We construct many structures by chosing a set of axioms and deriving everything else from them. As far as I remember we never proved in our lectures that those axioms do not contradict. So:
Is it always possible to prove that a set of axioms does not contain a contradiction? (So if not could we perhaps find a contradiction in a structure that we definde by axioms someday?) Or is it proven that doing so is impossible?
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For a first order theory, basically a set of "axioms" formulated in a first order language, you have Gödel's completeness theorem. The theorem establishes that a first order theory is consistent(i.e. non-contradictory) is and only if you have a model for that theory(i.e. a "real" mathematical object satisfying the axioms of the theory).
From this you can know that group theory, ring theory, field theory and, in general, every first order theory that has a model is a consistent theory.
To see this you proceed using the completeness theorem and observing, for example, that $(\mathbb{Z}_2,+)$ is a model for group theory, and $(\mathbb{Z}_2,+,.)$ is a model for field and ring theory.
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Proved from what?
Even logic is formalized in a mathematical fashion, so it has a mathematical backbone. We need to first assume that first-order logic does not include a contradiction. Then we need to decide on which system we are going to use to develop our mathematics. Arithmetical systems will suffice for syntactical manipulations of strings, and set theoretical systems will suffice for semantical proofs using models.
Whatever meta-theory, in which we formalized the logic and the proofs and whatnot, you are working in, if it is sufficiently strong, it will not prove its own consistency. This means that along with the assumption that first-order logic does not contain a contradiction, we implicitly assume that our meta-theory is consistent, whatever it might be. And the more we need, the more consistency we will have to prove.
Goedel proved that a theory which is both consistent, recursively enumerable, and interprets arithmetics, cannot prove its own consistency (and is therefore not complete). This means that we really have to assume the consistency of our meta-theory (which is usually a theory satisfying these conditions).
Of course, we can prove the consistency of arithmetic, using set theory, and the consistency of set theory using an even stronger set theory, and so on. But that's not what you're looking for in an answer. The answer, simply, is that at some point, you have to take some things for granted. The question is how much you are willing to take for granted, how much logic and foundational theories you are familiar with, to transform an argument form one system to another.
You can prove contradiction by constructing results which stem from them that contradict. Right? If A leads to Z and B leads to NOT Z then something is obviously fishy. Or equally showing that C leads to both Y and NOT Y simultaneously.
The inability to show contradiction does not prove anything. Indeed even the ability to demonstrate consistency in any given set of logic does not prove that the logic is valid.
Logics are constructed by logicians that are self-consistent but contradict other forms of logics, each based on its own set of axioms. Thus it is unfair and even irrational to judge one train of thought based in one set of logical principles against that of another. A logic can only be disproved through self-inconsistency, or inconsistency with the real world if you want to go there, but not through inconsistency between "worlds" of logic.
Even if you do show that a premise results in a contradiction. Does that necessarily prove that the premise is false? Ultimately though, and I think you have failed to realize this (many people do), but the Law of Non-Contradiction that prohibits contradiction in logic is itself an axiom.
Id love to ask a question. What should we do when mathematics or logic contradicts reality?
I want to point out the fact that reality has already contradicted mathematics and logic. Take Euclidean geometry, for example. It makes the assumption that space is flat (Euclidean). Most of modern mathematics is based atop of these fundamental assumptions about geometry, including the all-important calculus. All of modern physics, too. Except one day Einstein comes along and shows that real space is curved. What can we conclude but all of the mathematics we've been using all along is built on flat-space assumptions and is faulty at the fundamental level - at least to the real world we apply it in. How do you rectify that? There is the good-old "accurate enough" argument, and of course we have invented other forms of geometry, but... convolution is the key to success.
The law of non-contradiction is pretty steadfast in logic and mathematics. But in quantum physics - the reality from which the entire universe is built - allows a certain degree of contradictory truths. And gad-zooks! The law of non-contradiction might actually be fallible or non-applicable in certain contexts of reasoning.
If you can appreciate any of this from a pure mathematical or pure logical perspective, or the significance it has on physics... just imagine the implications in philosophy, religion, morality.