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The standard terminology is that an interpretation $I$ of a theory $U$ in a theory $T$ is faithful if for all sentences $\phi$ in the language of $U$, $$T\vdash\phi^I\iff U\vdash\phi.$$ (Here and below, I will assume all theories to be recursively axiomatized.) A classical result of Feferman, Kreisel, and Orey [1] states:

Theorem 1 (Feferman, Kreisel, Orey). Let $T$ be a reflexive theory which includes a modicum of arithmetic and which is $\Sigma_1$-sound. Then all theories interpretable in $T$ are faithfully interpretable in $T$.

They also show that conversely, if a $\Sigma_1$-sound theory $U$ is faithfully interpretable in a reflexive theory $T$, then $T$ is itself $\Sigma_1$-sound.

Corollary: PA is faithfully interpretable in ZFC if and only if ZFC is $\Sigma_1$-sound.

Further results on faithful interpretability in reflexive theories were obtained by Lindström [2]; for example, he proved the following characterization:

Theorem 2 (Lindström). Let $T$ and $U$ be essentially reflexive theories. Then $U$ is faithfully interpretable in $T$ if and only if $U$ is $\Pi_1$-conservative over $T$ (which is, by itself, equivalent to the interpretability of $U$ in $T$), and $T$ is $\Sigma_1$-conservative over $U$.

Generalizing Theorem 1 in a different direction, a theory $T$ is called trustworthy if all theories that are interpretable in $T$ are faithfully interpretable in $T$. Thus, Theorem 1 states that $\Sigma_1$-sound reflexive theories are trustworthy.

A complete characterization of trustworthiness was given by Visser [3], who in particular proves that the assumption of reflexivity is unnecessary. (Note that the criterion below does not rely on a fixed interpretation of arithmetic, unlike notions such as $\Sigma_1$-soundness or reflexivity.)

Theorem 3 (Visser). A theory $T$ is trustworthy if and only if it has an extension $T'$ (in the same language) such that Robinson’s arithmetic $Q$ has a $\Sigma_1$-sound interpretation in $T'$.

He also derives from this an earlier result attributed to (but apparently unpublished by) H. Friedman:

Corollary. A consistent finitely axiomatized sequential theory is trustworthy.

References

[1] Solomon Feferman, Georg Kreisel, and Steven Orey: 1-Consistency and faithful interpretations, Archiv für mathematische Logik und Grundlagenforschung 6 (1962), pp. 52–63, doi 10.1007/BF02025806.

[2] Per Lindström: On faithful interpretability, in: Computation and Proof Theory (Börger, Oberschelp, Richter, Schinzel, Thomas, eds.), Lecture Notes in Mathematics vol. 1104, Springer, 1984, doi 10.1007/BFb0099490.

[3] Albert Visser: Faith & falsity, Annals of Pure and Applied Logic 131 (2005), pp. 103–131, doi 10.1016/j.apal.2004.04.008.

The standard terminology is that an interpretation $I$ of a theory $U$ in a theory $T$ is faithful if for all sentences $\phi$ in the language of $U$, $$T\vdash\phi^I\iff U\vdash\phi.$$ (Here and below, I will assume all theories to be recursively axiomatized.) A classical result of Feferman, Kreisel, and Orey [1] states:

Theorem 1 (Feferman, Kreisel, Orey). Let $T$ be a reflexive theory which includes a modicum of arithmetic and which is $\Sigma_1$-sound. Then all theories interpretable in $T$ are faithfully interpretable in $T$.

They also show that conversely, if a $\Sigma_1$-sound theory $U$ is faithfully interpretable in an essentially reflexive theory $T$, then $T$ is itself $\Sigma_1$-sound.

Corollary: PA is faithfully interpretable in ZFC if and only if ZFC is $\Sigma_1$-sound.

Further results on faithful interpretability in reflexive theories were obtained by Lindström [2]; for example, he proved the following characterization:

Theorem 2 (Lindström). Let $T$ and $U$ be essentially reflexive theories. Then $U$ is faithfully interpretable in $T$ if and only if $U$ is $\Pi_1$-conservative over $T$ (which is, by itself, equivalent to the interpretability of $U$ in $T$), and $T$ is $\Sigma_1$-conservative over $U$.

