On mathoverflow I've posed the question in the title in connection to Muller's 2001 criteria for a founding theory of mathematics, which largely raised in connection to Category theory [see here]. Some aspects of this question are mathematical, specifically those concerning the consistency of RfST and whether the Counter-Reflection schema adds un-necessary strength beyond what is stipulated in Muller's criteria.

Here I want to discuss the philosophical aspect, that is: how far that theory fits Muller's criteria, and how far it's from H.Friedman's criteria, (pages:5-6) for founding mathematics (which largely raised in connection with large cardinal axioms)?

I personally think that RfST can prove ℘^ω(V), i.e. the union of all finite power iterations of V.

Muller's argument seem to suggest that this is not needed, specifically note this quote from him:

The conclusion of this brief informal analysis coincides partly with an assertion of Fraenkel, Bar-Hillel, Levy ´ and Van Dalen [1973, p. 143]: Category Theory involves only objects which are members of the classes V, ℘V, ℘℘V, . . ., ℘nV, where V is the class of all sets and n is some fixed finite number.

[end of page 10].

And also this quote from the very next page:

What is needed and sufficient is ‘only’ a denumerable sequence of increasing cumulation sets Ψ(i1 + 1) for ℘V, Ψ(i1 + 2) for ℘2V, until Ψ(i1 + ω), but certainly no more.

Anyhow, I think this theory might construct more classes, but it is a bit more and not mindbogglingly, flabbergastingly abundant as to constitute a violation of Bourbaki.

One of the points against simply upgrading class existence to class separation axiom as to get ARC of Muller's, is that it can be attacked as ad-hoc. I don't see a clear justification for it. By doing that we are replacing a very natural principle of class existence after predicates, by the technical fix of class separation. I'd prefer class separation to be a theorem rather than an axiom. And I think the plausible basis for it lies in Counter-Reflection, i.e. it is a part of a reflection process in the opposite direction, i.e. from sets to classes, this makes matters more thematic (to conform to Friedman's terminology) and less ad-hoc.

More generally speaking I see all axioms of RfST being very natural and nicely fit what's expected of a founding set\class theory for Category theory, and for most of mathematics in general.

Reflective Set Theory RfST is formulated in mono-sorted first order predicate logic with extra-logical primitives of equality ‘‘=", membership ‘‘∈", and a single primitive constant symbol V denoting the class of all sets.

The axioms are those of first order identity theory +

  1. Extensionality: ∀x(x∈a ↔ x∈b) → a=b
  2. Class comprehension: if φ(y) is a formula in which the symbol ‘‘y" occurs free, then all closures of: ∃x∀y(y∈x ↔ y∈V ∧ φ(y)), are axioms.
  3. Super-Transitivity: y⊂x ∧ x∈V → y∈V
  4. Reflection: if φ is a formula that doesn't use the symbol V, in which only y,x1,..,xn occur free, then: ∀x1,..,xn∈V[∃y(φ) → ∃y∈V(φ)], is an axiom.

  5. Counter-Reflection: if φ is a sentence that doesn't use the symbol V, and φV is the sentence obtained by merely bounding every quantifier in φ by V, then: (φV→φ) is an axiom.

  6. Foundation: ∃m∈x → ∃y∈x ∀z∈x(z∉y)
  7. Choice: ∀m,n∈X(∄k∈m(k∈n)) → ∃Y∀x∈X(x≠∅ → ∃!y∈Y(y∈x))
  • This question might be too technical for our users. You should probably focus on describing the Counter-Reflection schema informally, and the relevant naturalness criteria it might be violating (it is unlikely people will read through Muller's and Friedman's papers). A philosophical discussion of alternative set theories is Maddy's Believing the Axioms, she discusses naturalness of reflection principles, in particular (p.506ff). See also part II, p.750ff. – Conifold Dec 21 '18 at 19:54
  • @Conifold thanks a lot for the references, the pages you've pointed are about using reflection for higher large cardinals, however the system here is equivalent to ZFC itself and is not involved with any large cardinal, however if super-transitivity is strengthened into limitation of size axiom, specifically V is the set of all strictly smaller subsets of it, then we get to a Mahlo cardinal. As far as I can see Maddy has a positive opinion about reflection principles. – Zuhair Dec 22 '18 at 7:17

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