Re: infinity

On Thu, 25 Aug 2005 19:11:40 -0600, Virgil
<ITSnetNOTcom#virgil@xxxxxxxxxxx> wrote:

>In article <htisg153qtakcrjbs3n5j28msiddevefrn@xxxxxxx>,
> Martin Shobe <mshobe@xxxxxxxxxxxxx> wrote:
>
>> On 24 Aug 2005 08:23:13 -0700, "Jiri Lebl" <jirka@xxxxxx> wrote:
>>
>> >*** T. Winter wrote:
>> >> As aleph_0 is (in the cardinal numbers) not divisible by any natural
>> >> number,
>> >> it could be considered prime. Whether it is odd or even depends on your
>> >> definition. If you define a number even whenever it is divisible by 2,
>> >> aleph_0 is odd.
>> >
>> >I think this is taking it too far. The property of being prime, odd or
>> >even is a property of natural numbers, not of cardinal numbers
>> >(actually being prime is a property of a ring and the cardinal numbers
>> >are NOT a ring, they're not even a set).
>>
>> I consider this an example of where the reliance on set theory has
>> hurt mathematics. Whether or not the objects form a set, is rather
>> peripheral to what a ring is. (How often do you actually use the set
>> part of the definition?)
>
>The closure of the addition operation and multiplication operation of
>rings depend essentially on having a set of objects, as does the closure
>versus non-closure of division by non-zero objects in distinguishing
>betseen rings, for example the ring of integers versus the ring of
>ratiionals.

Sorry, but you don't need sets for that. Cardinal arithmatic is
closed, but the cardinals do not form a set.

Martin
.

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