Rating: **4.2**/5.0 (16 Votes)

Category: Essay

This may be a bit of a trivial question, but can one prove the reflexive, symmetric and transitive properties of equality and the transitive property of inequality of real numbers?(and if so, how? Is there a fairly straightforward, possibly algebraic method?) i.e. Is it possible to prove that $\forall\;a, b\in\mathbb

asked Feb 25 '15 at 23:07

Could one use the following *axiom of order*. Let $a>0$ and $b>0$ then $a+b>0$, and the definition that $a>b\iff a-b>0$. Also, $a<b\iff b>a$. Using this, let $a<b$ and $b<c$, it follows then, that $b-a>0$ and $c-b>0$. This implies that $(b-a) +(c-b) >0\Rightarrow c-a>0\Rightarrow c>a\iff a<c$. Therefore, if $a<b$ and $b<c$ then $a<c$. Would this be correct? If so, can a similar argument be used to prove that equality is transitive (i.e. By starting with $a=b$ and $b=c$ and then using that if $a=0$ and $b=0$ then $a+b=0$, and also, if $a=b$ then $a+(-b) =0$)? – Will Feb 25 '15 at 23:31

Thanks for your help Adam. Out of interest, do you have any notes/links to notes that discuss the equality relation from the sets point of view (I find the notation a bit confusing as to what it intuitively means and can't seem to find anything on it)? Does the equality relation follow from the idea of ordered pairs that $(x,y)=(y,x)\iff x=y$? – Will Feb 26 '15 at 12:37

Absolutely. The equality relation on the real line is stated formally as follows:

Naturally,we assume $S\neq \emptyset$.So let's check all the axioms for an equivalence relation.

(1) Reflexivity. Clearly for every $x \in R$. $(x,x)\in S$.

(2) Symmetry: Let a = b where $a,b\in R$. Then $(a,b)\in S$. 2 ordered pairs in a relation S are the same iff for $(a,b) ,(c,d) \in S$,then a=c and b=d i.e. (a,b) = <,> = <*, >. But this means $(b,a) \in S$ and b=a.*

(3) Transitivity: Let a=b and b=c where $a,b,c\in R$. That means $(a,b), (b,c) \in S$. By reflexivity, b=b. Since a=b, (b,c) = (a,c). So $(a,c) \in S$. Since $(b,c)\in S$, $(c,b)\in S$ by symmetry. Since a=b, $(c,a)\in S$. But now, since $(a,c) and (c,a)\in S$, then a=c and that does it. So equality on *R* is an equivalence relation.

For inequality, a stricter ordering relation then "=" is needed. You have the right idea with your proof,but you have to be a little more careful about the axioms and make sure the order relation is defined via ordered pairs as we've done above. You've got the right idea,though. See if you can finish it yourself.

answered Feb 25 '15 at 23:47

@Will The modern definition of set equality is usually attributed to K.Kuratowski,although there were equivalent definitions advocated by Zermelo,Hausdorff and others. There are some very nice,brief notes by our own Pete Clark here: math.uga.edu/

pete/3200relationsfunctions.pdf Paul Halmos' NAIVE SET THEORY is the standard reference for basic set properties and now it's available cheaply! – Mathemagician1234 Feb 26 '15 at 20:34

There is a construction whereby every real number is really a set of things. (For the purposes of proving the equality properties, it's not material what the construction is.) We then rely on basic properties of sets.

Now, for any given sets $x$ and $y$, if we can show they have the same elements, then they are equal. Let $a,b,c \in \mathbb

Further, if $a = b$, then $a$ and $b$ have the same elements. But then $b$ and $a$ have the same elements, and hence, $b=a$. (And vice-versa.)

You can imagine the transitive argument.

