# I What are the implications of infinity?

1. May 24, 2017

### david2

hi,

say you have an infinite distance. is half of that distance also infinite? or 1/10000000000000 of that distance. and so on and so on.

i suspect it must be because if you say that half of infinity is for example 10000 km then infinty must be 20000 km, which it is not.

so, if this proposition (that every part of infinity is infinite) is true then i come to the following conclusion:

the universe cannot be infinite because we can move around. if every bit of infinity is infinite then it would not be possible to move lets say 10 cm. it would take forever.

am i missing something?

2. May 24, 2017

### jfizzix

For starters, there are an infinite number of non-overlapping length intervals in one meter, but humans have no trouble crossing such distances.

3. May 24, 2017

### david2

so these intervals in real life (not a moot point) must have a finite lenght. not?

4. May 24, 2017

### phinds

This canard has been around for a couple of thousand years. Google Zeno's Paradox.

5. May 24, 2017

### david2

thx phinds , will check it out.

6. May 24, 2017

### Staff: Mentor

It is not.
Infinity is not a real number. There are extensions of the real numbers where you can work with infinity as numbers. In these extensions, infinity divided by a finite number is (edit: ±) infinity. You cannot cross 10% of an infinite universe. But you don't have to do that to move 10 cm. Infinity divided by infinity is undefined.

Last edited: May 24, 2017
7. May 24, 2017

### david2

interesting

8. May 24, 2017

### Janosh89

Riemann (misspelt?) used numbers and there correspondence ,gridwise , to quantify and even add infinities together.
thus the set of fractions _ 1/2 1/3 etc were not an unbounded infinity but the set of decimal equivalents were...

9. May 24, 2017

### jbriggs444

The set of fractions 1/2, 1/3, etc is bounded above by 1/2 and below by zero. The set is clearly bounded. It is equally clear that the number of fractions in the set exceeds any finite bound. The same applies for the set of equivalent decimal numerals. (0.5, 0.333..., 0.25, 0.2, ... )

It is not clear what you are trying to say. Perhaps that the set of all rational fractions is countably infinite but that the set of all infinite decimal strings is uncountably infinite? [That's Cantor, not Riemann]

Last edited: May 24, 2017
10. May 24, 2017

### Drakkith

Staff Emeritus
That's right. When I walk to the store, that distance is finite and it takes me a finite amount of time to cross it. It's important to understand that infinity describes, what I like to call, a process (I'm sure mathematicians have a proper term for it though). If I walk to the store at 1 meter per second, then every second I go a finite distance of 1 meter. That's obvious of course. But, what happens if I start dividing this length into equal finite segments? Well, if I divide it into 2 equal segments, then every second I cross 2 segments of 1/2 meter each. The total distance is still the same, and it still takes the same amount of time to go 1 meter. Also, each segment has a different start and end point. If the start of segment 1 is at $x=10$ then the end of segment 1 is at $x=10.5$. Segment 2 would then go from 10.5 to 11.

Now, what happens if we keep increasing the number of segments? The length of each segment decreases and the number of segments per meter increases. So I'd have to cross more and more segments as I divide the total distance up into more and more pieces. Note that each segment still has a start and an end point, each is still finite in size, and the number of segments is still a regular old number.

But let's keep going. Let's keep dividing it up, further and further. Am I forced to stop at some point? Is there some number of segments that I suddenly cannot go above? No, there is not. I can divide my 1 meter segment into more and more pieces and the number of segments increases without end. Infinity describes this process. It is the concept that something can continue happening without end. In this case, I can continue to increase the number of segments in my 1 meter length to any arbitrary amount. I don't suddenly have to stop at 1050 segments. I can continue to 1050 + 1 and even beyond.

In this case, we say that there are an infinite number of segments and the behavior of these segments is that they get smaller and smaller as their length goes to zero. They become point-like. Another description of these segments is that they become infinitesimal in size. Note that infinitesimal is not zero any more than infinity is a number. It merely describes the behavior of something as it gets smaller and smaller. If there is no minimum size, then the length of any segment can anything. We can make their lengths as small as we want as long as it's non-zero.

Be aware that just because we've started with a finite number of line segments and made them smaller and smaller, this doesn't mean that reality is "discrete". As far as we know, things like length and distance are continuums, meaning that there is no inherent minimum length that something can be. In our math, in order to properly go from a discrete number of line segments, where each line segment is non-zero in size, to a continuum, we need concepts like infinity and infinitesimals.

That's my understanding at least.

11. May 24, 2017

### jbriggs444

The modern mathematical notion of infinity does not include the notion of a process. Roughly speaking, the idea of a process is the idea of a "potential infinity". As you suggest, this is the notion that no matter how far you go, you can go farther or no matter how finely you divide, you can divide more finely.

Gregor Cantor put a foundation under the notion of a completed infinity. This is the idea of a collection of infinitely many things. For instance the set of all the natural numbers has infinitely many members. We talk about this set as a fixed thing, not as a continuously incrementing process.

It can be a difficult thing to wrap one's head around. It took me about a week in my first formal course on real analysis before the idea gelled and I could stop thinking of the Peano axioms as describing an unending process and start thinking of them as defining the properties of a completed whole.

12. May 25, 2017

### david2

hi again,

I also saw some other threads about the infinity subject. very interesting albeit a bit difficult for me to grasp instantly.

btw this is a great site. very informative.

13. May 25, 2017

### phinds

For more fun with infinity, Google "Hilbert's Hotel"

14. May 27, 2017

### david2

i will, thx again

15. May 28, 2017

### mustang19

The implications of infinity include:

1. 0.999... Equals 1
2. Infinity equals -1/12
3. The explosion principle applies to mathematics

16. May 28, 2017

### phinds

Why is #3 a result of infinity? Do you contend that it does not exist WITHOUT infinity?

17. May 28, 2017

### WWGD

...There is a bijection between an infinite set S and a proper/strict subset of S, e.g.., between the natural numbers and the even numbers.

18. May 28, 2017

### Drakkith

Staff Emeritus
19. Jun 1, 2017

### sumar

one question 10cm must be a percent of infinity right?
and if so does that mean that every time you move you are moving a percent of infinity?
at lest that's my understanding

20. Jun 1, 2017

### WWGD

Not in the standard sense of the word; a is x% of y if (a/y)*100= x, but , in the standard Reals, infinity is not a number, so an expression (a/$\infty$) has no meaning, unless you "massage it" with limits.

21. Jun 1, 2017

### Staff: Mentor

No.
No. A percentage is a ratio. What you are saying essentially is that $\frac {10} \infty$ is some positive number, but that isn't a meaningful fraction.

As a limit, $\lim_{n \to \infty} \frac {10} n = 0$
No matter what specific number you put in the numerator, the limit is still zero.

22. Jun 1, 2017

### sumar

thanks

23. Jun 1, 2017

### sumar

there are two types of infinity countable and uncountable right?

24. Jun 1, 2017

### WWGD

No; remember Cantor's theorem $|A| < |2^{A}|$

EDIT: For the sake of completeness, likely overkill for the question of whether there are levels of infinity between countable and uncountable, the issue is undecidable, though by above theorem there are more than two levels of infinity..

Last edited: Jun 1, 2017
25. Jun 1, 2017

### Staff: Mentor

Your original question about whether 10 was some percentage of infinity has nothing to do with sets being either countably infinite or uncountably infinite.