Without intending to blow my own didgeridoo,the purpose of this file is to correct that major over cite (sic!). Every paper accessible from this page is on Ferment Magazine. Each contains an idea, an insight, an invention, which the author believes originates with himself.
I am too busy to investigate alien priority claims.However, if readers of these articles come across the same ideas propounded elsewhere, please feel free to cry out to the plagiarist: "Hey! That idea comes from the Ferment Magazine website!"
To handle the problems associated with "counting", or "enumerating" an unstructured entity, Zermelo and others came up with the Axiom of Choice. This states that it is not necessary to know anything about the structure imposed on a set to be able to select elements from it.
This works very well at the beginning of the counting process;but what does one do when one has no information about how the process terminates? In particular, let C be an "unstructured" set that represents the first (countable) transfinite. By the Axiom of Choice, one can (without replacement), continue to choose elements from C until it is exhausted. As one makes the selection, one arrange these elements in a simple total linear ordering. However,by doing so,one is liable to end up with any one of the uncountable number of countable transfinite ordinals! There is no guarantee, at the beginning, that one arrives at ϖ, or ϖ +1, or 2ϖ.
In other words, one needs an Axiom of Choice at the beginning,and yet another Axiom of Choice at the end, one that which states that it is possible to "choose" a countable ordinal α from the set U of countable transfinite ordinals, and a method of selecting the elements of C, so that one will end up with α.
My paper, "Weakly Infinite Cardinals"proposes several possible solutions to this problem.Go to: