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."

Van Gogh (source public domain)

He asks if the axioms of separation are hoaxioms. Why van Gogh ?

This letter has been sent to

Josef Berger, LMU München

John P. Burgess, Princeton University

Sy-David Friedman, Universität Wien

Peter Koellner, Harvard University

Peter Koepke, Universität Bonn

Hannes Leitgeb, LMU München

Benedikt Löwe, Universität Hamburg

Penelope Maddy, University of California

Heike Mildenberger, ALU Freiburg

Helmut Schwichtenberg, LMU München

W. Hugh Woodin, Harvard University

A few of the contacted professors answered to the above letter, some quite kindly. However, no one saw any real problem. All of them took immediate refuge to model theory. No one discussed the fact that formula strings are enumerable and that in general formula strings do not represent properties and relations in a basis-incomplete system.