I wish to correct my example in
http://www.internationalskeptics.com/forums/showpost.php?p=13165612&postcount=100, since it does not satisfy what I actually wish to say.
A corrected example: N members have at least two common properties, which are quantity and/or order of the members of a given set (including N's members.)
One of the ways to construct N members in terms of sets is done by "von Neumann ordinals" (
http://www.internationalskeptics.com/forums/showpost.php?p=13163749&postcount=88) and it is equivalent to ZF axiom of infinity.
The Axiom of Infinity establishes the existence of a set with certain properties. The set contains the empty set and what I will call other base elements. The set contains the successors of all of its elements (which would be its base elements, and their successors, and their successors, and ...).
A set which can be considered the minimal set satisfying the axiom's conditions is the set I will call N. It includes only the empty set as a base element. The empty set is the only element of N that is not the successor another element.
N members are constructed by no members (the empty set) or by all their finitely many predecessors so, given any arbitrary N successor it is a finite set whether it is defined in terms of "von Neumann ordinals" or its equivalent expression, called ZF axiom of infinity.
So, no matter what names ("von Neumann ordinals" or "ZF axiom of infinity", etc.) are related to N, it has finitely many members.
The only way to provide
an illusion that N has infinitely many members is to define the cardinality of any arbitrary set, such that it is defined by N members, but not by any particular N member.
By doing so, we ignore any particular N member and focused only on the fact that no N successor is the largest possible construction.
So we have a starting point (the empty set) but no end point (because no N successor is the largest possible construction).
The lack of an end point does not actually guarantee infinitely many N members, therefore N is at best can be defined in terms of potential infinity, and it is done by an arbitrary set A cardinality (where one of these sets is N itself) by N members, but not by any particular N member.
As a result A finite cardinality is not satisfied, but it does not mean that "it opens the door" to actually infinite cardinality.
So being non-finite is not actually be infinite, as give by the following definition:
Definition 4: Set A is called non-finite iff given any n in N, |A| is not any particular n
For any give set A, which particular n (or not any) would that be?
Please look at definitions 3 and 4 in
http://www.internationalskeptics.com/forums/showpost.php?p=13164777&postcount=98.
I have. There were a lot of words in those posts, but nowhere did you define anything. How, for example, would one determine that |D| = 4 and not 7 for some set D?
By definition 3.
Definition 3: Set D is called finite iff given any n in N, |D| is any particular n
In means that |D| is defined by a particular n.
