[FOM] cardinality beyond regularity and Choice

Irving ianellis at iupui.edu
Wed Dec 16 14:29:00 EST 2009

Zuhair Abdul Ghafoor Al-Johar <zaljohar at yahoo.com> writes:

> Subject: Re: [FOM] cardinality beyond regularity and Choice
> To: fom at cs.nyu.edu
> Message-ID: <751108.7071.qm at web31401.mail.mud.yahoo.com>
> Content-Type: text/plain; charset=us-ascii
>> Irving H. Anellis wrote:
>> it was of course Georg Cantor who first introduced cardinal
> numbers and defined the concept of cardinality (as the "Machtigkeit" of
> a set)
> ........................
> I agree that Georg Cantor was the first
> who introduced Cardinality
> but he didn't define it! he only characterized it.
> Cantor's approach to Cardinality was more akin
> to the primitive approach of Cardinality that I
> mentioned in this post.
> It was Frege who presented a "DEFINED" concept
> of Cardinality, a definition that
> Russell revived later.
> Zuhair
> ------------------------------
In 1878, Cantor wrote (at S. 119in "Ein Beitrag zur 
Mannigfaltigkeitslehre", Journal fu"r die reinen und angewandte Math, 
84 (1878) 119)-133:

Wenn zwei wohldefinirte Manngfaltigkeiten M und N sich eindeutig un 
vollsta"/ndig, Element fu"/r Element, einander zuordnen lassen (was 
wenn es auf eine Art mo/glich ist, immer auch noch auf viele andere 
Weisen geschehen kann), so mo"/ge fu"r das Folgende die Ausdruckweise 
gestattet sein, dass diese Mannigfaltigkeiten {it/ gleiche 
Ma"chtigkeit/} haben, oder auch, dass sie {it/a"/quivalent/} sind.

Granted, in this passage Cantor does not explicitly say that the 
Machtigkeit of a set is its cardinality. In this sense, then Prof. 
Al-Johar is correct in asserting that (in this particular passage, at 
least), Cantor is characterizing cardinality in terms of comparability 
of the power of a set, rather than explicitly defining cardinality.

Irving H. Anellis
Visiting Research Associate
Peirce Edition, Institute for American Thought
902 W. New York St.
Indiana University-Purdue University at Indianapolis
Indianapolis, IN 46202-5159
URL: http://www.irvinganellis.info

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