PAI Private Arithmetics Institute       

Natural Numbers, Recursion, Geometry, Symmetry, Axioms, Logic, Metalanguage and more 


     Letter to the reader using conventional notation


An anthology of somewhat strange answers:

1.  "The expression 'true' in ZFC depends on what one understands by a model. This, however, is not unambiguous for ZFC."

2.  "The separation axioms just state that one can do the 'intersection' of a set with a formula, however, it does not claim that it actually says what this intersection set is"

3.  "ZFC is incomplete. We assume ZFC to be consistent. Due to incompleteness the separation axioms may lead to so-called Schrödinger-sets, i.e. sets whose status is unknown, just as one does not know if Schrödinger's cat is dead or alive. Progressing to a fixed ZFC model (i.e. a consistent completion) the Schrödinger phenomenon of definition disappears. But nobody knows which ZFC model one should favour. Already the first additional axiom is controversial, leave alone completion."

4.  "Axiomatic set theory usually is understood as a foundation theory, that carries a different status as compared to 'axioms' of a certain kind of mathematical structure, i.e. as stipulations in the definiens of the definition of a certain kind of mathematical structure (e.g. a certain geometry)."

5.  "Set theory also has pragmatic and normative features that transcend pure logic. The analysis of set theory rather belongs to the methodology of sciences."