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Johannes Hahn
Johannes Hahn
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The Truth Lemma

The result says that what's true in a forcing extension $M[G]$ is just what's forced to be true by the path of the generic filter $G$. More precisely:

Suppose $M$ is a countable transitive model of $\text{ZFC}$, $\mathbb{P} \in M$ a forcing poset, $\varphi(x_1, \dots, x_n)$ a set theoretical formula, and $\tau_1, \dots, \tau_n$ a sequence of $\mathbb{P}$-names in $M$. If a filter $G$ on $\mathbb{P}$ meets each dense subset of $\mathbb{P}$ in $M$, then $\varphi(\tau_1^G, \dots, \tau_n^G)$ holds in $M[G]$ iff $G$ hits a $p$ that forces $\varphi(\tau_1, \dots, \tau_n)$.

In fact, $M$ just needs to satisfy a rich enough finite subtheory of $\text{ZFC}$. (How rich depends on $\varphi$.) Therefore the existence of suitable $M$ follows from the Reflection Theorem, which can be proved in $\text{ZFC}$ without assuming that $\text{ZFC}$ has a countable transitive model.

With a result known as the Definability Lemma, the Truth Lemma implies that every partial order forces $\text{ZFC}$. That is, $\text{ZFC}$ holds in every $M[G]$—no matter the $M$, $\mathbb{P}$, or $G$. This is key to showing that $M[G]$ is always the smallest transitive extension of $M$ verifyingsatisfying $\text{ZFC}$ and containing $G$ as an element. Thus, in diverse circumstances one may build the partial order $\mathbb{P}$ of attempts to construct a desired object, argue that the object exists in any extension of $M$ containing a filter $G$ as in the lemma, and thereby obtain the actual existence of the desired object in a model where the axioms of ZFC still hold. Magic!

The Truth Lemma

The result says that what's true in a forcing extension $M[G]$ is just what's forced to be true by the path of the generic filter $G$. More precisely:

Suppose $M$ is a countable transitive model of $\text{ZFC}$, $\mathbb{P} \in M$ a forcing poset, $\varphi(x_1, \dots, x_n)$ a set theoretical formula, and $\tau_1, \dots, \tau_n$ a sequence of $\mathbb{P}$-names in $M$. If a filter $G$ on $\mathbb{P}$ meets each dense subset of $\mathbb{P}$ in $M$, then $\varphi(\tau_1^G, \dots, \tau_n^G)$ holds in $M[G]$ iff $G$ hits a $p$ that forces $\varphi(\tau_1, \dots, \tau_n)$.

In fact, $M$ just needs to satisfy a rich enough finite subtheory of $\text{ZFC}$. (How rich depends on $\varphi$.) Therefore the existence of suitable $M$ follows from the Reflection Theorem, which can be proved in $\text{ZFC}$ without assuming that $\text{ZFC}$ has a countable transitive model.

With a result known as the Definability Lemma, the Truth Lemma implies that every partial order forces $\text{ZFC}$. That is, $\text{ZFC}$ holds in every $M[G]$—no matter the $M$, $\mathbb{P}$, or $G$. This is key to showing that $M[G]$ is always the smallest transitive extension of $M$ verifying $\text{ZFC}$ and containing $G$ as an element. Thus, in diverse circumstances one may build the partial order $\mathbb{P}$ of attempts to construct a desired object, argue that the object exists in any extension of $M$ containing a filter $G$ as in the lemma, and thereby obtain the actual existence of the desired object in a model where the axioms of ZFC still hold. Magic!

The Truth Lemma

The result says that what's true in a forcing extension $M[G]$ is just what's forced to be true by the path of the generic filter $G$. More precisely:

Suppose $M$ is a countable transitive model of $\text{ZFC}$, $\mathbb{P} \in M$ a forcing poset, $\varphi(x_1, \dots, x_n)$ a set theoretical formula, and $\tau_1, \dots, \tau_n$ a sequence of $\mathbb{P}$-names in $M$. If a filter $G$ on $\mathbb{P}$ meets each dense subset of $\mathbb{P}$ in $M$, then $\varphi(\tau_1^G, \dots, \tau_n^G)$ holds in $M[G]$ iff $G$ hits a $p$ that forces $\varphi(\tau_1, \dots, \tau_n)$.

In fact, $M$ just needs to satisfy a rich enough finite subtheory of $\text{ZFC}$. (How rich depends on $\varphi$.) Therefore the existence of suitable $M$ follows from the Reflection Theorem, which can be proved in $\text{ZFC}$ without assuming that $\text{ZFC}$ has a countable transitive model.

