Karlis Podnieks podnieks at
Wed Apr 22 02:13:01 EDT 1998

Preparing my (hyper)textbook for students "Around the Goedel's
theorem" (see the current state of this work on my home page) I
arrived to the following problem.

In Section 2.3 "Axioms of ZF" the critical point is reached at
the moment when the axiom of extensionality and all
comprehension axioms (separation, pairing, union, power-set,
replacement, infinity) are already introduced, and the teacher
must explain, why the next - the regularity axiom should be
believed "natural".

At this point a student my ask: let us stop here, and adopt the
axiom "All sets can be built by using our comprehension axioms",
how could this be done? I am not professional in set theory, but
my first reaction would be to reply: this can be done by using
the method invented by Kurt Goedel in 1938. He replaced the
comprehension axioms by a fixed set of 7 (or so) operations.
Using these operations we can define the class L of all
"constructible" sets. After this the desired axiom can be
expressed as V=L. And after this an absolute nirvana comes to
you: you can  p r o v e  AC and GCH.

My feeling says that the assertion "He replaced..."  is somewhat
dangerous. May I ask the opinion of professionals?

Best wishes, K.Podnieks, podnieks at
University of Latvia, Institute of Mathematics and Computer

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