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Edit: I found in Arnold Miller's lecture notes an entertaining account of an easy proof of (a fake version of) Completeness from Compactenss (see page 58). His system amounts to the abstract formal system I describe above. Namely, he introduces the MM proof system (for Mickey Mouse), where the axioms are all logical validities, and the only rule of inference is Modus Ponens. In this system, one can prove Completeness from Compactness easily as follows: We want to show that $T$ proves φ if and only if every model of $T$ is a model of φ. The forward direction is Soundness, which is easy. Conversely, suppose that every model of $T$ is a model of $\phi$. Thus, $T+\lnot\phi$ has no models. By Compactness, there are finitely many axioms $\phi_0,\ldots,\phi_n$ in $T$ such that there is no model of them plus $\lnot\phi$. Thus, $\phi_0\land\ldots\land\phi_n\implies\phi$ is a logical validity. And from this, one can easily make a proof of $\phi$ from $T$ in MM. QED!

Edit: I found in Arnold Miller's lecture notes an entertaining account of an easy proof of (a fake version of) Completeness from Compactness (see page 58). His system amounts to the abstract formal system I describe above. Namely, he introduces the MM proof system (for Mickey Mouse), where the axioms are all logical validities, and the only rule of inference is Modus Ponens. In this system, one can prove Completeness from Compactness easily as follows: We want to show that $T$ proves φ if and only if every model of $T$ is a model of φ. The forward direction is Soundness, which is easy. Conversely, suppose that every model of $T$ is a model of $\phi$. Thus, $T+\lnot\phi$ has no models. By Compactness, there are finitely many axioms $\phi_0,\ldots,\phi_n$ in $T$ such that there is no model of them plus $\lnot\phi$. Thus, $\phi_0\land\ldots\land\phi_n\implies\phi$ is a logical validity. And from this, one can easily make a proof of $\phi$ from $T$ in MM. QED!

Edit: I found in Arnold Miller's lecture notes an entertaining account of an easy proof of (a fake version of) Completeness from Compactenss (see page 58). His system amounts to the abstract formal system I describe above. Namely, he introduces the MM proof system (for Mickey Mouse), where the axioms are all logical validities, and the only rule of inference is Modus Ponens. In this system, one can prove Completeness from Compactness easily as follows: We want to show that T proves φ if and only if every model of T is a model of φ. The forward direction is Soundness, which is easy. Conversely, suppose that every model of T is a model of φ. Thus, T+¬φ has no models. By Compactness, there are finitely many axioms φ₀, ..., φ_n in T such that there is no model of them plus ¬φ. Thus, (φ₀∧...∧φ_n implies φ) is a logical validity. And from this, one can easily make a proof of φ from T in MM. QED!

Edit: I found in Arnold Miller's lecture notes an entertaining account of an easy proof of (a fake version of) Completeness from Compactenss (see page 58). His system amounts to the abstract formal system I describe above. Namely, he introduces the MM proof system (for Mickey Mouse), where the axioms are all logical validities, and the only rule of inference is Modus Ponens. In this system, one can prove Completeness from Compactness easily as follows: We want to show that $T$ proves φ if and only if every model of $T$ is a model of φ. The forward direction is Soundness, which is easy. Conversely, suppose that every model of $T$ is a model of $\phi$. Thus, $T+\lnot\phi$ has no models. By Compactness, there are finitely many axioms $\phi_0,\ldots,\phi_n$ in $T$ such that there is no model of them plus $\lnot\phi$. Thus, $\phi_0\land\ldots\land\phi_n\implies\phi$ is a logical validity. And from this, one can easily make a proof of $\phi$ from $T$ in MM. QED!

Edit: I found in Arnold Miller's lecture notes an entertaining account of an easy proof of (a fake version of) Completeness from Compactenss (see page 58). His system amounts to the abstract formal system I describe above. Namely, he introduces the MM proof system (for Mickey Mouse), where the axioms are all logical validities, and the only rule of inference is Modus Ponens. In this system, one can prove Completeness from Compactness easily as follows: We want to show that T proves φ if and only if every model of T is a model of φ. The forward direction is Soundness, which is easy. Conversely, suppose that every model of T is a model of φ. Thus, T+¬φ has no models. By Compactness, there are finitely many axioms φ₀, ..., φ_n in T such that there is no model of them plus φ. Thus, (φ₀∧...∧φ_n implies φ) is a logical validity. And from this, one can easily make a proof of φ from T in MM. QED!

Edit: I found in Arnold Miller's lecture notes an entertaining account of an easy proof of (a fake version of) Completeness from Compactenss (see page 58). His system amounts to the abstract formal system I describe above. Namely, he introduces the MM proof system (for Mickey Mouse), where the axioms are all logical validities, and the only rule of inference is Modus Ponens. In this system, one can prove Completeness from Compactness easily as follows: We want to show that T proves φ if and only if every model of T is a model of φ. The forward direction is Soundness, which is easy. Conversely, suppose that every model of T is a model of φ. Thus, T+¬φ has no models. By Compactness, there are finitely many axioms φ₀, ..., φ_n in T such that there is no model of them plus ¬φ. Thus, (φ₀∧...∧φ_n implies φ) is a logical validity. And from this, one can easily make a proof of φ from T in MM. QED!