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If I understand correctly, axioms are those statements that we assume to be true, instead of proving to be true.

I have seen that in topology theory, various axioms of countability and separation axioms seem to be definitions of some concepts, instead of being assumptions. So I was wondering if "axioms" in topology theory are really axioms?

If no, is this kind of naming rules common in all other branches of mathematics?

Thanks and regards!

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Axioms in the time of Euclid, and up to the nineteenth century, were assumed to be true. Certainly by the twentieth century, this was no longer the case. – André Nicolas Jan 29 '12 at 20:00
Thanks! Then is there some particular reason that encourages such abuse of names in favor over causing confusion? – Tim Jan 29 '12 at 20:03
Have you studied (introductory) abstract algebra? What is your take on (e.g.) "group axioms"? (I guess this partly answers your final question.) – Srivatsan Jan 29 '12 at 20:04
@Srivatsan: I understand the definition of group, but I don't think its definition as axioms, and haven't paid attention to notice if others call it axioms. – Tim Jan 29 '12 at 20:07
@Tim: The meaning of words (and not only in mathematics!) changes over time. – André Nicolas Jan 29 '12 at 20:15
up vote 7 down vote accepted

If you think about it really deep, axioms are really just characteristics of the model we want to work with.

  • The model of the theory of rings is a field if it satisfies the axioms of a field, which require it not only to be a ring - but to be a commutative ring, without zero divisors and that every nonzero element is a unit.

  • The model of the theory of fields is algebraically closed if it satisfies the "closure" axiom, namely every polynomial formula has a solution in the field.

  • The model of ZF, $V$, is a model of ZFC if it also satisfies the axiom of choice, that every family of nonempty sets in the model has a choice function.

If you really think about these things, all those theorems starting with "Let $x$ be this object, with such and such properties. Then $x$ has property $\tau$." is really just to say that if $x$ is a model of some theory which we used, but is also a model of such and such additional axioms, then it is a model for the sentence $\tau$.

So to say that $X$ satisfies the second countability axiom, or the $T_3$ separation axiom is simply to say that we adjoin the theory two more axioms and that $X$ is a model of the larger theory and not just the original theory (topological space).

I once explained someone that if some property holds then we can prove the axiom of choice, he was quite baffled by the fact that we can "prove an axiom", but this is really all we do. We prove one property follows from another, or is equivalent to another, and we actually say "This additional axiom is consistent with the previous ones".

Added: What is a model? We start with a language, in the language we have all the things we want to have: functions, relations, constants, etc.

Now we can interpret this language in a structure. The structure is really just a nonempty set with the addition that we specify how to interpret every symbol of the language (this constant is this element; this predicate is this subset; etc.).

For example, we can interpret the language with a single binary relation as $\subseteq$ in the power set of $X$; or as $\le$ in the real numbers, those are obviously very different ways to understand the same language.

In the language we can write statements. Let us ignore for the moment from the first/second/higher order logic, and allow us to write pretty much anything. We can then write "Every collection of nonempty relations has a function which chooses elements such that these elements have such and such properties with the said relations", or we can write "There is some element that has a certain property".

Now consider the situation that we have a language, and we have an interpretation for this language, and this structure is such that some statement is true in it, then we say that it is a model for that statement.

So in short (after all this long introduction) a model for a theory is just a structure in which all the statements of the theory are true.

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+1, thanks! Could you explain how a "model" is defined? Intuitively, is it equivalent to a concept? – Tim Jan 29 '12 at 20:34
Thanks for the edit! It is really nice for me to approach the concept of a "model". If I understand correctly, a structure is a set of elements having meanings/semantics, a language is a set of elements without meanings/semantics, and the interpretation of a language by a structure is a mapping from the language to the structure? – Tim Jan 31 '12 at 1:20
Another question is since a language has no semantics/meanings assigned to its elements, are its elements really classified into constants, expressions, function, relation, or statements? If yes, how is the classification done? – Tim Jan 31 '12 at 1:36
@Tim: Indeed the language is some encoding of a set of function symbols; relation symbols; constant symbols; and free variables symbols. You also have a function assigning arity to the relations and functions and there is an interpretation function specifying to which relation on the universe set each of the symbol is going to be interpreted as. This is something for a whole other question, though. Not just a minor comment... :-) – Asaf Karagila Jan 31 '12 at 8:14

The definition of axioms as "those statements that we assume to be true, instead of proving to be true" is vague enough to cover a multitude of interpretation.

The classical one considers an axiom to be an absolute mathematical truth that we're convinced of for reasons other than having seen proof. This was how axioms were viewed from Euclid's time until somewhen in the middle 19th century. Many works on mathematics for a lay audience still present it as the only one.

However, modern mathematics view it differently. According to it, axioms are simply statements that we "assume to be true" for the purpose of the argument at hand, and then worry about whether they are true later (if at all). In that sense, defining as an axiom that (say) "for any two different points there is an open set that contains one but not the other" is not meant to mean to assert a universal truth. There may well be spaces for which this is not true. Assuming it as an axiom simply means that spaces that don't satisfy it are not what we're talking about for the moment.

Point-set topology and category theory seem to be particularly fraught with large numbers of possible axioms one is supposed to mix and match among in order to find a combination that suits what one is doing -- but the same concept really pervades all areas of modern mathematics. One typically specifies axioms for some subarea, which is just a customary way to define "the things we're speaking about in this course", namely things that happen to satisfy those axioms.

Even at the very foundation of mathematics, the axioms of set theory are nowadays often viewed as replaceable parts that one explicitly chooses to work with. Most mathematicians would probably assert that the axioms of number theory ought to express some kind of universal truth, but the way they are treated in practice is not much different from the other fields anyway.

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Axioms are merely background assumptions defining the objects that are to be under consideration. If you’re studying Abelian groups, you’ll want to take commutativity of the group operation as an axiom. If you’re studying manifolds, you may want to take second countability as an axiom. For most mathematical purposes you probably want to take the axiom of choice as an assumption. What axioms you choose for your geometry depends on what kind of geometry you want. And so on. The notion that axioms are true is, as André said in the comments, very dated.

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Re your first sentences, I like to think of axioms as "rules by which to play the game". I might have seen this somewhere in MSE, but I can't remember where. – Srivatsan Jan 29 '12 at 20:09

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