A Novice Argument for the Utility and Coherence of Transfinite Subtraction

Below I lay out a set of definitions that I currently understand to constitute the basic paradigm of set theory. I then construct an argument, using those definitions, for the coherence and utility of transfinite subtraction. I can identify no logical contradiction so far, and I find the argument of interest because Craig says that one "cannot do inverse operations like subtraction in transfinite arithmetic with infinite quantities". I happen to believe that my objection to this point in Craig's "Kalam Cosmological Argument" is not fatal, but I shall not explicate that position here.

DEFINITIONS
Set – A collection of members
Member or Element – An object in a set
Proper subset – A set X whose every member belongs to another set Y, while Y includes members who do not belong to X
Equality – The state of a set X with another set Y, while X contains every one of and only the members of Y
Cardinality – The size of a set
Finite – The state of a set X whose members may be put into a one to one correspondence with the set {1, 2, 3… n}
Potential infinity – The state of a set X whose members may be put into a one to one correspondence with the set {1, 2, 3… n}, but whose cardinality may always be increased
Infinite – The state of a set X whose members may not be put into a one to one correspondence with the set {1, 2, 3… n}
Aleph-null or Aleph-naught – The cardinality of the set containing all rational numbers
Aleph-one – The cardinality of the set containing all real numbers
Aleph-n – The cardinality of the set containing the next higher cardinality than the set whose cardinality is aleph-(n-1).
Union – The set of the members of a set X and the members of a set Y
Intersection – The set of the members of a set X who also belong to a set Y
Power set – A set X of all possible subsets of a set Y. The set of a powerset has a larger cardinality than any subset of the set Y.

AN EXAMPLE OF ADDITION UNDER THE SET THEORY PARADIGM
If the intersection of a set X and a set Y is empty, and the cardinality of X is 3, and the cardinality of Y is 2, then the cardinality of the union of X and Y is 5.

ARGUMENT FOR THE UTILITY AND COHERENCE OF TRANSFINITE SUBTRACTION
One has not exhaustively defined a set if one has merely described its cardinality. For example, the set X, containing the members {1, 2, 3} has a cardinality of 3, and the set Y, containing the members {4, 5, 6}, also has a cardinality of 3. However, X is not equal to Y. “Infinity” is not a number; it is the state of any infinite set. Thus, the union of the set A whose member is {-1} and the set B whose members include zero and all positive whole numbers {0, 1, 2, 3…}, is to produce a set C with the members {-1, 0, 1, 2, 3…}. B and C are both infinite sets. However, C includes one member who is not in B, namely, {-1}. Therefore, C has a larger cardinality. It is therefore not a logical contradiction to say ‘infinity plus one is infinity’. Such a statement is unclear however, as it does not specify the members of either infinite set referenced, or the specific member (e.g. 1, 2, or 3) who is being added to the first infinite set.

Thus informed, transfinite subtraction becomes possible. For example, the above mentioned infinite set C minus the member -1 is the infinite set B. Thus, ‘infinity minus one is infinity’ is merely the statement ‘there is some infinite set D, that contains some member E, such that E may be subtracted from D, and D remains in the state infinite’.

[Update: I know blog at Philosophia Swingrovia.]