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At first I was thinking about the axiom of choice, but let's keep it general. What motivates the inclusion of new axioms (or change the ones we already have in an already defined axiomatic theory?. It seems that one motivation could be a way to solve problems that couldn't be solved before and are proved to be impossible to solve without the additions of a new axiom.

But this doesn't seem to turn mathematics a little bit upside down?. This is, some axioms seem to be very intuitive -field axioms would be an example- but other axioms -like the ones for topology or some of the ZFC- are instead constructed by having a good idea of the theory that we want as a consequence of the axioms, and even if I agree that axioms should be constructed that way I cannot help to have an uneasy feeling if later the theory is changed, even if this means solving new problems (like if we're changing the rules of the game on the run). So, what conditions would have to satisfy a proposed axiom to be considered, besides consistency with the former theory?, what motivates any change in an already defined theory?

I know this is a somewhat weird question, I just hope having been clear about my question.

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This is a great question, with many people here capable of giving great answers. All those answers would be small philosophical essays, however. I don't believe this question is well-suited for MSE. You may want to phrase it in the form of a reference request for essays and blogs that address this question. – Amit Kumar Gupta Jan 21 '14 at 7:24
Is this about "axioms" or "definitions"? – Did Jan 21 '14 at 7:27
Love the question and really looking forward to the answers! +1 and favourited. Although I disagree with the contention that any of the axioms of ZFC are counter-intuitive. If you believe in infinite sets and accept the need for the existence of powersets, then ZFC appears very, very natural. – goblin Jan 21 '14 at 7:32
@user18921 Precisely what I asked. The examples in the post (mixing field "axioms" (?) and topology "axioms" (?) with ZFC axioms) and some phrases like "adding axioms", make me wonder whether this question is not based on a confusion between axioms and definitions. – Did Jan 21 '14 at 7:41
@Did, I disagree with the distinction. The field axioms are genuine first-order axioms, just like the ZFC axioms. Furthermore, just like we tend to consider models of the field axioms (i.e. fields), set-theorists often consider models of the ZFC axioms (that we might call "cumulative hierarchies" or "set-theoretic universes"). – goblin Jan 21 '14 at 7:51

2 Answers 2

up vote 2 down vote accepted

In general, new axioms now are included if you can prove their independence from foundations and you need the axiom to prove other interesting statements. For instance, when Cohen proved the independence of the continuum hypothesis from ZFC, it ceased to be a conjecture and instead it (and it's negation) became an axiom in the recipe book for certain strong theories.

There are a number of statements that have been proved independent of ZFC, many inconsistent with each other. These are mixed and matched with the foundations depending on the needs of the mathematician at the time, often to prove things well into the strange world of large cardinals and other monsters.

Occasionally we go the other way however and attempt to do work trying to prove what axioms are necessary to prove a statement, and then we work in the world of reverse mathematics, proving the axioms from the theorems; For instance determining the minimum requisite axioms to prove something about primes in the natural numbers chaining down to which axioms in ZFC are required. Here we may decide that the orthodox system isn't granular enough and then replace the foundations for something more granular, or perhaps something that is more suitable for mechanized reasoning such as NGB, with its finite axiomization.

Then we change the foundations themselves. We often use ZFC as a starting point because its a familiar starting ground that is roughly equivalent to other foundations, and it has a strong trending name recognition, but we switch foundations around when we want some other properties; Finite axiomization, granular weakness, or perhaps even a more compact representation, or extensibility with objects like classes or semantics of other formal systems like higher order logic.

For more specific examples that are less foundational, at other times we want a very weak axiom system that is decidable, such as Pressburger arithmetic, but we decide that we want more expressible statements so we make conservative extensions that preserve its satisfiability. This is where many satisfiable modulo theory axioms come from, convenient because we can use SAT solvers instead of first order theorem provers. At other times we want to weaken an undecidable theory like Peano arithmetic to make it finitely axiomizable, and now we have Robinson arithmetic.

In any case new axioms are added when they're demonstrated independent from the foundations or the foundations themselves are undesirable for the task at hand.

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Not the same question but maybe relevant: Believing the Axioms by Penelope Maddy.

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