Re: Alephs - Alephs

From: David C. Ullrich (ullrich_at_math.okstate.edu)
Date: 01/20/05 

Date: Thu, 20 Jan 2005 05:32:15 -0600

On 19 Jan 2005 20:51:29 -0800, "Butch Malahide" <bof@sunflower.com>
wrote:

>
>David C. Ullrich wrote:
>> On 19 Jan 2005 13:50:19 -0800, "Poly-Poly Man"
>> <pyrophobicman@yahoo.com> wrote:
>>
>> >Okay, is the difference between aleph 1 and 2 aleph 1?
>
>No, aleph_a - aleph_b = aleph_a if b < a, undefined otherwise.
>
>> Subtraction of infinite cardinals is undefined.
>
>Interestingly, chapter IX of Sierpinski's classic text "Cardinal and
>Ordinal Numbers" is titled "Difference of cardinal numbers". Quoting
>Sierpinski: "For the cardinal numbers m and n we say that the
>difference m - n exists if there exists one and only one cardinal
>number p such that m = n + p. We denote this number p by m - n."

Ok, I lied.

>Subtraction of infinite cardinals is usually ignored in elementary
>texts, and it is not very interesting if you are working in ZFC, or if
>the cardinals are alephs, as in the original poster's question.
>However, there are some interesting and nontrivial ZF-theorems about
>subtraction of cardinals. For example, Sierpinski proves the following
>theorem of Tarski:
>
>"We are able to prove without the axiom of choice that if m is a
>cardinal number greater than or equal to aleph_0, then
>2^m - m = 2^m."
>
>The meaning is that if, from a set of cardinality 2^m you remove an
>*arbitrary* subset of cardinality m, you are left with a set of
>cardinality 2^m; and the point is that this can be proved without the
>axiom of choice.

************************

David C. Ullrich

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