinitely far away is called "extended metric", but every extended metric can be converted into a normal metric which retains the important concepts of convergence and continuity, so its not studied that often.
I'll read up more on this point because I'm having trouble understanding the "metric" concept. Thanks for the topic!

 The continuum is just a model of space. We don't know what RL space actually looks like, and most scientists don't care.
I'm not sure what you're saying here. Is it untrue to claim that there can not be a point to begin with, since all points can be broken down further to infinity?

 See our FAQ on 0.999... and infinity: http://www.physicsforums.com/forumdisplay.php?f=207
Thank you! That does answer a few of my questions!

 This bugs me, for from a purely logical standpoint it seems ludicrous to claim that two different numerical representations are representative of the same quantitative value.
A "purely logical point of view" suggests no such thing. That's ridiculous.
Is the statement $2+3=4+1$ ludicrous? Is the statement $y=x^{2}$ ludicrous? That $0.999... = 1$ is a fundamental property of the real number system. It can be shows to be true with absolute metaphysical certitude given the axioms of the real numbers.

 A "purely logical point of view" suggests no such thing. That's ridiculous. Is the statement 2+3=4+1 ludicrous? Is the statement y=x2 ludicrous? That 0.999...=1 is a fundamental property of the real number system. It can be shows to be true with absolute metaphysical certitude given the axioms of the real numbers.
As I clarified above, I should have said that no two decimal representations are representative of the same value. Could you give another decimal representation of 3.14159... (∏) that is its equivalent?

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Recognitions:
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Staff Emeritus
 Quote by upgrade As I clarified above, I should have said that no two decimal representations are representative of the same value. Could you give another decimal representation of 3.14159... (∏) that is its equivalent?
No, the decimal representation of pi is unique. But why is unique decimal representation so important anyway?? I don't see why...

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 Quote by upgrade As I clarified above, I should have said that no two decimal representations are representative of the same value. Could you give another decimal representation of 3.14159... (∏) that is its equivalent?
You are missing the point. The fact that SOME numbers have only representation has no effect on the fact that some ALSO have more than one representation. As has been said already, this is fundamental to the real number system.
 Ah I see, this was just a dumb misconception I had going on in which I forgot there are rational and irrational numbers! Thanks, sorry for wasting your time on that tangent.
 Recognitions: Science Advisor It's actually no particular reason to allow an infinite sequence of 9's as part of a decimal expansion. It doesn't appear in any division process; it doesn't appear "naturally". But we put it in to make any infinite sequence of digits define a number. But some infinite sequences (which incidentally don't appear in the division process) coincide with others (which do).