# [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
```