Difference between revisions of "Transitive ZFC model"

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A ''transitive model of ZFC'' is a [[transitive]] set $M$ such that the structure $\langle M,\in\rangle$ satisfies all of the [[ZFC]] axioms of set theory. The existence of such a model is strictly stronger than [[Con ZFC | Con(ZFC)]] and stronger than an iterated [[Con ZFC#Consistency hierarchy | consistency hierarchy]], but weaker than the existence of an [[worldly]] cardinal, a cardinal $\kappa$ for which $V_\kappa$ is a model of ZFC, and consequently also weaker than the existence of an [[inaccessible]] cardinal. Not all transitive models of ZFC have the $V_\kappa$ form, for if there is any transitive model of ZFC, then by the Löwenheim-Skolem theorem there is a countable such model, and these never have the form $V_\kappa$.  
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{{DISPLAYTITLE: Transitive model of $\text{ZFC}$}}
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A ''transitive model of $\text{ZFC}$'' is a [[transitive]] set $M$ such that the structure $\langle M,\in\rangle$ satisfies all of the [[ZFC|$\text{ZFC}$]] axioms of set theory. The existence of such a model is strictly stronger than [[Con ZFC | $\text{Con(ZFC)}$]] and stronger than an iterated [[Con ZFC#Consistency hierarchy | consistency hierarchy]], but weaker than the existence of an [[worldly]] cardinal, a cardinal $\kappa$ for which $V_\kappa$ is a model of $\text{ZFC}$, and consequently also weaker than the existence of an [[inaccessible]] cardinal. Not all transitive models of $\text{ZFC}$ have the $V_\kappa$ form, for if there is any transitive model of $\text{ZFC}$, then by the Löwenheim-Skolem theorem there is a countable such model, and these never have the form $V_\kappa$.  
  
Nevertheless, every transitive model $M$ of ZFC provides a set-theoretic forum inside of which one can view nearly all classical mathematics taking place. In this sense, such models are inaccessible to or out of reach of ordinary set-theoretic constructions. As a result, the existence of a transitive model of ZFC can be viewed as a large cardinal axiom: it expresses a notion of largeness, and the existence of such a model is not provable in ZFC and has consistency strength strictly exceeding ZFC.  
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Nevertheless, every transitive model $M$ of $\text{ZFC}$ provides a set-theoretic forum inside of which one can view nearly all classical mathematics taking place. In this sense, such models are inaccessible to or out of reach of ordinary set-theoretic constructions. As a result, the existence of a transitive model of $\text{ZFC}$ can be viewed as a large cardinal axiom: it expresses a notion of largeness, and the existence of such a model is not provable in $\text{ZFC}$ and has consistency strength strictly exceeding $\text{ZFC}$.  
  
== Minimal transitive model of ZFC ==
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== Minimal transitive model of $\text{ZFC}$ ==
  
If there is any transitive model $M$ of ZFC, then $L^M$, the constructible universe as computed in $M$, is also a transitive model of ZFC and indeed, has the form $L_\eta$, where $\eta=\text{ht}(M)$ is the height of $M$. The ''minimal transitive model of ZFC'' is the model $L_\eta$, where $\eta$ is smallest such that this is a model of ZFC. The argument just given shows that the minimal transitive model is a subset of all other transitive models of ZFC.  
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If there is any transitive model $M$ of $\text{ZFC}$, then $L^M$, the constructible universe as computed in $M$, is also a transitive model of $\text{ZFC}$ and indeed, has the form $L_\eta$, where $\eta=\text{ht}(M)$ is the height of $M$. The ''minimal transitive model of $\text{ZFC}$'' is the model $L_\eta$, where $\eta$ is smallest such that this is a model of $\text{ZFC}$. The argument just given shows that the minimal transitive model is a subset of all other transitive models of $\text{ZFC}$.  
  