Generalizing Theorem 1 in a different direction, a theory $T$ is called trustworthy if all theories that are interpretable in $T$ are faithfully interpretable in $T$. Thus, Theorem 1 states that $\Sigma_1$-sound reflexive theories are trustworthy.

A complete characterization of trustworthiness was given by Visser [3], who in particular proves that the assumption of reflexivity is unnecessary. (Note that the criterion below does not rely on a fixed interpretation of arithmetic, unlike notions such as $\Sigma_1$-soundness or reflexivity.)

Theorem 3 (Visser). A theory $T$ is trustworthy if and only if it has an extension $T'$ (in the same language) such that Robinson’s arithmetic $Q$ has a $\Sigma_1$-sound interpretation in $T'$.

He also derives from this an earlier result attributed to (but apparently unpublished by) H. Friedman:

Corollary. A consistent finitely axiomatized sequential theory is trustworthy.

References

[1] Solomon Feferman, Georg Kreisel, and Steven Orey: 1-Consistency and faithful interpretations, Archiv für mathematische Logik und Grundlagenforschung 6 (1962), pp. 52–63, doi 10.1007/BF02025806.

[2] Per Lindström: On faithful interpretability, in: Computation and Proof Theory (Börger, Oberschelp, Richter, Schinzel, Thomas, eds.), Lecture Notes in Mathematics vol. 1104, Springer, 1984, doi 10.1007/BFb0099490.

[3] Albert Visser: Faith & falsity, Annals of Pure and Applied Logic 131 (2005), pp. 103–131, doi 10.1016/j.apal.2004.04.008.

The standard terminology is that an interpretation $I$ of a theory $U$ in a theory $T$ is faithful if for all sentences $\phi$ in the language of $U$, $$T\vdash\phi^I\iff U\vdash\phi.$$ (Here and below, I will assume all theories to be recursively axiomatized.) A classical result of Feferman, Kreisel, and Orey [1] states:

Theorem 1 (Feferman, Kreisel, Orey). Let $T$ be a reflexive theory which includes a modicum of arithmetic and which is $\Sigma_1$-sound. Then all theories interpretable in $T$ are faithfully interpretable in $T$.

They also show that conversely, if a $\Sigma_1$-sound theory $U$ is faithfully interpretable in a reflexive theory $T$, then $T$ is itself $\Sigma_1$-sound.

Corollary: PA is faithfully interpretable in ZFC if and only if ZFC is $\Sigma_1$-sound.

Further results on faithful interpretability in reflexive theories were obtained by Lindström [2]; for example, he proved the following characterization:

Theorem 2 (Lindström). Let $T$ and $U$ be essentially reflexive theories. Then $U$ is faithfully interpretable in $T$ if and only if $U$ is $\Pi_1$-conservative over $T$ (which is, by itself, equivalent to the interpretability of $U$ in $T$), and $T$ is $\Sigma_1$-conservative over $U$.

Generalizing Theorem 1 in a different direction, a theory $T$ is called trustworthy if all theories that are interpretable in $T$ are faithfully interpretable in $T$. Thus, Theorem 1 states that $\Sigma_1$-sound reflexive theories are trustworthy.

A complete characterization of trustworthiness was given by Visser [3], who in particular proves that the assumption of reflexivity is unnecessary. (Note that the criterion below does not rely on a fixed interpretation of arithmetic, unlike notions such as $\Sigma_1$-soundness or reflexivity.)

Theorem 3 (Visser). A theory $T$ is trustworthy if and only if it has an extension $T'$ (in the same language) such that Robinson’s arithmetic $Q$ has a $\Sigma_1$-sound interpretation in $T'$.

He also derives from this an earlier result attributed to (but apparently unpublished by) H. Friedman:

Corollary. A consistent finitely axiomatized sequential theory is trustworthy.

References

[1] Solomon Feferman, Georg Kreisel, and Steven Orey: 1-Consistency and faithful interpretations, Archiv für mathematische Logik und Grundlagenforschung 6 (1962), pp. 52–63, doi 10.1007/BF02025806.

[2] Per Linström: On faithful interpretability, in: Computation and Proof Theory (Börger, Oberschelp, Richter, Schinzel, Thomas, eds.), Lecture Notes in Mathematics vol. 1104, Springer, 1984, doi 10.1007/BFb0099490.

