So, I was thinking a lot about different ways to order the irrationals. Most of them had to do with infinite-dimensional row/columns, but nothing was really working (obviously - as if I'm going to prove Cantor wrong LOL). However, despite the clear history against me, I'm wondering if it's possible to order the irrationals(adsbygoogle = window.adsbygoogle || []).push({}); randomly.

It would work like this: You have a random number generator, which also contains a decimal point, which acts like another number, but can only be used once (i.e. 11 "numbers" in this machine: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, ".") You'd start the number generator, and when it spits out the second non-decimal, you use that as the first of a new number, and start a second number generator. You then repeat the process and have more and more number generators working (eventually an infinite number.) If the second number spit out is a ".", you use the third number spit out. It's not exactly an "ordering", but there is there any real reason why this ordering couldn't be random? EDIT: The ordering would be that the number associated with the first number generator is 1st, the second is 2nd, and so on.

The reason I started thinking about this at all was because, in my opinion, I found Cantor's proof that the irrationals can't be ordered flawed. I'm sure the vast majority of the mathematical community disagrees, but I'll bet more than a handful find his proof a bit hand-wavy, at least the popular version. I haven't read the original. Cantor's proof depends on the list having an infinite # of entries with which to construct his new number which isn't on the list. But if you must construct this number from an infinity of numbers in your list, how can you ever know whether this new number isn't already on the list? In other words, if you can never get to the end of the list, can you really ever know if that "new" number you're constructing isn't there? And if one says that you can use the order of the list to rule that possibility out, that would seem to contradict his proof right there because Cantor's proof is supposedly saying that there ISN'T such an ordering.

I know that was probably hard to follow, and sounds like crazy jibber-jabber, but I'm very interested in this stuff, and his proof doesn't ring true to me (for the reasons above).

So... thoughts? On the random number generator idea and my critique of Cantor's proof? I'd love to hear what all of you think, especially the sages around here.

- Chris

**Physics Forums - The Fusion of Science and Community**

Dismiss Notice

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# By Listing Them Randomly, Could we Count the Irrationals?

Loading...

Similar Threads for Listing Randomly Count | Date |
---|---|

How to build a random vector perpendicular to another vector in R3 | Aug 19, 2014 |

List the things f has to do to be a homomorphism | Feb 15, 2013 |

Gaps in sequantial list. | Feb 20, 2012 |

Span of a list of vectors | Feb 10, 2012 |

Listing the elements of a symmetry group of a frieze pattern | Apr 16, 2011 |

**Physics Forums - The Fusion of Science and Community**