Axiomatic set theory conundrum
The author has developed an approach to logics that comprises, but also goes beyond predicate logic. The FUME method contains two tiers of precise languages: object-language Funcish and metalanguage Mencish. It allows for a very wide application in mathematics from geometry, number theory, recursion theory and axiomatic set theory with first-order logic, to higher-order logic theory of real numbers etc. . The conventional treatment of axiomatic set theory (ZFC) is replaced by the abstract calcule sigma so that certain shortcomings can be avoided by the use of Funcish-Mencish language hierarchy:
1. precise talking about formula strings necessitates a formalized metalanguage
2. talking about open arities, general tuples, open dimensions of spaces, finite systems of open cardinality and so on necessitates a formalized metalanguage. 'dot dot dot … ' just will not do
3. the Axiom of Infinity is generalized to Axioms of Recursion in order to allow for certain other infinite sets besides the natural number representation according to von Neumann
4. the Axioms of Separation are modified as it seems more convenient
5. there are only enumerably many properties that can be constructed from formula strings, as these are finite strings of characters from a finite alphabet; this should be kept in mind in connection with the Axioms of Replacement
6. a new look at Cantor's continuum hypothesis in abstract axiomatic set theory leads to the question of so-called basis-incompleteness versus proof-incompleteness
7. the Axioms of Separation seem to have a flaw; there is a caveat for axiomatic set theory.
There is a wild mess in conventional notation of axiomatic set theory with small and capital letters (Latin, Greek and even Hebrew) and ad-hoc introduced symbols. Similar expressions may relate to completely different categories. Sometimes the same expressions are used with different meaning so that they can only be understood in local context. Some entities are just introduced metalingually without proper representation in object-language. And one should always be skeptical when 'virtual' entities come into play as it is the case in connection with classes. This mess makes it very complicated for the simple-minded reader and perhaps even hides problems. Axiomatic set-theory talks about well-ordering, but it certainly needs ORDER itself!
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