MHB What does it mean for G to be consistent with Q?

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The discussion centers on the relationship between Golbach's conjecture (G) and Robinson's Arithmetic (Q). It establishes that G is true if and only if Q cannot prove the negation of G (¬G). Additionally, it asserts that G is true if and only if G is consistent with Q. The term "consistent with Q" is clarified to mean that G and Q do not contradict each other, implying that it is impossible to derive both a statement and its negation using G and Q. The soundness of Q reinforces its consistency, providing a foundation for this relationship.
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My text if Smith's Godel book. We establish that G (Golbach's conjecture and a Π1 wff of Robinson's Arithmetic Q) is true if and only if Q cannot prove ¬G.

That much is clear. But then it goes on to say:

G is true if and only if G is consistent with Q.

We know that Q is sound (and thus consistent), so can someone please explain what "consistent with Q" means.

Thanks for all help.
 
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"G is consistent with Q' simply means that G and Q do not contradict each other. That, in turn, means that we cannot use G and Q to prove both statement "T" and statement "not T".
 
HallsofIvy said:
"G is consistent with Q' simply means that G and Q do not contradict each other. That, in turn, means that we cannot use G and Q to prove both statement "T" and statement "not T".

Greatly appreciate your help with this, agapito
 
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