Maths says that there are more real numbers than integers, even though there are infinitely many of both types (yes, there are different types of infinity). What if I could come up with a way of matching real numbers one to one with integers, would that prove there are equally many?
How about this method of matching integers with real numbers between 0 and 1:
Take an integer (123)
Order the digits backwards (321)
Put the resulting number to the right of the decimal point (0.321)
This way, the counting would go:
0.1
0.2
0.3
…
0.9
0.01
0.11
0.21
…
0.99
0.001
0.101
…
It’s not in any order, but doesn’t this match every single real number (0,1) to every integer?
It is possible to show a way to count the integers - they are countably infinite. The reals however are uncountably infinite. (I’m not arguing against you - I know you are essentially saying the same thing.)
I know that. I’m trying to point out the flaws in the OP’s argument. He claims to have produced a map from the naturals to the reals in (0,1), but his map is actually from the naturals to the rationals that can be written as terminating decimals on (0,1).
Asking him what integers map to numbers outside this set exposes this discrepancy.
Doesn’t the integers include infinite numbers? If so, then why isn’t 333… an integer?
Ah, but Cantor’s Diagonal requires you to arrange the numbers in ascending order. This method doesn’t. If you were to try the same, the integer that maps to it is the previous one + 111…
The point is, there are many schemes to map between 2 sets of numbers. Some work, and some don’t. As long as there is one scheme that works, isn’t that enough to show they’re equal?
To get an integer that maps to an irrational number in this scheme, you need to know the last digit of the irrational number, which sounds impossible. But… is 14159264… (the digits of pi after 3.) an integer?
Oh, if they don’t include infinitely long numbers then this scheme doesn’t work then. I guess there’s a difference between infinitely many and infinitely long.
It’s been well known for a long time that there are no from the integers bijections to the reals or any interval of the reals. The most famous proof is Cantor’s diagonalization argument.
As others have pointed out what the OP maps to is numbers with terminatining decimals in (0,1) i.e. a subset of the rational numbers in (0,1).
Here’s the problem: Your list only consists of terminating decimals. That means it leaves out all of the irrational, and some of the rational, real numbers between 0 and 1. For example, as Lance Turbo points out, 1/3 (= 0.33333…) never appears on your list. Neither does pi/4.
Since it has been proved (by Cantor) that there can be no way of listing (or denumerating, or matching integers to) the set of all real numbers between 0 and 1, any such scheme is doomed to failure. You’re like a person who claims to have a way to trisect the angle with only straightedge and compass.
I mean none of this as a dis on you. Thinking about such things, and asking “What’s wrong with this argument,” is a perfectly good thing to be doing.
By the way, for anyone reading this thread who’s wondering what we’re talking about (different sizes of infinity? Cantor’s diagonalization argument?), one really good online explanation can be found here (at the Platonic Realms website):
