Does anyone ever proved that by introducing new compatible axioms we always narrow down the set of possible valid deductions? I think this is intuitively correct (but the fact it is false may in fact be counterintuitive).

So given some axioms we prove theorem $\Bbb P$.

Adding further axioms can't increase the number of cases where we can apply the theorem. So if $\Bbb P$ is our result, adding new axioms to our "language" just reduce the result to something smaller.

If I understand correctly most times people try in fact to remove as much axioms as possible from the "language" and also to remove as much hypothesis as possible from theorems to make them applicable in most cases.

  • $\begingroup$ Can you clarify the question? I can't understand what you're asking. $\endgroup$ – Anthony P Dec 3 '16 at 4:15
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    $\begingroup$ May write a longer answer later, but briefly I'll say, I think you are conflating the things that could be true with the things that are provably true. New assumptions reduce (or keep the same) the number of things that could be true, but they increase (or keep the same) the number of things that can be proven to be true. $\endgroup$ – DanielV Dec 3 '16 at 15:27
  • $\begingroup$ Yes thanks, actually I wanted to know if someone proved that or that is just common sense. $\endgroup$ – GameDeveloper Dec 4 '16 at 16:01

You need to distinguish between axioms and assumptions of a theorem and between one theorem and the body of theorems derived from a set of axioms. A set of axioms, like the real numbers, the group axioms or Peano arithmetic, is used to prove many theorems. Some of those theorems have the form of implications. For example there is a theorem in analysis that if the function $f$ is continuous on a closed interval, it attains its supremum and infimum. You could strengthen the assumptions by asking that $f$ be differentiable. That weakens the theorem because it applies in fewer cases than the earlier version. If you add to the axioms, you increase the set of theorems you can prove because you have more at your disposal.

  • $\begingroup$ I was more going toward the fact that there are actually people that studies incompatibilities between axioms, and that have found sets of axioms that allow certain combinations that are mutually exclusive. Assume I add a new axiom, and that is not reduntant nore in contradiction, then are we actually weaking proven results? (if possible at all doing such thing). I'm speaking of math that does not necessarily have to make sense, but just have to be consistent. I don't know If I can explain it well (If Could explain, I probably don't have any need to ask it in first place) $\endgroup$ – GameDeveloper Dec 3 '16 at 4:45
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    $\begingroup$ In the sense that if you take all the group theory axioms and theorems, then add the commutativity axiom your theorems become restricted to Abelian groups. They may be more general than that. For certain "optional" axioms like this, we keep track of which results need them. $\endgroup$ – Ross Millikan Dec 3 '16 at 4:49
  • $\begingroup$ That's the kind of thing I wanted to know thanks. $\endgroup$ – GameDeveloper Dec 4 '16 at 16:00

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