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i.e. Does an axiom already exist, which prevents the addition of those new axioms which can contradict already existing axioms?

Also, who decides that something is an axiom?

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There is no such axiom. However, one can produce a sentence $\text{Con}$ in the language of first-order PA (Peano arithmetic) which can be interpreted as "saying" that PA is consistent. One can then study the consequences of PA plus $\text{Con}$. –  André Nicolas Feb 12 '13 at 7:51
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5 Answers 5

No, adding more axioms can't remove a contradiction, it can only produce more contradictions.

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I don't think more contradictions is the right term, rather a higher chance of a contradiction is better. –  dinoboy Feb 12 '13 at 7:00
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But, historically, adding more axioms to set theory did remove contradictions! –  Math Gems Feb 12 '13 at 7:04
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@MathGems: Not without removing axioms that people had been assuming they could use until that. –  Henning Makholm Feb 12 '13 at 7:16
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@MathGems: That's the same thing -- the instances of the old unrestricted axiom scheme that were excluded by the restriction are now not axioms anymore -- that is, they have been removed. –  Henning Makholm Feb 12 '13 at 7:34
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@MathGems: This is not a good analogy. Your analogy is like saying that adding more sugar to hemlock fixes the problem with drinking it. –  Asaf Karagila Feb 12 '13 at 9:06

In monotonic logic we have $ A \vDash B \Rightarrow A \cup A' \vDash B$. If we have contradiction then $A \vDash \bot$. Hence if we add axioms ($A' \neq \emptyset$) by monotonicity $A \cup A' \vDash \bot$. Therefore adding more would not allow to remove the contradiction.

While non-monotonic logic exists usually in formal systems a monotonic one is used.

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Since nobody would be forced to actually use that purported No-Contradiction-Axiom in a proof, any proof of a contradiction $P\land\neg P$ would still be valid.

Or: Assume you have a model for axioms $\mathbf A_1, \mathbf A_2, \ldots $ and $\mathbf{AxiomOfNoContradiction}$. Then it is also a model of $\mathbf A_1, \mathbf A_2, \ldots $ alone.

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Actually it's even worse than that. Any formal system, sufficiently rich for Gödel's second incompleteness theorem to apply, which has its own consistency as an axiom must be inconsistent. See en.wikipedia.org/wiki/… –  Robert Israel Feb 12 '13 at 20:14

The Law of Non-Contradiction is a fundamental axiom to logical reasoning, pioneered and formalized by Plato (I think), and logic is integral to mathematical and geometrical reasoning.


The result is that we build a system of knowledge and calculi that is self-consistent... but not necessarily true. The truth of all of mathematics (and of science that rests on mathematics) is ultimately dependent on the truth of the axioms it rests on.

You can build up two mutually exclusive logics which are self-consistent within themselves, but not when you compare or try to integrate the two.

The point is, when you have two assumptions (axioms) that produce a contradiction... within a certain "world" of logic... we can conclude that one of the two, if not both, are incorrect.

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I think we're talking about two separate issues. The "Law of noncontradiction" simply says "not(P and not(P))". It doesn't prevent contradictions. It just says contradictory statements can't both be true. –  Robert Israel Feb 12 '13 at 7:26
    
How is that a different thing? I think you fail to grasp the scope and meaning of the law of non-contradiction. –  CogitoErgoCogitoSum Feb 12 '13 at 13:42

All of mathematics operates in a "world" (e.g. Euclidean Geometry) defined by a set of axioms (e.g. the celebrated 5 axioms, including the Parallel Postulate), which are statements that are assumed to be true. All the of the investigations in that world boil down to deciding if certain statements can be derived logically from the axioms.

If an axiom is contradicted by others (i.e. the other axioms can derive the negation of the axiom), the entire theory would make no sense at all, and nobody discusses these theories.

Anyone can define their own world by setting their own axioms, and the commonplace worlds we see (e.g. Linear Algebra, Real Analysis, Number Theory etc) are axioms that have been chosen well to give rise to extremely interesting worlds.

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There's nothing wrong with axiom systems that are not minimal. For example, most presentations of ZFC set theory are not minimal, because they contain a Null Set axiom that can be derived from the axioms of Infinity and Subsets. (Also, technically almost any axiom system that contains an axiom scheme is non-minimal, because achieving minimality would be at the cost of being able to recognize axioms recursively). –  Henning Makholm Feb 12 '13 at 7:13
    
hmm noted, I've edited that in. –  Herng Yi Feb 12 '13 at 7:22

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