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François G. Dorais
François G. Dorais
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This is a question from Stephan Belniuk'sStefan Bilaniuk's very good free online book "A Problem Course in Mathematical Logic"A Problem Course in Mathematical Logic:

Problem 18.6. Suppose Sigma$\Sigma$ is a recursive set of sentences of LN. Find a sentence of LN, which we'll denote by Con(Sigma)$Con(\Sigma)$, such that Sigma$\Sigma$ is consistent if and only if A |- Con(Sigma)$A \vdash Con(\Sigma)$.

By Godel's second incompleteness theorem isn't this impossible?

This is a question from Stephan Belniuk's very good free online book "A Problem Course in Mathematical Logic":

Problem 18.6. Suppose Sigma is a recursive set of sentences of LN. Find a sentence of LN, which we'll denote by Con(Sigma), such that Sigma is consistent if and only if A |- Con(Sigma).

By Godel's second incompleteness theorem isn't this impossible?

This is a question from Stefan Bilaniuk's very good free online book A Problem Course in Mathematical Logic:

Problem 18.6. Suppose $\Sigma$ is a recursive set of sentences of LN. Find a sentence of LN, which we'll denote by $Con(\Sigma)$, such that $\Sigma$ is consistent if and only if $A \vdash Con(\Sigma)$.

By Godel's second incompleteness theorem isn't this impossible?

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Dave
Dave
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Statement of consistency in Godel's second incompleteness theorem

This is a question from Stephan Belniuk's very good free online book "A Problem Course in Mathematical Logic":

Problem 18.6. Suppose Sigma is a recursive set of sentences of LN. Find a sentence of LN, which we'll denote by Con(Sigma), such that Sigma is consistent if and only if A |- Con(Sigma).

By Godel's second incompleteness theorem isn't this impossible?