== Omega model of ZFC ==  
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== Omega model of $\text{ZFC}$ ==  
  
An ''$\omega$-model'' of ZFC is a model of ZFC whose collection of natural numbers is isomorphic to the actual natural numbers. In other words, an $\omega$-model is a model having no nonstandard natural numbers, although it may have nonstandard ordinals. (More generally, for any ordinal $\alpha$, an $\alpha$-model has well-founded part at least $\alpha$.) Every transitive model of ZFC is an $\omega$-model, but the latter concept is strictly weaker.
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An ''$\omega$-model'' of $\text{ZFC}$ is a model of $\text{ZFC}$ whose collection of natural numbers is isomorphic to the actual natural numbers. In other words, an $\omega$-model is a model having no nonstandard natural numbers, although it may have nonstandard ordinals. (More generally, for any ordinal $\alpha$, an $\alpha$-model has well-founded part at least $\alpha$.) Every transitive model of $\text{ZFC}$ is an $\omega$-model, but the latter concept is strictly weaker.
  
 
== Consistency hierarchy ==
 
== Consistency hierarchy ==
  
The existence of an $\omega$-model of ZFC and implies [[Con ZFC | Con(ZFC)]], of course, and also [[ Con ZFC#Consistency hierarchy | Con(ZFC+Con(ZFC))]] and a large part of the iterated [[ Con ZFC#Consistency hierarchy | consistency hierarchy]]. This is simply because if $M\models\text{ZFC}$ and has the standard natural numbers, then $M$ agrees that Con(ZFC) holds, since it  has the same proofs as we do in the ambient background. Thus, we believe that $M$ satisfies ZFC+Con(ZFC) and consequently we believe Con(ZFC+Con(ZFC)). It follows again that $M$ agrees with this consistency assertion, and so we now believe $\text{Con}^3(\text{ZFC})$. The model $M$ therefore agrees and so we believe $\text{Con}^4(\text{ZFC})$ and so on transfinitely, as long as we are able to describe the ordinal iterates in a way that $M$ interprets them correctly.
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The existence of an $\omega$-model of $\text{ZFC}$ and implies [[Con ZFC | $\text{Con(ZFC)}$]], of course, and also [[ Con ZFC#Consistency hierarchy | $\text{Con(ZFC+Con(ZFC))}$]] and a large part of the iterated [[ Con ZFC#Consistency hierarchy | consistency hierarchy]]. This is simply because if $M\models\text{ZFC}$ and has the standard natural numbers, then $M$ agrees that $\text{Con(ZFC)}$ holds, since it  has the same proofs as we do in the ambient background. Thus, we believe that $M$ satisfies $\text{ZFC+Con(ZFC)}$ and consequently we believe $\text{Con(ZFC+Con(ZFC))}$. It follows again that $M$ agrees with this consistency assertion, and so we now believe $\text{Con}^3(\text{ZFC})$. The model $M$ therefore agrees and so we believe $\text{Con}^4(\text{ZFC})$ and so on transfinitely, as long as we are able to describe the ordinal iterates in a way that $M$ interprets them correctly.
  
== Transitive models of ZFC fragments ==
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== Transitive models of $\text{ZFC}$ fragments ==
  
Every finite fragment of ZFC admits numerous transitive models, as a consequence of the [[reflection theorem]].  
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Every finite fragment of $\text{ZFC}$ admits numerous transitive models, as a consequence of the [[reflection theorem]].  
  
 
== Transitive models and forcing ==
 
== Transitive models and forcing ==
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Countable transitive models of set theory were used historically as a convenient way to formalize [[forcing]]. Such models $M$ make the theory of forcing convenient, since one can easily prove that for every partial order $\mathbb{P}$ in $M$, there is an $M$-generic [[filter]] $G\subset\mathbb{P}$, simply by enumerating the dense subsets of $\mathbb{P}$ in $M$ in a countable sequence $\langle D_n\mid n\lt\omega\rangle$, and building a descending sequence $p_0\geq p_1\geq p_2\geq\cdots$, with $p_n\in D_n$. The filter $G$ generated by the sequence is $M$-generic.  
 
