For easy reference, here are those two definitions:
Definition 1: |A| = |B| iff there is bijection from A to B
Definition 2: |A| < |B| iff (there is injection no surjection from A to B) OR (A=∅ ∧ B~=∅).
Neither is "direct" nor "indirect". They each define the meaning of a relation between the cardinality of two not necessarily distinct sets.
Neither is restricted to the members of any particular set.
Neither supports your current definition for infinite set.
Additionally, Definition 2 is problematic because "injection no surjection" is not a meaningful phrase, and if you meant "injection that is not a surjection" then it has other problems. Everything from the OR and on is unnecessary.
jsfisher, in order to establish the minimal set satisfying the requirements of the Axiom of Infinity, one first has to show that
all the natural numbers (which are defined by von Neumann's treatment of the natural numbers in terms of sets) are actually establish a set that satisfying the requirements of the Axiom of Infinity.
So at the first stage the following definitions are restricted only to
all the members of this minimal set, such that the initial member is the empty set (
v=∅) and the rest of the members are defined as
v⋃{
v}, which are all finite sets:
Definition 1: |
v| = |
v| iff there is bijection from
v to
v.
Definition 2: |
v| < |
v⋃{
v}| iff (there is an injective non-surjective function from
v to
v⋃{
v}) OR (
v=∅ ∧
v⋃{
v}(≠∅)).
Definitions 1 and 2 look trivial at first glance, but these are exactly the only two possible functions among the finite sets, which define
all the natural numbers by von Neumann's treatment, where also (
v=∅ ∧
v⋃{
v}(≠∅)) must be added since there is
no function from
v=∅ to
v⋃{
v}(≠∅), but we wish to include also the inequality |∅| < |
v⋃{
v}(≠∅)|.
B.T.W, |∅| < |
v⋃{
v}(≠∅)| is not covered by your mathematical framework, which is
entirely based on the injective function.
Actually, one does not need more than this uncovered case, in order to show that a mathematical framework (which is
entirely based on the injective function) does not satisfy the requirements of the Axiom of Infinity.
Since
∀v ∈ V(
v∪{
v} ∈ V ∧ |
v∪{
v}| can't be but < |V|) |V| is not established, simply because it can't be defined as the cardinality of
all V members, which are
all finite sets by von Neumann's treatment of
all the natural numbers.
=======================
A non-finite set is any set that its cardinality can't be defined.
There is bijection from a non-finite set to its non-finite proper subset exactly because their cardinalities can't be defined.
=======================
Classical mathematicians are using an ad hoc trick, and inconstantly establish out of nowhere λ :
(
https://en.wikipedia.org/wiki/Limit_ordinal )
In set theory, a limit ordinal is an ordinal number that is neither zero nor a successor ordinal. Alternatively, an ordinal λ is a limit ordinal if there is an ordinal less than λ, and whenever β is an ordinal less than λ, then there exists an ordinal γ such that β < γ < λ. Every ordinal number is either zero, or a successor ordinal, or a limit ordinal.
(
https://en.wikipedia.org/wiki/Limit_cardinal )
In mathematics, limit cardinals are certain cardinal numbers. A cardinal number λ is a weak limit cardinal if λ is neither a successor cardinal nor zero. This means that one cannot "reach" λ from another cardinal by repeated successor operations. These cardinals are sometimes called simply "limit cardinals" when the context is clear.
This means that one cannot "reach" λ from another cardinal by repeated successor operations.
This means that λ is beyond collections, as seen, for example, in
http://www.internationalskeptics.com/forums/showpost.php?p=12584400&postcount=3285.