With a result known as the Definability Lemma, the Truth Lemma implies that every partial order forces $\text{ZFC}$. That is, $\text{ZFC}$ holds in every $M[G]$—no matter the $M$, $\mathbb{P}$, or $G$. This is key to showing that $M[G]$ is always the smallest transitive extension of $M$ satisfying $\text{ZFC}$ and containing $G$ as an element. Thus, in diverse circumstances one may build the partial order $\mathbb{P}$ of attempts to construct a desired object, argue that the object exists in any extension of $M$ containing a filter $G$ as in the lemma, and thereby obtain the actual existence of the desired object in a model where the axioms of ZFC still hold. Magic!

Stated the lemma and linked it to extant stub.
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Cole Leahy
Cole Leahy
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The Truth Lemma

The result says that what's true in a forcing extension $M[G]$ is just what's forced to be true by the path of the generic filter $G$. More precisely:

Suppose $M$ is a countable transitive model of $\text{ZFC}$, $\mathbb{P} \in M$ a forcing poset, $\varphi(x_1, \dots, x_n)$ a set theoretical formula, and $\tau_1, \dots, \tau_n$ a sequence of $\mathbb{P}$-names in $M$. If a filter $G$ on $\mathbb{P}$ meets each dense subset of $\mathbb{P}$ in $M$, then $\varphi(\tau_1^G, \dots, \tau_n^G)$ holds in $M[G]$ iff $G$ hits a $p$ that forces $\varphi(\tau_1, \dots, \tau_n)$.

In fact, $M$ just needs to satisfy a rich enough finite subtheory of $\text{ZFC}$. (How rich depends on $\varphi$.) Therefore the existence of suitable $M$ follows from the Reflection Theorem, which can be proved in $\text{ZFC}$ without assuming that $\text{ZFC}$ has a countable transitive model.

With a result known as the Definability Lemma, the Truth Lemma implies that every partial order forces ZFC$\text{ZFC}$. That is magical, $\text{ZFC}$ holds in every $M[G]$—no matter the sense$M$, $\mathbb{P}$, or $G$. This is key to showing that $M[G]$ is always the smallest transitive extension of your question$M$ verifying $\text{ZFC}$ and containing $G$ as an element. Thus, since in diverse circumstances one may easily build the partial order $\mathbb{P}$ of atttemptsattempts to construct a desired object, argue that the object exists in any extension of $M$ containing a filter $G$ as in the lemma, and thereby obtain the actual existence of such anthe desired object in a forcing extension,model where the axioms of ZFC still hold. Magic!

The fact that every partial order forces ZFC is magical, in the sense of your question, since in diverse circumstances one may easily build the partial order of atttempts to construct a desired object, and thereby obtain the actual existence of such an object in a forcing extension, where the axioms of ZFC still hold. Magic!

The Truth Lemma

The result says that what's true in a forcing extension $M[G]$ is just what's forced to be true by the path of the generic filter $G$. More precisely:

Suppose $M$ is a countable transitive model of $\text{ZFC}$, $\mathbb{P} \in M$ a forcing poset, $\varphi(x_1, \dots, x_n)$ a set theoretical formula, and $\tau_1, \dots, \tau_n$ a sequence of $\mathbb{P}$-names in $M$. If a filter $G$ on $\mathbb{P}$ meets each dense subset of $\mathbb{P}$ in $M$, then $\varphi(\tau_1^G, \dots, \tau_n^G)$ holds in $M[G]$ iff $G$ hits a $p$ that forces $\varphi(\tau_1, \dots, \tau_n)$.

In fact, $M$ just needs to satisfy a rich enough finite subtheory of $\text{ZFC}$. (How rich depends on $\varphi$.) Therefore the existence of suitable $M$ follows from the Reflection Theorem, which can be proved in $\text{ZFC}$ without assuming that $\text{ZFC}$ has a countable transitive model.

With a result known as the Definability Lemma, the Truth Lemma implies that every partial order forces $\text{ZFC}$. That is, $\text{ZFC}$ holds in every $M[G]$—no matter the $M$, $\mathbb{P}$, or $G$. This is key to showing that $M[G]$ is always the smallest transitive extension of $M$ verifying $\text{ZFC}$ and containing $G$ as an element. Thus, in diverse circumstances one may build the partial order $\mathbb{P}$ of attempts to construct a desired object, argue that the object exists in any extension of $M$ containing a filter $G$ as in the lemma, and thereby obtain the actual existence of the desired object in a model where the axioms of ZFC still hold. Magic!

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Joel David Hamkins
Joel David Hamkins
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The fact that every partial order forces ZFC is magical, in the sense of your question, since in diverse circumstances one may easily build the partial order of atttempts to construct a desired object, and thereby obtain the actual existence of such an object in a forcing extension, where the axioms of ZFC still hold. Magic!