To address the inequality, simply note that if $x < y$ if and only if there exists $z \in \mathbb

*The following texts are the property of their respective authors and we thank them for giving us the opportunity to share for free to students, teachers and users of the Web their texts will used only for illustrative educational and scientific purposes only.*

*All the information in our site are for educational uses.*

*The information of medicine and health contained in the site are of a general nature and purpose which is purely informative and for this reason may not replace in any case, the council of a doctor or a qualified entity legally to the profession.*

*Reflexive axiom of equality- a number or expression is equal to itself (e.g. ab=ab).*

*The meaning and definition indicated above are indicative not be used for medical and legal purposes*

*Source web site to visit. http://engstrom.wikispaces.com/*

*Author. not indicated on the source document of the above text*

*If you are the author of the text above and you not agree to share your knowledge for teaching, research, scholarship (for fair use as indicated in the United States copyrigh low) please send us an e-mail and we will remove your text quickly.*

*Fair use**is a limitation and exception to the exclusive right granted by copyright law to the author of a creative work. In United States copyright law, fair use is a doctrine that permits limited use of copyrighted material without acquiring permission from the rights holders. Examples of fair use include commentary, search engines, criticism, news reporting, research, teaching, library archiving and scholarship. It provides for the legal, unlicensed citation or incorporation of copyrighted material in another author's work under a four-factor balancing test.**(source: http://en.wikipedia.org/wiki/Fair_use)*

*Google key word. reflexive axiom of equality*

*If you want to quickly find the pages about a particular topic as reflexive axiom of equality use the following search engine:*

I'm carrying out some experiments in theorem proving with combinator logic, which is looking promising, but there's one stumbling block: it has been pointed out that in combinator logic it is true that e.g. I = SKK but this is not a theorem, it has to be added as an axiom. Does anyone know of a complete list of the axioms that need to be added?

Edit: You can of course prove by hand that I = SKK, but unless I'm missing something, it's not a theorem within the system of combinator logic with equality. That having been said, you can just macro expand I to SKK. but I'm still missing something important. Taking the set of clauses p(X) and

p(X), which easily resolve to a contradiction in ordinary first-order logic, and converting them to SK, performing substitution and evaluating all calls of S and K, my program generates the following (where I am using ' for Unlambda's backtick):

''eq ''s ''s ''s 'k s ''s ''s 'k s ''s 'k k 'k eq ''s ''s 'k s 'k k 'k k ''s 'k k 'k false 'k true 'k true

It looks like maybe what I need is an appropriate set of rules for handling the partial calls 'k and ''s, I'm just not seeing what those rules should be, and all the literature I can find in this area was written for a target audience of mathematicians not programmers. I suspect the answer is probably quite simple once you understand it.

asked Sep 3 '09 at 17:44

Some textbooks define *I* as mere alias for *((S K) K)*. In this case they are identical (as terms) *per definitionem*. To prove their equality (as functions), we need only to prove that equality is reflexive, which can be achieved by a reflexivity axiom scheme:

- Proposition ``
*E*=*E*'' is deducible (*Reflexivity*axiom scheme, instantiated for each possible terms denoted here by metavariable*E*)

Thus, I suppose in the followings, that Your questions investigates another approach: when combinator *I* is not defined as a *mere alias* for compound term *((S K) K)*. but introduced as a *standalone basic combinator* constant on its own, whose operational semantics is declared *explicitly by axiom* scheme

I suppose Your question asks

whether we can deduce formally (remaining inside the system), that such a standalone-defined *I* behaves exactly as *((S K) K)*. when used as functions in reductions?

I think we can, but we must resort to stronger tools. I conjecture that the usual axiom schemes are not enough, we have to declare also the extensionality property (equality of functions), that's the main point. If we want to formalize extensionality as an axiom, we have to augment our object language with *free variables* .

I think, we have to adopt such an approach for building combinatory logic, that we have to allow also the use of variables in the object langauge. Oof course, I mean "just" *free* valuables. Using bound variables would be cheating, we have to remain inside the realm of combinatory logic. Using free varaibles is not cheating, it's a honest tool. Thus, we can do the formal proof You required.

Besides the straightforward equality axioms and rules of inference (transitivity, reflexivity, symmetry, Leibniz rules), we must add an *extensionality* rule of inference for equality. Here is the point where free variables matter.

In Csörnyei 2007: 157-158, I have found the following approach. I think this way the proof can be done.

Most of the axioms are in fact *axiom schemes*. consisting of infinitely many axiom instances. The instances must be instantiated for for every possible *E*. *F*. *G* terms. Here, I use italics for metavariables.

The superficial infinite nature of axiom schemes won't raise computability problems, because they can be tackled in a finite time: our axiom system is *recursive*. It means that a clever parser can decide in a finite time (moreover, very effectively), whether a given proposition is an instance of an axiom scheme, or not. Thus, the usage of axiom schemes does not raise neither theoretical nor practical problems.

