John Collins wrote: >
There do exist consistent approaches to set theory where you do have a universal set and can therefore consider taking complements to be a sinle-argument operation. to bypass the obvious paradox (that any set can be used to make a necessarily larger powerset) you need to concoct a map from the universal set onto its own powerset.
I was not thinking of that one but rather to the inconsistency that appears when one wants to consider things like "the set of all sets that do not containe themselves".
The easiest way to do this is to have lots of 'urelements' or' indivisible but somehow different sets, which can then be mapped to larger sets in the powerset. If you find urelements philosophically objectionable (which most computationally-minded people do)
This is the first time I heard of such things as 'urelements' and I haven't that faintest idea of what that might be but, for sure, I must be severely "computationally-minded".
then there exist other more difficult approaches: Try a google search for "Alonzo Church", "Willard Quine" or "Thomas Forster" to see some people who have tried...
I have heard of the first two but not on that topic.