[3] Albert Visser: Faith & falsity, Annals of Pure and Applied Logic 131 (2005), pp. 103–131, doi 10.1016/j.apal.2004.04.008.

The standard terminology is that an interpretation $I$ of a theory $U$ in a theory $T$ is faithful if for all sentences $\phi$ in the language of $U$, $$T\vdash\phi^I\iff U\vdash\phi.$$ (Here and below, I will assume all theories to be recursively axiomatized.) A classical result of Feferman, Kreisel, and Orey [1] states:

Theorem 1 (Feferman, Kreisel, Orey). Let $T$ be a reflexive theory which includes a modicum of arithmetic and which is $\Sigma_1$-sound. Then all theories interpretable in $T$ are faithfully interpretable in $T$.

They also show that conversely, if a $\Sigma_1$-sound theory $U$ is faithfully interpretable in a reflexive theory $T$, then $T$ is itself $\Sigma_1$-sound.

Corollary: PA is faithfully interpretable in ZFC if and only if ZFC is $\Sigma_1$-sound.

Further results on faithful interpretability in reflexive theories were obtained by Lindström [2]; for example, he proved the following characterization:

Theorem 2 (Lindström). Let $T$ and $U$ be essentially reflexive theories. Then $U$ is faithfully interpretable in $T$ if and only if $U$ is $\Pi_1$-conservative over $T$ (which is, by itself, equivalent to the interpretability of $U$ in $T$), and $T$ is $\Sigma_1$-conservative over $U$.

Generalizing Theorem 1 in a different direction, a theory $T$ is called trustworthy if all theories that are interpretable in $T$ are faithfully interpretable in $T$. Thus, Theorem 1 states that $\Sigma_1$-sound reflexive theories are trustworthy.

A complete characterization of trustworthiness was given by Visser [3], who in particular proves that the assumption of reflexivity is unnecessary. (Note that the criterion below does not rely on a fixed interpretation of arithmetic, unlike notions such as $\Sigma_1$-soundness or reflexivity.)

Theorem 3 (Visser). A theory $T$ is trustworthy if and only if it has an extension $T'$ (in the same language) such that Robinson’s arithmetic $Q$ has a $\Sigma_1$-sound interpretation in $T'$.

He also derives from this an earlier result attributed to (but apparently unpublished by) H. Friedman:

Corollary. A consistent finitely axiomatized sequential theory is trustworthy.

References

[1] Solomon Feferman, Georg Kreisel, and Steven Orey: 1-Consistency and faithful interpretations, Archiv für mathematische Logik und Grundlagenforschung 6 (1962), pp. 52–63, doi 10.1007/BF02025806.

[2] Per Lindström: On faithful interpretability, in: Computation and Proof Theory (Börger, Oberschelp, Richter, Schinzel, Thomas, eds.), Lecture Notes in Mathematics vol. 1104, Springer, 1984, doi 10.1007/BFb0099490.

[3] Albert Visser: Faith & falsity, Annals of Pure and Applied Logic 131 (2005), pp. 103–131, doi 10.1016/j.apal.2004.04.008.

The standard terminology is that an interpretation $I$ of a theory $U$ in a theory $T$ is faithful if for all sentences $\phi$ in the language of $U$, $$T\vdash\phi^I\iff U\vdash\phi.$$

  A classical result of Feferman, Kreisel, and Orey [1] (generalized by Lindström [2]) states:

Theorem 1 (Feferman, Kreisel, Orey). Let $T$ be a reflexive theory which includes a modicum of arithmetic and which is $\Sigma_1$-sound. Then all theories interpretable in $T$ are faithfully interpretable in $T$.

They also show that conversely, if a $\Sigma_1$-sound theory $U$ is faithfully interpretable in a reflexive theory $T$, then $T$ is itself $\Sigma_1$-sound.

Corollary: PA is faithfully interpretable in ZFC if and only if ZFC is $\Sigma_1$-sound.

In general, a theory $T$ is called trustworthy if all theories that are interpretable in $T$ are faithfully interpretable in $T$. Thus, Theorem 1 states that $\Sigma_1$-sound reflexive theories are trustworthy.

