Consider the topological set theory $GPK^{+}_{\infty}$ +$AC_{WF}$, where $GPK^{+}_{\infty}$ is defined as follows (this from Olivier Esser's papers, "An Interpretation of the Zermelo-Fraenkel Set Theory and the Kelly-Morse Set Theory in a Positive Theory" and "On the Consistency of a Positive Theory"):

Bounded Positive Formulas: Within the language $\mathscr L$ = ($\in$, =) of set theory, the class $BPF$ of bounded positive formulas is defined as follows (as the smallest class of formulas of $\mathscr L$ such that, or, in the alternative, the class of formulas obtained by applying a finite number of times the following rules):

(1) If $x$ and $y$ are variables, then $x$$\in$$y$ and $x$=$y$ are $BPF$ formulas.

(2) If $\varphi$ and $\psi$ are $BPF$ formulas, then so are $\varphi$$\land$$\psi$, $\varphi$$\lor$$\psi$, $\exists$$x$$\varphi$, and ($\forall$$x$$\in$$y$)$\varphi$.

Definition. If $\Sigma$ is a collection of formulas (possibly with parameters, but where the variable '$a$' does not appear), $comp({\Sigma})$ is the following scheme:

$\exists$$a$$\forall$$x$($x$$\in$$a$$\Leftrightarrow$$\alpha$), for each $\alpha$$\in$$\Sigma$.

Definition. A class is definable (with parameters) collection of sets.

Definition. A set can be defined in one of two (equivalent) ways:

(1) If $a$, $y$ are classes, then $a$ is a set iff $\exists$$y$($a$$\in$$y$), or

(2) A class {$A$|$\alpha$} is said to be a set if $\exists$$a$$\forall$$x$($x$$\in$$a$ $\Leftrightarrow$ $\alpha$); in this case one identifies $a$ and $A$.

$GPK^{+}_{\infty}$ consists of the following axioms:

(1) Extensionality. $\forall$$x$$\forall$$y$[($\forall$$z$($z$$\in$$x$$\Leftrightarrow$$z$$\in$$y$) $\Rightarrow$ $x$=$y$];

(2) Empty Set. $\exists$$x$$\forall$$y$($y$$\notin$$x$);

(3) Comprehension. $comp({BPF})$ (i.e. the universal closures of the formulas $\exists$$a$$\forall$$x$($x$$\in$$a$$\Leftrightarrow$$\varphi$), where $\varphi$ is a $BPF$;

(4) Closure. For any formula $\Gamma$($z$, $y_1$,...,$y_n$) whose free variables are among $z$,$y_1$,...,$y_n$,

$\forall$$y$,...,$\forall$$y_n$ $\exists$$x$[$\forall$$z$($\Gamma$($z$,$y_1$,...,$y_n$) $\Rightarrow$ $z$$\in$$x$) $\land$ $\forall$$y$($\forall$$z$($\Gamma$($z$,$y_1$,...,$y_n$) $\Rightarrow$ $z$$\in$$y$) $\Rightarrow$ $x$$\subset$$y$)]. [The scheme Closure says that for each class $A$={$z$|$\Gamma$($z$,$y_1$,...,$y_n$)} there is a smallest set (for the inclusion) which contains it; this set is denoted by $\bar A$ or $cl({A})$.]

(5) Infinity. There exists a limit ordinal.

Definition. $AC_{WF}$$=_{df}$ $\exists$$f$[($\forall$$x$$\in$$WF$)$\exists$$!$($x$,$y$)$\in$$f$ $\land$ ($\forall$$x$$\in$$WF$)$\forall$$y$(($x$$\neq$$\emptyset$ $\land$ ($x$,$y$)$\in$$f$) $\Rightarrow$ $y$$\in$ $x$)] [$AC_{WF}$ says that there is a set $f$ such that $f$$\cap$$WF$ is a class-function $F$ such that ($\forall$$x$$\in$$WF$)($x$$\neq$$\emptyset$ $\Rightarrow$ $F$($x$)$\in$$x$), where $WF$ is the class of well-founded sets of $GPK^{+}_{\infty}$.]

Furthermore, Esser, in "An Interpretation of the Zermelo-Fraenkel Set Theory and the Kelly-Morse Set Theory in a Positive Theory", characterizes Kelly-Morse 'set theory' (class theory, actually) as follows:

Recall that the Kelly-Morse theory is the class theory of G$\ddot o$del-Bernays to which we add the following scheme: For any formula $\varphi$ (with parameters) the collection {$x$| ($x$ is a set) $\land$ $\varphi$} is a class (in G$\ddot o$del-Bernays class theory we have only this if $\varphi$ is a formula containing no class variables).

