# Alternative axioms for NBG or MK

While I was thinking about NBG and MK I had the idea for two alternative axioms. As usual $$V$$ is the class of sets.

## The first one:

For a boolean function $$f : \{T,F\}^n \to \{T,F\}$$ let $$\varphi_f(x_1,\dots,x_n)$$ be a formal representation of $$f$$. That means for $$a_1,\dots, a_n \in \{T,F\}$$ we have $$f(a_1,\dots,a_n) = T \ \Leftrightarrow\ \models\varphi_f(a_1,\dots,a_n)$$. (I'm identifying T, F with $$\top$$, $$\bot$$). So for example if $$f$$ is the AND-function we have $$\varphi_f = (x_1 \wedge x_n)$$.

Axiom 1 (scheme):

For all boolean functions $$f : \{T,F\}^n \to \{T,F\}$$ and all $$R_1, \dots, R_n \in \{=,\in,\subseteq\}$$:

For all $$b_1,\dots, b_n \in V$$ we have

$$\{x; \varphi_f(x R_1 b_1, \dots, x R_n b_n)\} \in V \quad \Leftrightarrow \quad \neg\varphi_f(\bot, \dots, \bot)$$ $$\$$ (or semantically $$f(F,\dots,F) = F$$)

This axiom implies

• EmptySet $$\quad$$ (choose $$n=0$$ and $$f(\langle\rangle) = F$$; $$\ \langle\rangle \in \{T,F\}^0$$ is the empty sequence)
• Pairing $$\quad$$ (choose $$n=2$$, "$$f = AND$$" and $$R_1,R_2 =\; =$$)
• Powerset $$\quad$$ (choose $$n=1$$, "$$f =$$ identity" and $$R_1 =\; \subseteq$$)
• SmallUnion $$\quad$$ (choose $$n=2$$, "$$f = OR$$" and $$R_1, R_2 =\; \in$$)

and others... (Let $$a \in^2 b :\Leftrightarrow \exists c (a \in c \wedge c \in b)$$. If we allow the $$R_i$$ to be $$\in^2$$ we have Union too. )

Further: the axiom states that many classes are proper (without the help of other axioms).

And I'm quite sure that this axiom follows from NBG/MK.

If we choose Extensionality, (Foundation), Class Comprehension, Limitation of Size, Infinity and our Axiom 1 we have a version of NBG resp. MK which is easy to remember. What do you think?

## The second:

Axiom 2:

If $$X$$ is a class of non-empty disjoint sets, then $$X$$ is a set iff there is a choice set for $$X$$.

(I think the formalisation is clear)

This axiom is a fusion of choice and (at least) a part of replacement. So for example if we have a class function $$f: A \to B$$ and $$A$$ is a set that contains no pairs, we could build the class $$X = \{ \{x, \langle x, f(x)\rangle\}; x \in A\}$$ and use our axiom to conclude that $$X$$ is a set (since $$A$$ is obviously a choice set of $$X$$). With Union and Separation we get that the image of $$f$$ is a set too.

My first question:

Is Axiom 2 equivalent to choice and replacement (modulo other standard axioms)?

My second question: Are similar axioms studied somewhere?

• With "choice set of X" I mean a set $A$, so that $A \cap x$ is a singleton for all $x \in X$. – Popov Florino Nov 29 '18 at 12:08
• I see, thanks... – Carl Mummert Nov 29 '18 at 12:25

Let $$f : A \to B$$ be a class function and $$A$$ a set
(i) If $$f$$ is injective we define $$C := \{a \in A; f(a) \notin A\}$$. With separation it is a set. Now we build the class $$X = \{ \{a,f(a)\}; a \in C \}$$. Since $$C$$ is a choice set of $$X$$ (and $$X$$ is a class of non-empty disjoint sets) we get with Axiom 2 that $$X$$ is a set. If we have small union, union and separation this implies, that $$\operatorname{im} f = f(C) \cup (A \cap \operatorname{im} f)$$ is a set.
(ii) Now consider an arbitrary class function $$f$$. $$f$$ defines an equivalence relation $$a \sim b :\Leftrightarrow f(a) = f(b)$$ on $$A$$. With powerset and separation we get, that the class of equivalence classes $$A/{\sim} := \{ [a]_\sim; a \in A\}$$ is a set. Since $$f_{\sim}: A/{\sim} \to B,\ [a]_{\sim} \mapsto f(a)$$ is injective, we can use (i) to show that $$\operatorname{im} f = \operatorname{im} f_{\sim}$$ is a set.