Can we determine the number of theorems that can be proven with X axioms?

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In a consistent system with X axioms, determining the number of theorems or lemmas that can be derived is complex and may not yield a specific count. Theorems can be infinite, especially when considering the implications of axioms and rules of inference. Axioms themselves can be viewed as theorems since they can be derived from the axioms of the system. The relationship between axioms, theorems, and tautologies varies across different logical systems, with some systems allowing every theorem to be a tautology and vice versa. Ultimately, the number of theorems is influenced by the axioms and rules, but no definitive method exists to calculate an exact number based solely on the axioms.
  • #31
My * symbol means the same thing as --> (regular implies, not strict implies), and my post doesn't suggest otherwise. Yes, sorry, I missed that you were using --> in the ordinary way, but then I wonder what the point of your post was, and how it was a response to what lqg was saying. Well you said, "just thinking aloud" so perhaps it wasn't meant to be a response to what he was saying?
 
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  • #32
AKG said:
I wonder what the point of your post was, and how it was a response to what lqg was saying. Well you said, "just thinking aloud" so perhaps it wasn't meant to be a response to what he was saying?
That's exactly what I had intended, you read my thoughts. :smile:
 

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