## A Flaw in Separation Axioms

**Authors:** Hannes Hutzelmeyer

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:
- precise talking about formula strings necessitates a formalized metalanguage
- 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
- the Axiom of Infinity is generalized in order to allow for certain other infinite sets besides the natural
number representation according to von Neumann (i.e. general recursion)
- the Axioms of Separation is modified as it seems more convenient
- 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
- 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
- the Axioms of Separation seem to have a flaw; there is a caveat for axiomatic set theory.

**Comments:** 18 Pages.

**Download:** **PDF**

### Submission history

[v1] 2019-09-24 08:00:29

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