<blockquote id="quote"><font size="2" face="Verdana, Arial, Helvetica" id="quote">quote:<hr height="1" noshade id="quote"><i>Originally posted by tvanflandern</i>
<br /><blockquote id="quote"><font size="2" face="Verdana, Arial, Helvetica" id="quote">quote:<hr height="1" noshade id="quote"><i>Originally posted by 1234567890</i>
<br />On a number line from 1 to 5, since infinite divisibility is allowed, there are an infinite number of points between 1 and 2, an infinite number of points between 2 and 1, 3 and 1 , 4 and 1, and 5 and 1. So which distance is larger? If you say 5 and 1 then there are more points between 5 and 1 than the other pairs. But if that is the case then the distance between the other pairs must not be infinitely divisible, no?<hr height="1" noshade id="quote"></blockquote id="quote"></font id="quote">The mathematics of infinities is a very interesting subject. It has specific rules, it has good reasons for those rules, and the set of rules is consistent; i.e., no contradictions (such as your example above) can arise. You will really enjoy learning it.
Sticking to the lowest order of infinity (there are higher orders, as jrich mentioned), here are a few basics, where "inf" = infinity, "con" = any positive finite constant, and "indet" = indeterminate (can be anything):
inf+con = inf
inf-con = inf
inf*con = inf
inf/con = inf
inf^con = inf
inf+inf = inf
inf-inf = indet
inf*inf = inf
inf/inf = indet
inf^inf = a higher order of infinity
When you asked above "which distance is larger?", distance is not based on counting points or else all line segments would have infinite length. The number of points in all line segments, however large or small, is infinite. And by the above rules, the difference between the number of points in any two line segments is indeterminate.
So when you say "the distance between the other pairs must not be infinitely divisible, no?", the answer is indeed "no". They are all infinitely divisible, and that is consistent with the rules for the mathematics of infinities.
If your next question is "why", that is where Larry's advice about reading a book or two, and my earlier recommendation about a visit to your local library or ordering a book from the internet, come into play.
The math of infinities is a <font color="orange"><i>big</i></font id="orange"> subject. [8D] -|Tom|-
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Thx for the advice.
