hakui said:
Infinite is an interesting concept. Is there anyone who can understand it ?
:souka:
Hmm. Interesting question Hakui.
I have heard many interesting legends about 'infinity.' First let me begin with the etymology of in-fin-it-y from M-W.
infinity (14c) n. < infinite (14c) adj. ME infinit < MF infinit < L infinitus: in+finitus 'finite'
finite (15c) adj. < ME finit < L finitus: pp of L finire 'to finish'
1a. having definite or definable limits <~ no. of possibiities>
1b. having a limited nature of existence <~ beings>
2. completely determinable in theory in theory or in fact by counting, measurement, or thought <the ~ velocity of light>
3a. less than an arbitrary positive integer and greater than the negative of that integer
3b. having a finite number of elements <a ~ set>
4. of, relating to, or being a verb or verb form that can function as a predicate or as the initial element of one and that is limited (as in tense, person, and number)
According to above definitions and the usage in mathematical/scientific contexts, there appears to be two defnitely distinquishable senses in which the notion of 'infinity' can be used.
1. counting: "There are an infinite number of natural numbers." Mathematically speaking, for whatever big natural number n one can propose, another can always beat that number by counter offering n + 1. So the statement would be identical to "There is no greatest number in natural numbers." or "That there are a finite number of natural numbers is false."
2. measurement: "The maximum value in a set of all possible natural numbers is infinite." Mathematically the argument would follow a similar pattern; but this time counting is not an issue. For any given big natural number, its magnitude can always be exceeded by the magnitude of another natural number."
A notable case where a confusion of the two have caused an interesting paradox in logic is Zeno's Paradox in which Hercules and the tortoise have a racing match of say 5 stadia, or 1,000 m; the tortoise being hard working but naturally slow, say 1/10 as fast as Hercules, is given a head start of 900 m. Let us say we take measurments every time Hercules catches up to the previous position of the tortoise.
Hercules: 0.....900..990..999....on & on
Tortoise: 900..990..999..999.9..on & on
Zeno's Paradox says, "No matter how hard Hercules runs, he can never outrun the tortoise. As can be seen in the illustration, Hercules will always be a little behind the tortoise."
What is wrong with Zeno's argument ? What can be said about infinity from this example ?