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To prove a theorem in mathematics we usually use our axioms, definitions and other already proved theorems. Suppose we wante to prove a specific theorem and we haven't prove any other theorem and also we don't use any other definition except the definitions that are contained in the statement we want to prove. That is we want to prove from scratch (using just the axioms) the statement "If P then Q" where we have only defined what P, Q and the conditional mean. Are the axioms (using also rules of inference) enough to prove that statement in the system?

Does adding definitions change the system?

Suppose the statement "The numbers 2,4 are even" where we have defined "even=any number greater than 2". This clearly shows that just by changing one definition we have at least change the truth value of a statement. In other words theorems in an axiomatic system are also sensitive to definitions. That is the system changes not only by axioms but also from the definitions. So does this mean that we are not allowed to add definitions in a system or we can just only such definitions that don't contradict the previous irrespective of the fact that these new definitions may alter the number of theorems that can be proved?

  • Theorems are not sensitive to definitions, labeling something else "even" does not change the theorems, only relabels their abbreviated formulations. The role of definitions in a formal system is mostly abbreviation for convenience. – Conifold Jul 28 at 20:53
  • Yes, they are. A definition is a "formal abbreviation": it does not "add" nothing new to what can be already proved from the axioms. – Mauro ALLEGRANZA Jul 29 at 5:54
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    In your example, if we stay with the definition of "Even", we have that not-Even(2), because we can easily prove that "2 is not greater than 2". Thus, if we add to the theory the statement Even(2), we have a contradiction. – Mauro ALLEGRANZA Jul 29 at 7:30
  • @MauroALLEGRANZA So what I can prove depends only on the axioms? But to define new term should I first prove that it somehow makes sense to add this "abbrevation"? For example first we show that there exists a unique number that can be obtained by adding and multiplying the entries of the matrix and we call it determinant. Can we do the same with every other definition? In other words is every defined entity proved to exist by the axioms before we define it? – ado sar Jul 29 at 9:01
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    What we like only matters for which theorems we choose to focus on, and how we wish to present them. But all derivative terms can be eliminated from statements by replacing them according to definitions, and theorems reformulated in basic terms only. In particular, all theorems in ZFC can be reduced to statements about sets and elements, be they about ellipses, numbers, vector spaces or trunctions. Of course, the result will not be attractive to a human, but the difference is purely decorative. All you can do "arbitrarily" is pick theorems and abbreviations that are more appealing intuitively. – Conifold Jul 30 at 10:45

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