# infinity understanding problem?

between 0 meter -> 1 meter there are 100 cm.

but each cm has infinite numbers :

for example between 0..1 cm there are :

0.000000000001
..
0.00000000000111
..
0.000000000001111111
and more numbers and combinations...
..
..
..
1.0

to each number I can add another digit to the right

there are infinity of numbers

question :

how can a person walk 3 cm if he had to go through an infinite series of numbers ?

it not seems logic

any help ?

-
This is Xeno's paradox. The name is misleading though, as it's not a paradox, just something Xeno thought was strange. – Alex Becker Jun 22 '12 at 17:53
– talmid Jun 22 '12 at 17:54
@AlexBecker is there any explanation to this paradox ? – Royi Namir Jun 22 '12 at 17:54
@Alex: "Paradox" means "contrary to received opinion; a statement that is seemingly contradictory or opposed to common sense, but is perhaps true; contrary to expectation or common sense". So the name is accurate; it's the perception that "paradox" means "logical contradiction" that is incorrect. – Arturo Magidin Jun 22 '12 at 17:58
@RoyiNamir: There are three Xeno paradoxes. (i) Achilles never catches up to the turtle. (ii) An arrow never moves. and (iii) The dichotomy paradox. Your problem is a version of (iii), not of (i). – Arturo Magidin Jun 22 '12 at 18:10

## 1 Answer

This is an old problem. Zeno of Elea is credited with some classical pointed formulations of it about 2500 years ago. (Note: not "Xeno" as one commenter above spelled him).

There are infinitely many different places to be at between 0 cm and 1 cm, but by the same token there are also infinitely many different instants in, say, one second, so they match up nicely.

Now, whether space and time can physically be subdivided infinitely finely is not a mathematical question. It's just that the most common mathematical model of them allows arbitrarily fine divisions, because that is much easier to deal with than the alternative and consistently seems to lead to useful results in practice. It is perfectly conceivable that actual time or space cannot be divided indefinitely; that would just mean that the mathematical model is not an accurate description at small enough scales. (Again, this would not be a mathematical problem. The model might describe something else, or describe no physical situation, and it would be no worse as mathematics for that).

As a physical question, the last few hundred year's physics has shown that matter cannot be subdivided indefinitely; a a scale of around 0.00000001 cm you find atoms that cannot be cut up without fundamentally changing what they are. However, the atoms are still thought to move around in a fundamentally continuous space. That might change with the next unpredictable revolution in physics, of course.

-
so when im moving my hand 3 cm , i actually moving through all the atoms within the 3cm aligned side by side , but im doing it pretty fast so its fine.....correct ? – Royi Namir Jun 22 '12 at 18:12
I don't see which possible problem you're referring to by "it's fine" here. – Henning Makholm Jun 22 '12 at 18:13
still....physically its settled.... but mathematically it doesnt....:( – Royi Namir Jun 22 '12 at 18:13
"possible "....i meant.....sorry. – Royi Namir Jun 22 '12 at 18:14
You're making completely no sense, I'm afraid. – Henning Makholm Jun 22 '12 at 18:17