Can axioms be false?

Question

I have often wondered, can axioms be false? For example, I could take as an axiom that "Dogs don't exist", but that is false. To give a more mathematical example, I could take as an axiom that "2 is not equal to 2", but that is, again, false. So, then, does it make sense to talk of axioms being false?

Answer 1

It depends. An axiom is a starting point or foundation for reasoning. The term has been used in different ways historically. It used to mean a proposition that is self-evidently true, or at least so widely accepted that it does not need to be argued for, hence its role as a starting point.

In modern logic, the axioms of a theory are sentences from which all the other theorems of the theory are deducible. Typically there are many ways to axiomatise a given theory, so the choice of axioms is somewhat arbitrary. Axioms in this sense do not have to be true. If a theory is unsound, then some of its theorems and at least one of its axioms are false. More generally, for a given theory, its axioms might be true under some interpretations and false under others. Or the theory may be used in an abstract way with uninterpreted sentences, in which case the axioms don't have a truth value.

A theory that has axioms that are false does not automatically make it inconsistent. An inconsistent theory explodes if it is closed under the deducibility relation of classical logic, so being inconsistent is highly undesirable. Depending on exactly which system of logic you are using, "2 is not equal to 2" might qualify as a contradiction, rather than merely being false. In which case, it would make a theory inconsistent. But "Dogs don't exist" is contingently false, not contradictory.

Answer 2

You may choose any statement as an axiom. But to choose a contradiction as an axiom makes the whole subsequent theory a useless enterprise - much ado about nothing, see the comment of @user4894.

Added. IMO an axiom has to satisfy at least the following requirements:

• It captures in a formal way the conception one has in mind.
• It is independent from the other axioms.
• It is useful - possible even necessary - to derive the propositions of the theory.
• No inconsistencies with the other axioms of the theory are known.

Answer 3

In mathematical logic you consider syntax and sematics separately: an axiom from the syntactic point of view is just a meaningless formula, neither true or false. You can give a semantics to your language and this will give meaning to any formula. A semantics that makes all your axioms true will be called a "model" of the axioms (or a model of the axiomatic theory), but there is no guarantee in general that the axioms will have a model, in principle. Also even if a model exists that makes all axioms true you can still consider alternative semantics where some axioms are false.

Answer 4

An axiom was originally understood as a sentence self-evidently or "universally" recognised as true. However, the notion was borrowed by mathematicians who redefined it to mean any fundamental assumption.

Speakers of one natural language usually understand each other, and they have dictionaries to make sure they do. This even extends to speakers of different natural languages as long as they have bilingual dictionaries to help them. Consequently, most humans would agree on the axiom that the sun rises from behind the Eastern horizon and sets down on the Western one.

Another example of a false mathematical axiom is the assertion that the implication α → β is true when α is false. We understand the axiom once we understand that mathematicians have decided to redefine the notion of implication such that the implication α → β is just the horseshoe ¬α ∨ β! The horseshoe ¬α ∨ β obviously is true if ¬α is true, and so if α is false, so, under the mathematicians' redefinition of the implication as the horseshoe, the implication α → β is indeed true if α false! Et voilà. This does not change the fact that in the real world an implication is not necessarily true if its first term is false, but it allows mathematical textbooks to teach students very nearly everywhere in the world that the implication α → β is true when α is false...

There is no absolute semantic so that we can all choose to redefine the words we use. Axioms in the ordinary sense might be false, but they are normally difficult to falsify because very nearly everyone accepts them as true, but science certainly did manage to falsify a good number of previously "universally" accepted axioms.

Mathematical axioms, however, are a mixed bunch. Some are also axioms in the original sense, most are not, and some are patently false on the face of them, notwithstanding that mathematicians may pretend otherwise.

So, axioms, which sense do you mean?

Irrespective of how we define our notion of axioms, if an axiom is a statement about the real world, then it can of course be false. However, a statement B that an axiom A is false would itself be possibly false, if the axiom is, well, true in fact.

However, this all depends on how you choose to define "true". I quote from one answer here:

for a given theory, its axioms might be true under some interpretations and false under others

This statement can only be true if you redefine the notion of truth itself! In the ordinary sense of "true", a statement is either true or false, but its truth doesn't depend on any interpretation. Its truth only depends on what reality is, and, according to our standard notion of reality, there is just one reality. The notion of "interpretation" makes it as if there were alternative realities from which we could pick and choose. Thus, the quoted statement can only be true if we assume that there are different realities, which is in itself a redefinition of the notion of truth. Thus, an axiom a > a would be true if we assumed an "interpretation such that the symbol ">" just mean equal! Easy. Or, while an axiom x > 0 would be false on the face of it and according to our shared semantic, it becomes "true" if you redefine the notion of truth such that the axiom is "true" once you assume an interpretation such that x is redefined as belonging for example to {1, 2, 3}.

