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edit approved Oct 9, 2021 at 22:31
mrtaurho
edit approved Oct 9, 2021 at 22:31
mrtaurho
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Why do people say Godel'sGödel's sentence is true when it is true in some models but false in others?

I am a beginner, so this question may be naive.

Suppose we have a (sufficiently strong) consistent first order logic system. Godel'sGödel's first incompleteness theorem says there exists a GodelGödel sentence g$g$ which is unprovable, and its negation is also unprovable. By Godel'sGödel's completeness theorem, g$g$ can't be a logical consequence of the axioms, which means there are models of the system that makes g$g$ false. So my question is: then why do people say g$g$ is true when viewed outside the system?

PS: apparently there are "non-standard models" that makes g$g$ false according to wikipedia, then why don't people say g$g$ is true in standard models, which is more accurate? Also, do non-standard models work with natural numbers anymore?

Why do people say Godel's sentence is true when it is true in some models but false in others?

I am a beginner, so this question may be naive.

Suppose we have a (sufficiently strong) consistent first order logic system. Godel's first incompleteness theorem says there exists a Godel sentence g which is unprovable, and its negation is also unprovable. By Godel's completeness theorem, g can't be a logical consequence of the axioms, which means there are models of the system that makes g false. So my question is: then why do people say g is true when viewed outside the system?

PS: apparently there are "non-standard models" that makes g false according to wikipedia, then why don't people say g is true in standard models, which is more accurate? Also, do non-standard models work with natural numbers anymore?

Why do people say Gödel's sentence is true when it is true in some models but false in others?

I am a beginner, so this question may be naive.

Suppose we have a (sufficiently strong) consistent first order logic system. Gödel's first incompleteness theorem says there exists a Gödel sentence $g$ which is unprovable, and its negation is also unprovable. By Gödel's completeness theorem, $g$ can't be a logical consequence of the axioms, which means there are models of the system that makes $g$ false. So my question is: then why do people say $g$ is true when viewed outside the system?

PS: apparently there are "non-standard models" that makes $g$ false according to wikipedia, then why don't people say $g$ is true in standard models, which is more accurate? Also, do non-standard models work with natural numbers anymore?

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Why do people say Godel's sentence is true when it is true in some models but false in others?

I am a beginner, so this question may be naive.

Suppose we have a (sufficiently strong) consistent first order logic system. Godel's first incompleteness theorem says there exists a Godel sentence g which is unprovable, and its negation is also unprovable. By Godel's completeness theorem, g can't be a logical consequence of the axioms, which means there are models of the system that makes g false. So my question is: then why do people say g is true when viewed outside the system?

PS: apparently there are "non-standard models" that makes g false according to wikipedia, then why don't people say g is true in standard models, which is more accurate? Also, do non-standard models work with natural numbers anymore?