4 ms·

But I'm saying if you evaluate the axiom symbols two different ways, it's two different axioms.

• catnaroek 11y ago
  What you're saying is more or less equivalent to “two C implementations targeting different [architectures / operating systems / whatever] are actually implementations of two different programming languages”.
  • plutooo 11y ago
    No, I'm saying the meaning behind statements are different. On some architectures, an int is 16-bit, on others 32-bit. Anyway, this analogy was pushed too far a long time ago.
    • catnaroek 11y ago
      Ah, lovely! You just arrived exactly where I was trying to get. > I'm saying the meaning behind statements are different. Yes, exactly! And, just like a single C program can have two different meanings under two different implementations, the same axiomatic system can have two different meanings when deducing its consequences (proving theorems) using different rules of inference.