Re: Different size infinities?

From: David C. Ullrich (ullrich_at_math.okstate.edu)
Date: 10/29/04 

Date: Fri, 29 Oct 2004 08:32:28 -0500

On Fri, 29 Oct 2004 10:28:27 +0200, Han de Bruijn
<Han.deBruijn@DTO.TUDelft.NL> wrote:

>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";

This is very funny. It's like this was sci.chemistry
and I quoted four famous scientists from a century
or so ago doubting the existence of atoms. (Don't
know whether you've noticed, but those predictions
about how future generations would recoil in shock
and horror turned out to be simply wrong. What would
your point be in quoting people from a century ago
who turned out to be simply wrong about something?)

>As quoted from:
>
> http://www.mlahanas.de/Greeks/Infinite.htm
>
>Han de Bruijn

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

David C. Ullrich

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