A complete characterization of trustworthiness was given by Visser [3], who in particular proves that the assumption of reflexivity is unnecessary. (Note that the criterion below does not rely on a fixed interpretation of arithmetic, unlike notions such as $\Sigma_1$-soundness or reflexivity.)

Theorem 2 (Visser). A recursively axiomatizable theory $T$ is trustworthy if and only if it has an extension $T'$ (in the same language) such that Robinson’s arithmetic $Q$ has a $\Sigma_1$-sound interpretation in $T'$.

He also derives from this an earlier result attributed to (but apparently unpublished by) H. Friedman:

Corollary. A consistent finitely axiomatized sequential theory is trustworthy.

References

[1] Solomon Feferman, Georg Kreisel, and Steven Orey: 1-Consistency and faithful interpretations, Archiv für mathematische Logik und Grundlagenforschung 6 (1962), pp. 52–63, doi 10.1007/BF02025806.

[2] Per Linström: On faithful interpretability, in: Computation and Proof Theory (Börger, Oberschelp, Richter, Schinzel, Thomas, eds.), Lecture Notes in Mathematics vol. 1104, Springer, 1984, doi 10.1007/BFb0099490.

[3] Albert Visser: Faith & falsity, Annals of Pure and Applied Logic 131 (2005), pp. 103–131, doi 10.1016/j.apal.2004.04.008.

The standard terminology is that an interpretation $I$ of a theory $U$ in a theory $T$ is faithful if for all sentences $\phi$ in the language of $U$, $$T\vdash\phi^I\iff U\vdash\phi.$$ (Here and below, I will assume all theories to be recursively axiomatized.) A classical result of Feferman, Kreisel, and Orey [1] states:

Theorem 1 (Feferman, Kreisel, Orey). Let $T$ be a reflexive theory which includes a modicum of arithmetic and which is $\Sigma_1$-sound. Then all theories interpretable in $T$ are faithfully interpretable in $T$.

They also show that conversely, if a $\Sigma_1$-sound theory $U$ is faithfully interpretable in a reflexive theory $T$, then $T$ is itself $\Sigma_1$-sound.

Corollary: PA is faithfully interpretable in ZFC if and only if ZFC is $\Sigma_1$-sound.

Further results on faithful interpretability in reflexive theories were obtained by Lindström [2]; for example, he proved the following characterization:

Theorem 2 (Lindström). Let $T$ and $U$ be essentially reflexive theories. Then $U$ is faithfully interpretable in $T$ if and only if $U$ is $\Pi_1$-conservative over $T$ (which is, by itself, equivalent to the interpretability of $U$ in $T$), and $T$ is $\Sigma_1$-conservative over $U$.

Generalizing Theorem 1 in a different direction, a theory $T$ is called trustworthy if all theories that are interpretable in $T$ are faithfully interpretable in $T$. Thus, Theorem 1 states that $\Sigma_1$-sound reflexive theories are trustworthy.

A complete characterization of trustworthiness was given by Visser [3], who in particular proves that the assumption of reflexivity is unnecessary. (Note that the criterion below does not rely on a fixed interpretation of arithmetic, unlike notions such as $\Sigma_1$-soundness or reflexivity.)

Theorem 3 (Visser). A theory $T$ is trustworthy if and only if it has an extension $T'$ (in the same language) such that Robinson’s arithmetic $Q$ has a $\Sigma_1$-sound interpretation in $T'$.

He also derives from this an earlier result attributed to (but apparently unpublished by) H. Friedman:

Corollary. A consistent finitely axiomatized sequential theory is trustworthy.

References

[1] Solomon Feferman, Georg Kreisel, and Steven Orey: 1-Consistency and faithful interpretations, Archiv für mathematische Logik und Grundlagenforschung 6 (1962), pp. 52–63, doi 10.1007/BF02025806.

[2] Per Linström: On faithful interpretability, in: Computation and Proof Theory (Börger, Oberschelp, Richter, Schinzel, Thomas, eds.), Lecture Notes in Mathematics vol. 1104, Springer, 1984, doi 10.1007/BFb0099490.

[3] Albert Visser: Faith & falsity, Annals of Pure and Applied Logic 131 (2005), pp. 103–131, doi 10.1016/j.apal.2004.04.008.