Countable transitive models of set theory were used historically as a convenient way to formalize [[forcing]]. Such models $M$ make the theory of forcing convenient, since one can easily prove that for every partial order $\mathbb{P}$ in $M$, there is an $M$-generic [[filter]] $G\subset\mathbb{P}$, simply by enumerating the dense subsets of $\mathbb{P}$ in $M$ in a countable sequence $\langle D_n\mid n\lt\omega\rangle$, and building a descending sequence $p_0\geq p_1\geq p_2\geq\cdots$, with $p_n\in D_n$. The filter $G$ generated by the sequence is $M$-generic.  
  
For the purposes of consistency proofs, this manner of formalization worked quite well. To show $\text{Con}(\text{ZFC})\to \text{Con}(\text{ZFC}+\varphi)$, one fixes a finite fragment of ZFC and works with a countable transitive model of a suitably large fragment, producing $\varphi$ with the desired fragment in a forcing extension of it.
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For the purposes of consistency proofs, this manner of formalization worked quite well. To show $\text{Con}(\text{ZFC})\to \text{Con}(\text{ZFC}+\varphi)$, one fixes a finite fragment of $\text{ZFC}$ and works with a countable transitive model of a suitably large fragment, producing $\varphi$ with the desired fragment in a forcing extension of it.
  
 
== Transitive model universe axiom ==  
 
== Transitive model universe axiom ==  
  
The ''transitive model universe axiom'' is the assertion that every set is an element of a transitive model of ZFC. This axiom makes a stronger claim than the [[reflecting#The Feferman theory | Feferman theory]], since it is asserted as a single first-order claim, but weaker than the [[universe axiom]], which asserts that the universes have the form $V_\kappa$ for inaccessible cardinals $\kappa$.  
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The ''transitive model universe axiom'' is the assertion that every set is an element of a transitive model of $\text{ZFC}$. This axiom makes a stronger claim than the [[reflecting#The Feferman theory | Feferman theory]], since it is asserted as a single first-order claim, but weaker than the [[universe axiom]], which asserts that the universes have the form $V_\kappa$ for inaccessible cardinals $\kappa$.  
  
The transitive model universe axiom is sometimes studied in the background theory not of ZFC, but of [[ZFC-P]], omitting the power set axiom, together with the axiom asserting that every set is countable. Such an enterprise amounts to adopting the latter theory, not as the fundamental axioms of mathematics, but rather as a background meta-theory for studying the [[multiverse]] perspective, investigating how the various actual set-theoretic universe, transitive models of full ZFC, relate to one another.
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The transitive model universe axiom is sometimes studied in the background theory not of $\text{ZFC}$, but of [[ZFC-P]], omitting the power set axiom, together with the axiom asserting that every set is countable. Such an enterprise amounts to adopting the latter theory, not as the fundamental axioms of mathematics, but rather as a background meta-theory for studying the [[multiverse]] perspective, investigating how the various actual set-theoretic universe, transitive models of full $\text{ZFC}$, relate to one another.

Revision as of 12:57, 11 November 2017

A transitive model of $\text{ZFC}$ is a transitive set $M$ such that the structure $\langle M,\in\rangle$ satisfies all of the $\text{ZFC}$ axioms of set theory. The existence of such a model is strictly stronger than $\text{Con(ZFC)}$ and stronger than an iterated consistency hierarchy, but weaker than the existence of an worldly cardinal, a cardinal $\kappa$ for which $V_\kappa$ is a model of $\text{ZFC}$, and consequently also weaker than the existence of an inaccessible cardinal. Not all transitive models of $\text{ZFC}$ have the $V_\kappa$ form, for if there is any transitive model of $\text{ZFC}$, then by the Löwenheim-Skolem theorem there is a countable such model, and these never have the form $V_\kappa$.

Nevertheless, every transitive model $M$ of $\text{ZFC}$ provides a set-theoretic forum inside of which one can view nearly all classical mathematics taking place. In this sense, such models are inaccessible to or out of reach of ordinary set-theoretic constructions. As a result, the existence of a transitive model of $\text{ZFC}$ can be viewed as a large cardinal axiom: it expresses a notion of largeness, and the existence of such a model is not provable in $\text{ZFC}$ and has consistency strength strictly exceeding $\text{ZFC}$.

