1
$\begingroup$

(Note: the 'Second-order $ZFC$' ($ZFC^2$) I am talking about is the theory [in the second order language of set theory consisting of a single non-logical symbol $\in$ ] consisting of the axioms Extensionality, Union, Pairing, Power Set, Infinity, Regularity, and Choice as in first-order $ZFC$, but replacing the axiom schema of Separation and Replacement with their second-order analogues, e.g. for Separation

$\mathbf Sep$: $\forall$$X$$\forall$$x$$\exists$$y$($\forall$$z$($z$$\in$$y$ $\leftrightarrow$ ($z$$\in$$x$ $\land$ $X$$z$)))

and for the axiom scheme of Replacement, a single second-order axiom that quantifies over every class function $f$ and says that the image of any set under any class function is again a set [the section describing Second-Order $ZFC$ comes from Vaananan and Wang's paper, "Internal Categoricity in Arithmetic and Set Theory", and the section describing the Second-Order Axiom of Replacement comes from Carl Mummert's answer to Keshav Srinivasan's mathstackexchange question, "Is there a Second-Order Axiomatization of $ZF(C)$ which is categorical?"]. I am specifying the 'Second-Order $ZFC$' in question due to the $FOM$ thread, "What is second order $ZFC$?". Note also that the deductive system of second-order $ZFC$ is "the extension of the usual logical axioms with the Comprehension Axiom

$CA$: $\exists$$X$$\forall$$\vec x$($X$$\vec x$ $\leftrightarrow$ $\varphi$($\vec x$)) for any second-order formula $\varphi$ not containing $X$ free" [this also from "Internal Categoricity in Arithmetic and Set Theory" (pg. 2)].)

As Vaananen and Wang state in their paper (pg. 2), this deductive system is normal second-order logic and that the natural semantics of this logic has as its class of models all normal models, namely Henkin models satisfying $CA$. They also state that second-order $ZFC$ is internally categorical. Their intuitive definition of internally categorical is as as follows:

We say a theory $T$ is internally categorical if all models of $T$ within a common normal model [Henkin model] are witnessed isomorphic by the model. We will make this definition more intelligible through examples in what follows.

Here is a more precise definition of internal categoricity (this from Vaananen's paper, "Second-Order Logic and Set Theory":

A second-order theory $T$ is internally categorical if $\forall$($\mathfrak M$, $\mathcal G$), $\mathfrak M_0$ $\in$ $\mathcal G$, $\mathfrak M_1$ $\in$ $\mathcal G$((($\mathfrak M_0$, $\mathcal G$) $\vDash$ $T$ $\land$ ($\mathfrak M_1$, $\mathcal G$) $\vDash$ $T$) $\rightarrow$ $\exists$$f$ $\in$ $\mathcal G$($f$: $\mathfrak M_0$$\cong$$\mathfrak M_1$)), where ($\mathfrak M$, $\mathcal G$) is a Henkin model of $T$ [that is, $\mathfrak M$ is a "usual" first order structure (Vaananen used the term 'usual' in his paper) and $\mathcal G$ is a set of subsets, relations and functions on $M$].

Vaananen and Wang prove that second-order $ZFC$ is internally categorical:

Theorem 3. Let $L$=($V$, $E$, $V^{'}$, $E^{'}$) be two copies of the language of set theory. Let ($M$, $\mathcal G$) be a normal [Henkin] model. suppose

  1. ($M$, $\mathcal G$) $\vDash$ $ZFC^2$($V$, $E$)

  2. ($M$, $\mathcal G$) $\vDash$ $ZFC^2$($V^{'}$, $E^{'}$)

  3. $V$, $V^{'}$, $E$, $E^{'}$ $\in$ $\mathcal G$

  4. ($M$, $\mathcal G$) $\vDash$ ($\exists$$\pi$) $ISO$($\pi$, $Ord$, $Ord^{'}$). $Ord$ and $Ord^{'}$ denote ordinals in ($V$, $E$) and ($V^{'}$, $E^{'}$) respectively.

Then

($M$, $\mathcal G$) $\vDash$ ($\exists$$R$) $ISO$($R$, $V$, $E$, $V{'}$, $E^{'}$).

Recall that normal (Henkin) models can be models of any cardinality.

With this in mind, consider the following result of Gitman and Hamkins:

[Assume $ZFC$ consistent.] [Then] if there are saturated models of $ZFC$ of cardinality $\kappa$, then the collection of these satisfies all the multiverse axioms [see their paper, "A Natural Model of the Multiverse Axioms" for these].

Questions:

  1. For $ZFC^2$, do there exist saturated Henkin models (if so, then there exists a collection of all such models of cardinality $\kappa$)?

  2. Does the collection of all the saturated Henkin models of $ZFC^2$ (if such exist) of cardinality $\kappa$ satisfy the Gitman-Hamkins multiverse axioms?

  3. If the answer to (2) is "yes", then does the 'Universe' of the universe view of set theory correspond to the 'canonical'(?) universe abstracted from the universes of the "internally isomorphic" Henkin models of $ZFC^2$?

(Thanks to Asaf Karagila for implicitly raising this issue in a comment in one of Prof. Hamkins' blogposts--I forget which.)

$\endgroup$
  • $\begingroup$ @MikhailKatz: Would you (for example) consider some of the work of the late Ed Nelson as 'Cargo Cult Mathematics'? Why or why not? $\endgroup$ – Thomas Benjamin Jun 25 '18 at 14:11
  • $\begingroup$ No problem, feel free to post an answer about Nelson. $\endgroup$ – Mikhail Katz Jun 26 '18 at 13:35

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.