🚀 go-pugleaf

Greg d. Moore wrote:
> 
> jon murray wrote:
> >
> > William Chops Westfield wrote:
> > >
> > >     Which is the larger set of numbers: All positive integers, or all integers?
> > >     Both are infinity, but is one set twice as large as the other?
> > >
> > > They're the same size.  The set of all rational number (fractions) is
> > > the same size too.  But the set of irrational numbers is bigger...
> > >
> > > That should be obvious.
> > >
> >
> > Well, not being a mathematitian, it isn't obvious to me. Is there any
> > imaginable situation where my lack of insight is likely to cause me a
> > problem?
> >
> 
>         No, but the explanation is rather simple.
> 
> Integers        1 -1 2 -2 3 -3 4 -4 etc... (This will obviously go to
> infinity)
> Positive Int    1  2 3  4 5  6 7  8 etc.  (This too will go to infinity).
> 
> Basically you can count to infinity with either one.
> 
> The reason irrationals are a "bigger" infinity is as follows.
> 
> Count from 1 to 2 with every irrational inbetween...
> 
> All Number set:   1 1.1 1.11 1.111 1.1111 1.11111 ...... up to 2.
> Integers          1  2   3    4     5      6   up to infinity.
> 
> So, (wrap your mind around this :-) you get to infinity with the
> integers before you get to 2.  So what happens when you finally get to
> 2?  You're out of inifite integers, but you've got an inifity to go. :-)
> 
> In other words, it takes an infinite number of "counters" to get from
> 1.0 to 2.0.  So, it must take a BIGGER infinite number of counters to
> get from 1.0 to infinity.0.
> 
> Make sense?

Not exactly:  The same argument can be conceivably used to try and
"Prove" that the rational numbers are larger than the integers:

take the integers n from 1 to infinity, and map them to the rational
numbers 0 + 1/n between 0 and 1.  You can reach infinity and not even
have mapped all the rational numbers between 0 and 1.  All this shows is
that you have chosen the wrong map.  The way to prove that one set is
different from the other is to prove that no map exists.  One exists for
the integers<->rationals, but none exists for integers<->reals.
> 
> > Jon
> >
> > *********
> > My address has been spam-protected.
> > It's 'netconnect' & I live in Australia (.au)
> > *********
> 
> --
> Greg D. Moore   President       mooregr@greenms.com
> Green Mountain Software         http://www.greenms.com/
> 518-273-8602


GuildMaster Arrataz
(Math Concentrator, UofC)