It should also be noted that in "On the Consistency of a Positive Theory", Esser proves that $GPK^{+}_{\infty}$ + $AC_{WF}$ and $KMC$ + "On is ramifiable" are mutually interpretable , where $KMC$ is Kelly- Morse with Global Choice, and "On is ramifiable" is defined as follows:

A class of $KMC$ is sais to be On-finite iff it is a set, On-infinite if not. We mean by On is ramifiable the following sentence: "Every On-infinite tree with On-finite levels has an On-infinite branch".

A tree $T$ is a class of sets (called the nodes of the tree) together with an ordering relation $\lt_{T}$ such that for each node of the tree the class {$x$| $x$$\lt_{T}$$a$} is a well-ordered set. A branch of a tree is a well ordered subclass of the tree. The length of a branch is the ordinal of the well ordered type of the branch if it is a set, and On if it is not a set . A successor of a branch $B$ is a node $x$ of $T$ such that {$y$|$y$$\lt_{T}$$x$}$\supseteq$$B$. A node $x$ is said to be an immediate successor of $B$ iff {$y$| $y$$\lt_{T}$ $x$}=$B$. The $\alpha^{th}$ level ($\alpha$$\in$On) of a tree $T$ is the class of the $x$$\in$$T$ such that {$y$| $y$$\lt_{T}$ } is a branch of length $\alpha$. A tree ($T$,$\lt_{T}$) is said to be a binary tree iff

(i) each element of $T$ has at most two immediate successors;

(ii) every branch of limit length has at most one immediate successor.

Finally, in "On the Consistency of a Positive Theory", Esser makes the following claim:

...we can give a natural model of $KMC$ + "On is ramifiable" inside of "ZF$ + "there exists a weakly compact strongly inaccessible cardinal".

Now for the questions (these refer to the Enayat-Hamkins preprint, "$ZFC$ proves that the class of ordinals is not weakly compact for definable classes" [arXiv:1610.02729v1 [mathLO]]):

Consider the two main theorems of the Enayat-Hamkins paper:

Theorem $A$. Let $\mathcal M$ be any model of $ZFC$.

(1) The definable tree property fails in $\mathcal M$: There is an $\mathcal M$-definable Ord-tree with no $\mathcal M$-definable cofinal branch.

(2) The definable partition property fails in $\mathcal M$: There is an $\mathcal M$-definable 2-coloring $f$: $[$$X$$]^2$ $\rightarrow$ $2$ for some $\mathcal M$-definable proper class $X$ such that no $\mathcal M$-definable proper class is momochromatic for $f$.

(3) The definable compactness property fails in $\mathcal M$: There is a definable theory $\Gamma$ in the logic $\mathcal L_{Ord,\omega}$ (in the sense of $\mathcal M$) of size Ord such that every set-sized subtheory of $\Gamma$ is satisfiable in $\mathcal M$, but there is no $\mathcal M$-definable model of $\Gamma$.

Theorem B. The definable $\lozenge_{Ord}$ principle holds in a model of $ZFC$ iff $\mathcal M$ carries an $\mathcal M$-definable global well-ordering.

Instead of assuming $ZFC$ as the metatheory (as Enayat and Hamkins do), let me assume $GPK^{+}_{\infty}$ + $AC_{WF}$ as the metatheory. Can $GPK^{+}_{\infty}$ +$AC_{WF}$ prove that $ZFC$ proves Theorems $A$ and $B$ in a manner that is less "unexpected" in the sense that given an interpretation of $ZFC$ in $GPK^{+}_{\infty}$ + $AC_{WF}$, that interpretation of $ZFC$ proves Theorems $A$ and $B$ more 'naturally' in that $ZFC$ was interpreted in a positive theory?

Question 2: How are the definitions of "On is ramifiable" and "Tree" as Esser defined them related to the corresponding definitions in the Enayat-Hamkins paper?

Querstion 3: Does $ZF$ (with AC excluded) prove Theorems $A$ and $B$?

Your Answer


By clicking "Post Your Answer", you acknowledge that you have read our updated terms of service, privacy policy and cookie policy, and that your continued use of the website is subject to these policies.

Browse other questions tagged or ask your own question.