[FOM] Questions on Cantor

Vaughan Pratt pratt at cs.stanford.edu
Wed Jan 30 05:36:05 EST 2013

On 1/29/2013 11:56 AM, Arik Hinkis wrote:
> Isn't any well-order relation also a well-founded relation?

Yes.  That's what I meant by "a well-ordered set (X, <) can be defined 
as a linearly (totally) ordered set whose ordering relation < 
(understood as strict) is a well-founded relation."

> In his 1883
> Grundlagen Cantor characterized the order relation between his
> transfinite numbers (later ordinals) in terms of its well-founded
> properties.

"Well-founded" is a property of an arbitrary binary relation R on a set 
X, one definition of which is that there exists an infinite sequence x0, 
x1, x2, ... of elements of X such that x_{i+1}Rx_i holds for all i >= 0. 
  Where does Cantor refer to this property, however defined?

It's been my understanding that every partially ordered set considered 
in Grundlagen is linearly ordered.  Have I misunderstood Cantor?

> Well-order he characterized by a different property and he
> did not show there that the properties are equivalent.

In Grundlagen Cantor exhibited a linearly ordered set (strictly 
speaking, class) of cardinalities that he was able to show were all 
distinct.  He believed, incorrectly, that he had shown that all 
cardinalities belonged to that class.  Is that what you meant by "did 
not show?"

According to Dauben (the source I've been relying on) Cantor defined a 
well-ordered set as "any well-defined set in which the elements are 
arranged in a definite, given succession such that there is a first 
element of the set, and for every other element there is a definite 
successor, unless it is the last element of the succession." At some 
point (certainly by 1890) Cantor further required that every subset 
having a successor had an immediate successor (in order to rule out the 
case of the nonnegative integers followed by the negative integers each 
standardly ordered).

Cantor believed, again incorrectly, that he had shown that the 
cardinalities were well-ordered.  (One nice feature of the modern 
definition of a well-ordered set, namely as a partially ordered set 
every nonempty subset of which has a least element, is that "set" can be 
replaced by "class" without having to change "subset" to "subclass.") 
Perhaps you had that in mind instead of, or as well as, the above.

Vaughan Pratt

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