Answer 5

At least to a certain way of thinking, mathematical concepts have no meaning beyond their axiomatic definition. So, for instance, a "line" in Euclidean geometry is something that satisfies the properties of lines as set out in the axioms/postulates of Euclidean geometry. If you find something for which the axioms are false, then that something isn't a line. In this formulation, an axiom is true by definition.

It is possible for axioms to be contradictory; that is, it may be possible to prove both a statement and its negation from the set of axioms. In that case, the system is inconsistent. There is then some sense in which the axioms are "false".

As Marco Disce says, mathematicians speak of "models" for a system of axioms. Basically, a model is something that follows that system. So physical space can be treated as a model of three dimensional Euclidean geometry. If we find that the axioms don't perfectly describe physical space, such as with the curvature from general relativity, then that just shows that physical space is an imperfect model, it doesn't show that the axioms are "false".

To give a more mathematical example, I could take as an axiom that "2 is not equal to 2"

But what do "2" and "not" and "equal" mean? You'd have to rigorously define those terms before you could say "2 is not equal to 2" is "false". And you wouldn't be able to say it's false, other than be appealing to other axioms. So you can't say that a particular axiom is false, only that a system of axioms, taken as whole, is inconsistent.

"Dogs don't exist"

Again, you'd have to rigorously define "dogs" and "exist". To "exist" means "to be in the real world", but how do you define "the real world" mathematically? Mathematical axioms speak only of mathematical things, and "the real world" isn't a mathematical thing.

Answer 6

An axiom is a statement that is asserted to define the basis of a logical system. Axioms are neither true not false; they merely are, as underlying rules for the system. To use the Wittgensteinian analogy, the way that chess pieces move are axioms. It is merely presented to us that a knight moves like this and a rook moves like that, and the game itself (in all its variety) derives from those pre-given assertions. Thus, science, logic, and math are all unique games played out under the specific axioms they assert as true.

Some axioms are logically necessary for a system to be self-consistent; these axioms cannot be changed or removed without breaking the system as a whole. Other axioms are contingently necessary, meaning that they can be altered to produce new logical systems (e.g., the non-Euclidian geometries produced by changing Euclid's parallel postulate). But the point is that axioms come to light through the effort to reduce a logical system to the minimal necessary and sufficient conditions for the logical system to be functional and self-consistent.

Axioms are (in an odd way) like breathing. Breathing isn't true, or false, or right, or wrong; it's merely a necessity for life to sustain itself. Axioms are the necessary elements that define a logical system. If you discard them and choose others the ones you discarded aren't false, you've just discarded one valid system in favor of another.

Answer 7

Sorry, not personal, but the selected answer is incorrect: a false axiom must always be rejected, because it derives always into a paradoxical state (true and false). Logic "does not depend". For logic to be useful, it needs complete consistency. See the following explanation, the logic is simple. In simple words...

1. a simple false axiom makes all other axioms true and false at the same time.
2. in consequence, the false axiom contradicts itself: it becomes true and false SIMULTANEOUSLY. That is, it becomes paradoxical.

An axiom is part of a formal system (which is a formal expression of a set of axioms). The point to consider here is that an axiom is normally part of a logically consistent set. All axioms need to be logically consistent for the set to be valid (that is, for the set to be).

Now, what happens when you introduce a falsehood in a set of propositions? The set becomes fully contradicting due to the principle of explosion. This means that every axiom in the set is true and false at the same time.

See the following example:

 1. A=B
 2. B=C
 3. A=True
 4. C=False

• Apply 3 to 1 and 2: A, B and C are TRUE
• Apply 4, C is FALSE, to 2 and 1: B is FALSE
  • and therefore A is FALSE

As you see, all statements are contradictory, and the whole system loses all consistency. Let's say the "false" axiom is 4. Well, it is a paradox, because it expresses truth and falsehood at the same time.

But this does not end here: things become worse from here on:

All axiomatic systems are consistent with Logic. Therefore, it is mutually dependent on logic (that is, Logic itself DEPENDS on this axiomatic set). And what happens here?

• All previous axioms lead to this: ** FALSE = TRUE**

Which destructs not only the full system, but destroys ALL PROPOSITIONAL LOGIC (ergo, ALL LOGIC). This means that a statement like...

• Poison is bad for health

... is TRUE and FALSE at the same time.

All of this due to a simple false axiom. That is why it is necessary for ALL axiomatic systems to be consistent.

