# Does this axiomatic system satisfy requirements for founding mathematics?

In this article, the author, F.A.Muller, suggests criteria for a founding theory of mathematics (pp:14-16). The author proposes $$ARC$$ Class Theory to embody these requirements. The motivation is largely connected to set theoretic basis for category theory.

The first Question: how far the following theory $$\mathsf{RfST}"$$, is from fulfilling these requirements?

I mean the Class Separation schema he speaks of seems to be in some sense out of touch with what class theories are about, one needs in my opinion to speak of a more general theme that explains it. In other words it needs to be a theorem schema of some more basic principle that rise in connection with a theory about Classes and Sets. Hence the following theory suggested. The point is that the axiomatic system here is more elegant, and I think it serves the purpose.

However I'm not sure if the Counter-Reflection principle is too strong for the purposes outlined in Muller's article, specifically regarding adhering to Bourbaki without Super-abundancy, since I tend to think that Counter-Reflection doesn't add strength. Hence the question.

The next question: how much the axioms of $$\mathsf{RfST}$$ are far from Harvey Friedman's criteria for new axioms for mathematics, written in this article (see 5. CIRCUMSTANCES SURROUNDING ACTUAL ADOPTION OF NEW AXIOMS. pp:5-6)?

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

The axioms are those of first order identity theory +

1. Extensionality: $$\forall x (x \in a \leftrightarrow x \in b) \to a=b$$

2. Class comprehension: if $$\varphi(y)$$ is a formula in which the symbol $$y"$$ occurs free, then all closures of: $$\exists x \forall y (y \in x \leftrightarrow y \in V \wedge \varphi(y))$$, are axioms.

3. Super-Transitivity: $$y \subset x \wedge x \in V \to y \in V$$

4. Reflection: if $$\varphi(y, x_1,..,x_n)$$ is a formula that doesn't use the symbol $$V$$, in which only $$y,x_1,..,x_n$$ occur free, then:

$$\forall x_1,..,x_n \in V \\ [\exists y (\varphi(y,x_1,..,x_n)) \to \exists y \in V (\varphi(y,x_1,..,x_n))]$$

$$\hspace{12pt}$$ is an axiom.

1. Counter-Reflection: if $$\varphi$$ is a sentence that doesn't use the symbol $$V$$, and $$\varphi^V$$ is the sentence obtained by merely bounding every quantifier in $$\varphi$$ by $$V$$, then: $$(\varphi^V \to \varphi)$$ is an axiom.

2. Foundation: $$\exists m\in x \to \exists y \in x \forall z \in x (z \not \in y)$$

3. Choice: $$\forall m,n \in X ( \not \exists k \in m (k \in n)) \to \\\exists Y \forall x \in X (x \neq \emptyset \to \exists! y \in Y (y \in x ))$$

• Do you want some classes to have other classes as elements, or do you want it to be like in NGB/MK where the classes are the top-rank objects? – Monroe Eskew Dec 20 '18 at 15:35
• I think there was a reason that the paper was published in a journal of philosophy. – Harry Gindi Dec 20 '18 at 15:58
• It’s about philosophy of mathematics. It’s not a well-posed mathematical problem. – Monroe Eskew Dec 20 '18 at 16:00
• @ZuhairAl-Johar what kind of answer do you expect on this forum? – Monroe Eskew Dec 20 '18 at 16:13
• If you pick a particular well specified axiomatic system of Friedman's (call it HF) and ask how HF relates mathematically to the (assuming it is well-)specified system RfST, that is a mathematical question. To ask if any axiomatic system is a foundation for all of mathematics is a philosophical question, to which I believe the answer is no. Gerhard "A Kind Of Job Security" Paseman, 2018.12.20. – Gerhard Paseman Dec 20 '18 at 18:36