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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 ?

Arturo Magidin
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Royi Namir
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    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
  • See http://math.stackexchange.com/questions/142932/achilles-and-the-tortoise-paradox – talmid Jun 22 '12 at 17:54
  • @AlexBecker is there any explanation to this paradox ? – Royi Namir Jun 22 '12 at 17:54
  • @RoyiNamir Plenty, but Wikipedia's treatment is more thorough than any I could give here. – Alex Becker Jun 22 '12 at 17:55
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    @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
  • @ArturoMagidin Really? Huh, I guess I'm just used to the way a certain group of philosophers use it. – Alex Becker Jun 22 '12 at 18:03
  • @AlexBecker i dont see the connection to the turtule and the runner....im talking about the turtle only which has to wolk throught infinite numbers to go 3 cm... – Royi Namir Jun 22 '12 at 18:04
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    @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

2 Answers2

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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.

hmakholm left over Monica
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Infinity is not a number. However, even in mathematics textbook, infinity is used as a number: e.g. the set of natural numbers has infinity members. Therefore, infinity as commonly used even in mathematics is ambiguous. So when asking questions that involve infinity, the first clarification should be: what is the meaning of infinity (in the question)?

If we treat infinity as not a number, then we can't really understand its meaning in a number context. I.e. "go through an infinite series of numbers" is a question with no sensible interpretation.

If we treat infinity as a number, it must mean an arbitrary big number that beyond that big number, any effects are negligible and should be ignored. This interpretation has "arbitrary" embedded, which is still not defined in general but can be well defined in practical context. And it is up to the person who poses the question to define the context of his question before his question to be answered (or studied). Posing question without an acknowledgment of where the big number lies where beyond which the effect can be ignored is essentially asking questions about moving target, and it is ill-defined.

Now to the question: "how can a person walk 3 cm if he had to go through an infinite series of numbers ?"

We need to establish an arbitrary but definite number we all agree on that beyond which has no effect (regarding the context). Let's assume an arbitrary N, which can be very big, and we can easily see that no matter what N is, a person can go through N numbers in finite time. And if we define a constant speed and assume the N intervals are uniformly defined, we can calculate the time a person will take to cross the entire 3cm, and we need understand that we already assumed that any number beyond N (be it 100 or 1 million) will have no real effect (in this case, the crossing time) and can be ignored. So with the establishment of this context, we can meaningfully deal with all practical calculations that involve infinity (arbitrary big or small but finite).

Of course, the classical mathematics treatment is to use the concept of limit. Limit has essentially the same problem (as infinity) of ambiguity around finite/infinite divide, which is why Zeno of Elea posed the question in the first place! By merely invoking limit, we merely paraphrased the question without actually addressing the problem (the ambiguity around finite/infinite divide).

To extend my argument, a "real" number should not be treated as a number if its definition is rooted in "infinity". A number should always be assumed to have a precision attached (without better practical context, let's assume it to be arbitrary, but finite).

In physics, this arbitrariness is always well defined by the physical resolution in question.

Hui Zhou
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