You will reply that there are in fact, false axioms in common reasoning (e.g. the earth is flat). Yes, and such false axioms lead to contradictions, which lead to fouled judgements that risk our very existence (logic is not useless: it rules our behavior). In simple words, lack of logic due to false axioms can lead to death (e.g. Challenger Space Shuttle disaster, Titanic sinking, Salem Witch Trials, etc.). Survival depends on logic, breaking logic means risking existence.

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Added, about paraconsistent logics and dialetheism:

A key feature of formal systems is they provide a strong foundation for inferences. So, paraconsistent logics and dialetheism are ruled out from the start, since the OP question targets a monoletheia, not a dialetheia. In case of axiomatic systems, this consideration is critical, since paraconsistent logic allows weak and short inferences, which makes them useless in formal systems. Axiomatic systems need to accept long inference sequences. An axiomatic system that allow few -and dialetheical- derivations is essentially useless. See @JoWehler's answer and comments (not mine).

Answer 8

https://math.stackexchange.com/questions/1231447/what-separates-an-axiom-from-a-proposition

Propositions are true statements about the mathematical structure that can be derived from the axioms.

An axiom in one domain of reason is an assertion taken to be neither true nor false. The same assertion may be contemplated as a true or false proposition in some other domain of reason.

That which has no part

https://intellectualmathematics.com/blog/that-which-has-no-part-euclids-definitions

The first two lines of Euclid’s Elements are the most misunderstood. They define the concepts of point and line. “A point is that which has no part” and “a line is a length without breadth.” “A point is that which has no part” and “a line is a length without breadth.”

The concepts of the line and point are axiomatic in the context of Euclid's geometry. In actual life there may be no examples of items with length without breadth or that have no part. One might argue that the axioms are false but that would be outside the context of the knowledge domain we call geometry.

Answer 9

In a logic system you start with axioms which are taken as true, and then you use the rules of logic to derive other statements that you can prove are true or false.

If you pick an axiom that isn’t really reasonable, then you may be able to come up with a proof that this axiom is false. Well, that means your logic system is more than useless and you throw it away.

So an axiom that people are actually using won’t be false. There is a tiny possibility that a proof it is false hasn’t been found yet.

Answer 10

There are many levels of confusion in your question (and many answers).

Contingent vs Necessary

Dogs don't exist was true say a billion years in the past. It will also be true somewhere between now and the heat death of the universe.

So it's almost trivially true, just not true in/of our current world.

And when you contemplate all possible worlds, it's even wider than the above. Eg. A world in which owls carry significant messages to young bespectacled awkward boys is not true of our world and is almost certainly false for the past-future gamut described above, but is true of Harry Potter stories and fans' imaginations.

Further there are historical distortions. Since axiom was first and most famously used by Euclid 2 millenia ago in his geometry treatise, it's worth going back to Euclid. However since Greek is, ahem, Greek to me I asked google bard.

How did Euclid distinguish postulate from axiom?

In Euclid's famous work, "Elements," he made a subtle distinction between axioms and postulates, though its significance and interpretation have been debated over time. Here's how he used these terms:

Axioms

General Principles: Euclid used axioms for general assumptions considered self-evident truths, applicable not just to geometry, but to any logical system. These included statements like:

• Things which are equal to the same thing are equal to each other.
• If equals are added to equals, the wholes are equal.
• The whole is greater than the part.

Postulates

Geometry-Specific Assumptions: Postulates, on the other hand, were specific assumptions applicable only to geometry. They dealt with the construction and properties of geometric objects like points, lines, and circles.

Examples include:

• A straight line segment can be drawn joining any two points.
• Through any point not on a straight line, exactly one line parallel to the given line can be drawn.¹

Distinguishing between Axioms and Postulates

The precise distinction between axioms and postulates in Euclid's work is somewhat unclear. Some historians argue it was primarily a matter of scope: axioms being broader and postulates being more specific to geometry. Others suggest it might have been based on level of "self-evidence": axioms being more universally accepted truths, while postulates being slightly less evident, requiring some acceptance within the context of geometry.

Modern Interpretation

Today, the terms "axiom" and "postulate" are often used interchangeably. However, the distinction Euclid made can still be valuable.

It highlights the importance of both general logical principles and specific domain-dependent assumptions when building a mathematical system.

Summary

Alternate postulates produce alternate universes. Alternate axioms likely produce nonsense. ('Likely' because, as bard reminds, there's a difference between the two but not a very clear one)

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¹ This one has a long colorful history

Answer 11

To keep it as concise as possible: Axioms shouldn't be false if you're trying to use it to make further claims, but it could still be false without you knowing it.