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

  • MHB
  • Thread starter agapito
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In summary, The conversation discusses the relationship between G (Golbach's conjecture and a Π1 wff of Robinson's Arithmetic Q) and its consistency with Q. It is established that G is true if and only if Q cannot prove ¬G. The concept of "consistent with Q" is clarified as meaning that G and Q do not contradict each other, which also implies that they cannot be used to prove both a statement and its negation.
  • #1
agapito
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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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  • #2
"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".
 
  • #3
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
 

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

What is G and Q?

G and Q refer to two separate theories or systems of knowledge. G is typically used to represent a general theory or set of principles, while Q represents a specific theory or set of statements within that general theory.

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

For G to be consistent with Q means that the two theories do not contradict each other. In other words, there are no statements in G that directly conflict with the statements in Q.

How do you determine if G is consistent with Q?

This can be determined through logical analysis and comparison of the statements and principles in G and Q. If there are no direct contradictions between the two, then G can be considered consistent with Q.

Why is it important for G to be consistent with Q?

Consistency between theories is important because it ensures that our understanding of the world is coherent and logical. If there are contradictions between theories, it can lead to confusion and errors in our knowledge.

Can G and Q be consistent but not equivalent?

Yes, it is possible for G and Q to be consistent but not equivalent. This means that while the two theories do not contradict each other, they may still have different principles or statements that are not directly related or interchangeable.

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