Minimal transitive model of $\text{ZFC}$

If there is any transitive model $M$ of $\text{ZFC}$, then $L^M$, the constructible universe as computed in $M$, is also a transitive model of $\text{ZFC}$ and indeed, has the form $L_\eta$, where $\eta=\text{ht}(M)$ is the height of $M$. The minimal transitive model of $\text{ZFC}$ is the model $L_\eta$, where $\eta$ is smallest such that this is a model of $\text{ZFC}$. The argument just given shows that the minimal transitive model is a subset of all other transitive models of $\text{ZFC}$.

Omega model of $\text{ZFC}$

An $\omega$-model of $\text{ZFC}$ is a model of $\text{ZFC}$ whose collection of natural numbers is isomorphic to the actual natural numbers. In other words, an $\omega$-model is a model having no nonstandard natural numbers, although it may have nonstandard ordinals. (More generally, for any ordinal $\alpha$, an $\alpha$-model has well-founded part at least $\alpha$.) Every transitive model of $\text{ZFC}$ is an $\omega$-model, but the latter concept is strictly weaker.

Consistency hierarchy

The existence of an $\omega$-model of $\text{ZFC}$ and implies $\text{Con(ZFC)}$, of course, and also $\text{Con(ZFC+Con(ZFC))}$ and a large part of the iterated consistency hierarchy. This is simply because if $M\models\text{ZFC}$ and has the standard natural numbers, then $M$ agrees that $\text{Con(ZFC)}$ holds, since it has the same proofs as we do in the ambient background. Thus, we believe that $M$ satisfies $\text{ZFC+Con(ZFC)}$ and consequently we believe $\text{Con(ZFC+Con(ZFC))}$. It follows again that $M$ agrees with this consistency assertion, and so we now believe $\text{Con}^3(\text{ZFC})$. The model $M$ therefore agrees and so we believe $\text{Con}^4(\text{ZFC})$ and so on transfinitely, as long as we are able to describe the ordinal iterates in a way that $M$ interprets them correctly.

Transitive models of $\text{ZFC}$ fragments

Every finite fragment of $\text{ZFC}$ admits numerous transitive models, as a consequence of the reflection theorem.

Transitive models and forcing

Countable transitive models of set theory were used historically as a convenient way to formalize forcing. Such models $M$ make the theory of forcing convenient, since one can easily prove that for every partial order $\mathbb{P}$ in $M$, there is an $M$-generic filter $G\subset\mathbb{P}$, simply by enumerating the dense subsets of $\mathbb{P}$ in $M$ in a countable sequence $\langle D_n\mid n\lt\omega\rangle$, and building a descending sequence $p_0\geq p_1\geq p_2\geq\cdots$, with $p_n\in D_n$. The filter $G$ generated by the sequence is $M$-generic.

For the purposes of consistency proofs, this manner of formalization worked quite well. To show $\text{Con}(\text{ZFC})\to \text{Con}(\text{ZFC}+\varphi)$, one fixes a finite fragment of $\text{ZFC}$ and works with a countable transitive model of a suitably large fragment, producing $\varphi$ with the desired fragment in a forcing extension of it.

Transitive model universe axiom

The transitive model universe axiom is the assertion that every set is an element of a transitive model of $\text{ZFC}$. This axiom makes a stronger claim than the Feferman theory, since it is asserted as a single first-order claim, but weaker than the universe axiom, which asserts that the universes have the form $V_\kappa$ for inaccessible cardinals $\kappa$.

The transitive model universe axiom is sometimes studied in the background theory not of $\text{ZFC}$, but of ZFC-P, omitting the power set axiom, together with the axiom asserting that every set is countable. Such an enterprise amounts to adopting the latter theory, not as the fundamental axioms of mathematics, but rather as a background meta-theory for studying the multiverse perspective, investigating how the various actual set-theoretic universe, transitive models of full $\text{ZFC}$, relate to one another.