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    Joined: Sep 2007
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    We'll I'm laughing now, 'Neato! grin Crickets! LOL! You're totally cracking me up!

    And Dottie got it. So I figure I'm the problem with this one, not you!


    Kriston
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    Well, I think Dottie was jumping in to save my butt, she didn't want me to be embarrassed.

    TOO LATE!

    So kind though, don't you think?

    Must...........find bed........brain shutting down.......fingers typing...trying to supercede executive funtion.............

    smile

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    You misspelled "function" there, so I think you went one post too far for your brain...

    :p

    I should shut up and shut down, too. G'night, 'Neato. smile


    Kriston
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    There are different levels of infinity. That is for sure.

    A simple way to get to infinity is the set of all numbers N+1. For any number I can name a number bigger by adding 1. This is a very small infinity. But its much bigger than all the stars in the universe.

    An intermediate set is the set of all numbers derived by a set of functions where 1/N^K where k is all the numbers from 1..N.

    Then there are the numbers from the set of ALL functions that are continous, which is also infinite.

    In each above case, you can show that each set is infinitely larger than the other.

    And it gets even more interesting than that.

    It gives me vertigo when I think about it.




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    LMom and Austin are right that infinities come in different sizes (different "cardinalities"). One fun fact, which corresponds to the fact that (for a given cardinality) infinity plus infinity equals infinity is this: the set of all the even numbers is exactly as large as the set of all the odd numbers. Furthermore, each of these is exactly as large as the set of all the integers. You can tell that the evens are the same size as the odds by lining them up (putting them in "one-to-one correspondence"). Across from 1 put 2, across from 3 put 4, across from 5 put 6, and so on. Since it never happens that you have an odd number that has no even pair (or vice-versa), you've proven that the two sets have the same number of members. Then you can do this with either evens or odds on one side and all the integers on the other. It's one of the weird paradoxes of infinite sets that there are as many even numbers as there are integers.

    But not all infinite sets are the same size. Georg Cantor, in the late 19th c., used the notion of one-to-one correspondence to prove that there are more real numbers than integers. It's called Cantor's diagonal argument. It's a pretty simple argument - many of the mathy kids will be able to understand it. What's neat is that it's not known whether there are any sets that are bigger than the natural numbers but smaller than the reals. Cantor hypothesized that there was no set whose size is strictly between that of the integers and that of the reals. This was called the Continuum Hypothesis (abbreviated CH, and referring to the fact that the set of reals form a continuum). In 1940 Goedel showed that using the standard axioms of set theory CH cannot be proved. Then in 1963 Paul Cohen proved that using those same standard axioms CH cannot be disproved. So CH is said to be "independent" of the standard axioms of set theory. One of the things that set theorist do is to look for ways to augment the existing axioms of set theory that allow us to decide the truth of otherwise independent hypotheses.

    Contemporary set theory is a very hoary subject, and I'm at best someone who enjoys watching from the sidelines. But the introduction to the problems is very recent - starting with Cantor in the late 19th c. - and the moves made in the first half century or so are all pretty understandable. Anyone with mathy kids might find it an interesting project to pursue.

    BB

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    Interestingly, the rational numbers are countably infinite as well.

    Mathy kids may have fun finding a way to map the rational numbers onto the integers.

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    You guys are cracking me up... in a very positive way, mind you.

    And we wonder why our kids are obsessed with numbers! Sheeeeesh.
    The apple doesn't fall too far from the tree!

    By complete synchronicity, I just got the following joke in an email...

    "Aleph-null bottles of beer on the wall. Aleph-null bottles of beer.
    Take one down pass it around. Aleph-null bottles of beer on the wall.
    Aleph-null bottles of beer on the wall..."

    (Only here could I share that...)


    Mom to DS12 and DD3
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    Is there a scratching head icon w/ confused expression on face?

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    Originally Posted by ebeth
    "Aleph-null bottles of beer on the wall. Aleph-null bottles of beer.
    Take one down pass it around. Aleph-null bottles of beer on the wall.
    Aleph-null bottles of beer on the wall..."

    (Only here could I share that...)

    laugh laugh laugh

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    aleph-null equals infinity.

    Infinity minus one is still infinity.

    Is that what you were confused about Dazed?

    (I only really laughed hard at the email after reading the reference to aleph-null in a previous post. I read my email the first time fast enough that I didn't get the repeating 99 bottles of beer on the wall part of the joke!)

    Okay, okay... maybe it wasn't that good of a joke! But I tried!!


    Mom to DS12 and DD3
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