Re: Different size infinities?

From: Han de Bruijn (Han.deBruijn_at_DTO.TUDelft.NL)
Date: 10/29/04 

Date: Fri, 29 Oct 2004 10:28:27 +0200

Michael Barr wrote:
> Dave Seaman <dseaman@no.such.host> wrote in message news:<clpnj1$e3c$1@mozo.cc.purdue.edu>...
>
>>On 28 Oct 2004 02:07:05 GMT, R3769 wrote:
>>
>>>>I was exposed to the idea in class today (in a cursory manner), that
>>>>there are smaller and larger infinities.
>>>>
>>>>The explanation had to do with the infinity of "space" (or partitions)
>>>>of the interval [0,1] as compared with infinity of integers 1, 2,
>>>>3....
>>>>
>>>>There is a proof of an infinity being larger than another type of
>>>>infinity?
>>>>
>>>
>>>Yes. In fact it's not to hard to see there are an infinite number of different
>>>types of infinities.
>>
>>Which infinite number would that be? :-)
>
>
> And the answer is: None. More precisely, the _class_ of infinite
> cardinal numbers is not a set. There are more of them than can be
> counted with _any_ infinite number! And the variety of large infinite
> cardinals is just astonishing.

GAUSS: "I must protest most vehemently against the use of
the infinite as something consummated, as this is never
permitted in mathematics";

KRONECKER: "I don't know what predominates in Cantor's
theory - philosophy or theology, but I am sure that there
is no mathematics there";

POINCARE: "There is no actual infinity; Cantorians forgot
that and fell into contradictions. Later generations will
regard Mengenlehre as a disease from which one has to be
recovered ";

BROUWER: Cantor's theory as a whole is "a pathological
incident in history of mathematics from which future
generations will be horrified";

As quoted from:

        http://www.mlahanas.de/Greeks/Infinite.htm

Han de Bruijn

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