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Transfinite Arithmetic


Date: 10/28/97 at 17:08:19
From: MARK SIMAKOVSKY
Subject: Transfinite Arithmetic

What is transfinite arithmetic? I pretty much know what it means, but 
I am having trouble applying it to aleph-null. What is the relation 
between aleph-null, aleph-one, and aleph-two? How do you put these 
into sets?


Date: 10/28/97 at 17:37:19
From: Doctor Tom
Subject: Re: Transfinite Arithmetic

Hi Mark,

I'll give you a start here, but you've basically asked a question
that requires a college course to cover entirely.  Find a book on
set theory for details.

The easiest construction of the ordinal numbers is to define zero as
the empty set, and if you've already defined the number n, the number
n+1 = n union { n }

So (using "U" to indicate "union"):

0 = { }
1 = 0 U { 0 } = { } U { 0 } = { 0 }
2 = 1 U { 1 } = { 0 } U { 1 } = { 0, 1 }
3 = 2 U { 2 } = { 0, 1 } U { 2 } = { 0, 1, 2 }

and so on.

For the finite ordinals, you basically see that:

n = { 0, 1, 2 ,3 ..., (n-1) }

The ordinal numbers are the smallest set that:

1) contains the number zero = { }.
2) if it contains n, it contains n+1 (as defined above)
3) if S is a set of ordinal numbers, so is the union of all
   the elements of S.

If you leave out condition 3, you just get the natural numbers,
which is the same as the finite ordinals.

But the set { 0, 1, 2, 3, ... } is a set of ordinals, so
the union of all those sets is also an ordinal, and you'll see
that it is { 0, 1, 2, ... } -- the same thing.

But notice that every ordinal is a set containing the "right"
number of things -- 3 is a set with 3 items in it: 0, 1, and 2.

The set above is not any of the finite ordinals -- it's larger
than any of them -- it is infinite, the smallest size of
infinity, and is called aleph-0, where aleph is the first letter
of the Hebrew alphabet.

But since aleph-0 is an ordinal, so is aleph-0 + 1:

aleph-0 + 1 = aleph-0 U { aleph-0 } = { aleph-0, 0, 1, 2, ... }

it is different from aleph-0.  Similarly, you can get
aleph-0 + 2, aleph-0 + 3, ... and so on.  Then you can union
all of those and get aleph-0 * 2, and on, and on, and on ...

Eventually, you get to a set so large that it can't be matched
one-to-one with aleph-0, and the first time that happens, call
it aleph-1 -- the second smallest size of infinity.  This goes
on and on, too.

Arithmetic can be defined on the ordinals, and that's what you
should look up in a set-theory book.

You've probably seen a proof (Cantor's diagonalization proof) that
there are more real numbers than integers.  That means that C, the
size of the real numbers, is larger than aleph-0,  Up until about
1965 or so, nobody knew if C was equal to aleph-1 or not.  Then
Paul Cohen of Stanford proved that the theorem "C = aleph-1" is
independent of the other axioms of set theory.  In other words, it
can be neither proved nor disproved.  This is only the first of
a large set of amazing facts about the ordinal numbers.

But find a set-theory book -- I'm too lazy to type in chapter after
chapter. :^)

-Doctor Tom,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/   
    
Associated Topics:
College Definitions
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High School Definitions
High School Number Theory
High School Sets

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