Now let us seem our framework:

I added the constant *I* only because Your question presupposes that we have not defined the combinator *I* as an mere *alias/macro* for compound term *S**K**K*. but it is a standalone constant on its own.

I shall denote constants by boldface roman capitals.

*Sign of application*. A sign @ of ``application'' is enough (prefix notation with arity 2). As syntactic sugar, I use here parantheses instead of the explicit application sign: I shall use the explicit both opening ( and closing ) signs.

*Variables*. Although combinator logic does not make use of bound variables, scope etc, but we can introduce free variables. I suspect, they are not only syntactic sugar, they can strengthen the deduction system, too. I conjecture, that Your question will require their usage. Any enumerable infinite set (disjoint of the constants and parenthesis signs) will serve as the alphabet of variables, I will denote them here with unformatted roman lowercase letters x, y, z.

Terms are defined inductively:

- Any constant is a term
- Any variable is a term
- If
*E*is a term, and*F*is a term too, then also (*E**F*) is a term

I sometimes use practical conventions as syntactic sugar, e.g. write

The usual axioms are complete for beta equality, but do not give eta equality. Curry found a set of about thirty axioms to the usual ones to get completeness for beta-eta equality. They're listed in Hindley & Seldin's *Introduction to combinators and lambda-calculus*.

Roger Hindley, Curry's Last Problem. lists some additional desiderata we might want from mappings between the lambda calculus and notes that we don't have mappings that satisfy all of them. You likely won't care much about all of the criteria.

answered Feb 1 '10 at 13:31

In this section, we will outline eight of the most basic axioms of equality.

The first axiom is called the reflexive axiom or the reflexive property. It states that any quantity is equal to itself. This axiom governs real numbers, but can be interpreted for geometry. Any figure with a measure of some sort is also equal to itself. In other words, segments, angles, and polygons are always equal to themselves. You might think, what else would a figure be equal to if not itself? This is definitely one of the most obvious axioms there is, but it's important nonetheless. Geometric proofs, as well as proofs of all kinds, are so formal that no step goes unwritten. Thus, if perhaps two triangles share a side and you wish to prove those two triangles congruent using the SSS method, it is necessary to cite the reflexive property of segments to conclude that the shared side is equal in both triangles.

PARGRAPH The second of the basic axioms is the transitive axiom, or transitive property. It states that if two quantities are both equal to a third quantity, then they are equal to each other. This holds true in geometry when dealing with segments, angles, and polygons as well. It is an important way to show equality.

The third major axiom is the substitution axiom. It states that if two quantities are equal, then one can be replaced by the other in any expression, and the result won't be changed. It seems natural enough, but is necessary to form the foundation of higher math.

The fourth axiom is often called the partition axiom. It states that a quantity is equal to the sum of its parts. Likewise, in geometry, the measure of a segment or an angle is equal to the measures of its parts.

- The addition axiom states that when two equal quantities are added to two more equal quantities, their sums are equal. Thus, if
*a*=*b*and*y*=*z*. then*a*+*y*=*b*+*z*. - The subtraction axiom states that when two equal quantities are subtracted from two other equal quantities, their differences are equal.
- The multiplication axiom states that when two equal quantities are multiplied with two other equal quantities, their products are equal.
- The division axioms states axiom states that when two equal quantities are divided from two other equal quantities, their resultants are equal.

During the XXth century, a Russian mathematician, Andrei Kolmogorov, proposed a definition of probability, which is the one that we keep on using nowadays.

If we do a certain experiment, which has a sample space $$\Omega$$, we define the probability as a function that associates a certain probability, $$P(A)$$ with every event $$A$$, satisfying the following properties.

The probability of any event $$A$$ is positive or zero. Namely $$P(A)\geq 0$$. The probability measures, in a certain way, the difficulty of event $$A$$ happening: the smaller the probability, the more difficult it is to happen.

The probability of the sure event is $$1$$. Namely $$P(\Omega)=1$$. And so, the probability is always greater than $$0$$ and smaller than $$1$$: probability zero means that there is no possibility for it to happen (it is an impossible event), and probability $$1$$ means that it will always happen (it is a sure event).

The probability of the union of any set of two by two incompatible events is the sum of the probabilities of the events. That is, if we have, for example, events $$A, B, C$$, and these are two by two incompatible, then $$P(A\cup B \cup C)=P(A)+P(B)+P(C).$$

Note: In mathematics, an axiom is a result that is accepted without the need for proof. In this case, we say that this is the axiomatic definition of probability because we define probability as a function that satisfies these three axioms. Also, we might have chosen different axioms, and then probability would be another thing.

That is, the probabilities of complementary events add up to $$1$$. Often we will use this property to calculate probability of the complementary set: $$P(\overline)=1-P(A)$$.

Let's see why. We know that, on the one hand, $$A$$ and $$\overline$$ are incompatible, and on the other that $$A\cup \overline= \Omega$$, since one is the opposite of the other. This is another way of understanding what we already knew, i.e. that the event $$A\cup \overline$$ is a sure event, and therefore, because of axiom 2 $$P(A \cup \overline)=1$$, it always happens. Then, for the axiom 3 $$P(A \cup \overline)=P(A)+P(\overline)$$. But $$P(A \cup \overline)=P(\Omega)=1$$, thus $$P(A)+P(\overline)=1$$.

This property, which turns out to be very useful, can be generalized:

If we have three or more events, two by two incompatible, and such that their union is the whole sample space, that is to say, $$A, B, C$$ two by two incompatible so that $$A\cup B \cup C = \Omega$$, then $$P(A)+P(B)+P(C)=1$$, for axioms 2 and 3.

We say in this case that $$A, B, C$$ form an events complete system. Let's observe that whenever we express $$\Omega$$ as a set of elementary events, in fact we are giving a complete system of events.

As a result $$P(\emptyset)=0$$, that is to say, the probability of the impossible event is $$0$$, since, as we know that the event opposite to the impossible one is the sure event, then we can replace this in the equality of the property $$P(\emptyset)+P(\Omega)=1$$. Therefore, as for the second axiom of the probability $$P(\Omega)=1$$, we have $$P(\emptyset)+1=1$$, thus $$P(\emptyset)=0$$.

The notation "if $$A\subset B$$" reads "if the event $$A$$ is included in event $$B$$" that is to say, if all the possible results that satisfy $$A$$ also satisfy $$B$$.

This property is quite logical: if, after throwing a dice, we want to compare the probability of $$A =$$"to extract $$2$$" with $$B =$$"to extract an even number", then, the probability of $$A$$ has to be smaller or the same as that of $$B$$ since if we extract $$2$$, we are extracting an even number. In other words, when $$A$$ is satisfied, $$B$$ is also satisfied, therefore it should be more difficult to satisfy $$A$$ than $$B$$. Namely $$P(A) \leq P(B)$$.

This result, which is very important to remember, is a consequence of something that you can see in the sets cell: given two sets, A and B, you can express its union as $$A\cup B = (A-B)\cup (A\cap B) \cup (B-A),$$ which are two by two incompatible. Then, for axiom 3 $$P(A\cup B)=P(A-B)+P(A\cap B)+P(B-A)$$.

In the Sets Teory we have that $$A=(A-B) \cup (A\cap B)$$, which are two incompatible events, and therefore, for axiom 3 $$P(A)=P(A-B)+P(A\cap B)$$, that is to say, $$P(A-B)=P(A)-P(A\cap B)$$.

$$B=(B-A) \cup (B\cap A) = (B-A) \cup (A\cap B)$$, by which $$P(B-A)=P(B)-P(A\cap B)$$.

Replacing these probabilities in the equality, we find

$$P(A\cup B)= P(A-B)+P(A\cap B)+P(B-A)=$$ $$=P(A)-P(A\cap B)+P(A\cap B)+(P(B)-P(A\cap B))=$$ $$=P(A)+P(B)-P(A\cap B)$$

Now, we can solve some problems.

A dice of six faces is tailored so that the probability of getting every face is proportional to the number depicted on it.

1 What is the probability of extracting a $$6$$?

In this case, we say that the probability of each face turning up is not the same, therefore we cannot simply apply the rule of Laplace. If we follow the statement, it says that the probability of each face turning up is proportional to the number of the face itself, and this means that, if we say that the probability of face $$1$$ being turned up is $$k$$ which we do not know, then:

Now, since $$\<1\>,\<2\>,\<3\>,\<4\>,\<5\>,\<6\>$$ form an events complete system. necessarily

Therefore $$$k+2k+3k+4k+5k+6k=1$$$ which is an equation that we can already solve: $$$21k=1$$$ thus $$$k=\dfrac<1><21>$$$

And so, the probability of extracting $$6$$ is $$P(\<6\>)=6k=6\cdot \dfrac<1><21>=\dfrac<6><21>.$$

2 What is the probability of extracting an odd number?

The cases favourable to event $$A =$$ "to extract an odd number" are: $$\<1\>,\<3\>,\<5\>$$. Therefore, since they are incompatible events,

Tomorrow there is an exam. Esther has studied really hard, and she only has $$\dfrac<1><5>$$ probability of not passing the exam.

David has studied less, and he has $$\dfrac<1><3>$$ probability of not passing the exam. We know that the probability of both not passing the exam is $$\dfrac<1><8>$$.

What is the probability that at least one of them does not pass the exam?

The first thing that we must do is express the problem as we know how, i.e. with events. We define the events $$A = $$"Esther does not pass the exam", $$B =$$"David does not pass the exam".

From the statement, we know that $$P(A\cap B)=\dfrac<1><8>$$.

We might think that if Esther has probability $$\dfrac<1><5>$$ of not passing the exam, and David $$\dfrac<1><3>$$ of not passing the exam, then the probability of at least one of them not passing, that is to say $$P(A\cup B)$$, should be $$\dfrac<1><5> + \dfrac<1><3> = \dfrac<8><15>$$, but this is false.

If we compute it this way, we are assuming that the events $$A$$ and $$B$$ are incompatible, that is to say, that they cannot happen simultaneously, when the statement says that they could both not pass (simultaneously).

Therefore, the correct way of calculating this probability is using the formula that we have seen before: $$$P(A\cup B)=P(A)+P(B)-P(A\cap B)$$$

by replacing with the results that we know, we get $$$P(A\cup B)=\dfrac<1><5>+\dfrac<1><3>-\dfrac<1><8>=\dfrac<24><120>+\dfrac<40><120>-\dfrac<15><120>=\dfrac<49><120>$$$

or what amounts to the same, $$40,8\widehat<3>\%$$.

Everybody knows that equality is reflexive: $\forall(x)(x=x)$. But should reflexivity of equality be taken as an axiom of logic or as a theorem of set theory?

If you choose the former then you probably need the axiom of extensionality: $\forall(x)\forall(y)(x=y\leftrightarrow\forall(z)(z\in x\leftrightarrow z\in y))$.

If you choose the latter then probably $x=y$ is just an abbreviation for $\forall(z)(z\in x\leftrightarrow z\in y)$.

What I'm trying to do is to write down proofs for basic facts about set theory, but I'm not so sure which logical axioms and rules of inference should I take for granted.

We all want reflexivity to be provable in any first order theory, not just set theory. So this question has nothing particularly to do with set theory or with extensionality. Surely we want x=x in all groups, all rings, all graphs, and so on in any mathematical context.

So the answer is that either we add it as an axiom, or we make sure to have other logical axioms that can prove it. What you seem to want or need are the explicit details of your formal proof system. You may choose among many logical systems out there (see the Wikipedia page on proof theory http://en.wikipedia.org/wiki/Proof_theory ). In particular, the Hilbert calculus includes reflexivity explicitly as a logical axiom.

Ultimately, it is an irrelevant choice whether you have it as an axiom or as a theorem, unless you are proof theorist, who is studying the proofs themselves as mathematical objects, rather than using proof to understand its mathematical content, as you seem to be doing.

answered Jan 29 '10 at 14:33

I would just like to point out that the usual properties of equality relation, including reflexivity, follow from a single two-way proof rule by Lawvere (expressed in natural-deduction style):

This should be read as a *two-way rule* (I don't know how to produce a double horizontal line), i.e. from the top we may infer the bottom and vice versa. The rule has the form of an adjunction between functors, or a Galois connection if you will (equality is left adjoint to contraction is what the rule says). I personally find this more illuminating that wondering whether equality is axiomatized or derived.

For example, reflexivity follows when we take $\phi$ to be the formula $x = y$ and we read the rule bottom-up: because $x = y \vdash x = y$ it is also the case that $\vdash x = x$.

Reference: Bart Jacobs, "Categorical logic and type theory", Lemma 4.1.7, page 229, available in Google books .

answered Jan 29 '10 at 17:34

I find these proof-theoretic considerations to cast the most light: the stremgth of proof theory is showing how elementary the concepts we are looking at really are. – Charles Stewart Jan 30 '10 at 9:55

Of course, I very much doubt that in this particular case Lawvere was going after proof-theoretic analysis of equality. He was expressing the proof rules for equality in a category-theoretic way. This can be done with all logical connectives and the quantifiers: each receives exactly one two-way proof rule which is an adjunction. – Andrej Bauer Jan 31 '10 at 10:02

I think it is important to consider. The axiom of extensionality is provided as a means of defining equality between sets. Roughly stated we say *"A set X is equal to set Y if and only if they both have exactly the same elements."* A more precise statement, taking into account that we'd like to quantify over domains that we can define, is this:

1) $X=Y\leftrightarrow \ \forall x \in X\ (x\in Y) \ \wedge \ \forall y \in Y\ (y\in X).$

This is just defining an equality relation for sets, and reflexivity comes from the definition rather than having to be be put forth as an axiom. For example taking *X=X* ,

$\forall x \in X\ (x\in X) \ \wedge \ \forall x \in X\ (x\in X),$

We can say that reflexivity is a theorem here. It's one of the properties of an *Equivalence Relation*. which must obey reflexivity, symmetry, and transitivity. Our 'equals' relation on sets happily obeys these rules. I like to consider equality as a possible relation we can define about objects, with some properties that we want it to hold. If we were coming at things from the perspective of logic we might make these properties axioms and prove the existence of such relations as theorems. I'm not completely sure on this. But from the perspective of axiomatized set theory, the extensionality axiom implies the rest of the equality properties very concretely.

answered Jan 29 '10 at 17:06

I managed to write down a proof for the reflexivity of equality using only the definition of equality in terms of membership and the rules of natural deduction.

- Premise: $\forall x_0\forall x_1\left(\left(x_0=x_1\right)\leftrightarrow\forall x_2\left(\left(x_2\in x_0\right)\leftrightarrow\left(x_2\in x_1\right)\right)\right)$
- Assumption (1): $\forall x_0\forall x_1\left(\left(x_0=x_1\right)\leftrightarrow\forall x_2\left(\left(x_2\in x_0\right)\leftrightarrow\left(x_2\in x_1\right)\right)\right)$
- Universal elimination (2): $\forall x_1\left(\left(k_0=x_1\right)\leftrightarrow\forall x_2\left(\left(x_2\in k_0\right)\leftrightarrow\left(x_2\in x_1\right)\right)\right)$
- Universal elimination (3): $\left(\left(k_0=k_0\right)\leftrightarrow\forall x_2\left(\left(x_2\in k_0\right)\leftrightarrow\left(x_2\in k_0\right)\right)\right)$
- (Subproof 1) Premise: $\left(k_2\in k_0\right)$
- (Subproof 1) Assumption (5): $\left(k_2\in k_0\right)$
- (Subproof 2) Premise: $\left(k_2\in k_0\right)$
- (Subproof 2) Assumption (7): $\left(k_2\in k_0\right)$
- Biconditional introduction (5, 6, 7, 8): $\left(\left(k_2\in k_0\right)\leftrightarrow\left(k_2\in k_0\right)\right)$
- Universal introduction (9): $\forall x_2\left(\left(x_2\in k_0\right)\leftrightarrow\left(x_2\in k_0\right)\right)$
- Biconditional elimination (4, 10): $\left(k_0=k_0\right)$
- Universal introduction (11): $\forall x\left(x=x\right)$

In a similar way I also wrote down the proof for the symmetry and for the transitivity of equality.

answered Feb 5 '10 at 18:22

For what it's worth (it's been a while), the treatments I have seen take the fundamental properties of equality (reflexivity, symmetry, and transitivity, plus axioms stating that you cannot tell the difference between equal objects) as axioms, though not really as axioms of logic. Rather, there are first order theories, and among them are first order theories with equality, meaning they have a binary relation “=” and associated axioms. Then one spends just a page or two working out the consequences of this, and later, set theory is just another example of a first order theory with equality.

I wonder, though: If you drop equality and the axiom of extensionality, treating equality just as an abbreviation as you suggest, how do you prove that, if every member of *x* is equal to a member of *y* and vise versa, then *x* =_y_?

answered Jan 29 '10 at 14:34

Let alone proving that if x = y, then x belongs to z if and only if y belongs to z (which I always regarded as the true meaning of the axiom of extensionality). – Andrea Ferretti Jan 29 '10 at 14:40

@Andrea: I am not sure why you consider that statement being the true (or any) meaning of extensionality. To me, it's an equality axiom: The one that says you can't tell the difference between equal objects. Apart from that, I agree with your comment. – Harald Hanche-Olsen Jan 29 '10 at 16:24

An *axiom* is any sentence, proposition, statement or rule that forms the basis of a formal system. Unlike theorems, axioms are neither derived by principles of deduction, nor are they demonstrable by formal proofs. Instead, an axiom is taken for granted as valid, and serves as a necessary starting point for deducing and inferencing logically consistent propositions. In many usages, "axiom," "postulate ," and "assumption " are used interchangeably.

In certain epistemological theories, an axiom is a self-evident truth upon which other knowledge must rest, and from which other knowledge is built up. An axiom in this sense can be known before one knows any of these other propositions. Not all epistemologists agree that any axioms, understood in that sense, exist.

In logic and mathematics. an axiom is not necessarily a self-evident truth, but rather a formal logical expression used in a deduction to yield further results. To axiomatize a system of knowledge is to show that all of its claims can be derived from a small, well-understood set of sentences. This does not imply that they could have been known independently; and there are typically multiple ways to axiomatize a given system of knowledge (such as arithmetic ). Mathematics distinguishes two types of axioms: logical axioms and non-logical axioms.

The word "axiom" comes from the Greek word αξιωμα (*axioma* ), which means that which is deemed worthy or fit or that which is considered self-evident. The word comes from αξιοειν (*axioein* ), meaning to deem worthy, which in turn comes from αξιος (*axios* ), meaning worthy. Among the ancient Greek philosophers an axiom was a claim which could be seen to be true without any need for proof.

The logico-deductive method whereby conclusions (new knowledge) follow from premises (old knowledge) through the application of sound arguments (syllogisms, rules of inference), was developed by the ancient Greeks, and has become the core principle of modern mathematics. Tautologies excluded, nothing can be deduced if nothing is assumed. Axioms and postulates are the basic assumptions underlying a given body of deductive knowledge. They are accepted without demonstration. All other assertions (theorems. if we are talking about mathematics) must be proven with the aid of these basic assumptions. However, the interpretation of mathematical knowledge has changed from ancient times to the modern, and consequently the terms *axiom* and *postulate* hold a slightly different meaning for the present day mathematician, than they did for Aristotle and Euclid.

The ancient Greeks considered geometry as just one of several sciences. and held the theorems of geometry on par with scientific facts. As such, they developed and used the logico-deductive method as a means of avoiding error, and for structuring and communicating knowledge. Aristotle's posterior analytics is a definitive exposition of the classical view.

An “axiom”, in classical terminology, referred to a self-evident assumption common to many branches of science. A good example would be the assertion that *When an equal amount is taken from equals, an equal amount results.*

At the foundation of the various sciences lay certain additional hypotheses which were accepted without proof. Such a hypothesis was termed a *postulate*. While the axioms were common to many sciences, the postulates of each particular science were different. Their validity had to be established by means of real-world experience. Indeed, Aristotle warns that the content of a science cannot be successfully communicated, if the learner is in doubt about the truth of the postulates.

The classical approach is well illustrated by Euclid's elements, where a list of axioms (very basic, self-evident assertions) and postulates (common-sensical geometric facts drawn from our experience), are given.

- Axiom 1: Things which are equal to the same thing are also equal to one another.
- Axiom 2: If equals be added to equals, the wholes are equal.
- Axiom 3: If equals be subtracted from equals, the remainders are equal.
- Axiom 4: Things which coincide with one another are equal to one another.
- Axiom 5: The whole is greater than the part.

- Postulate 1: It is possible to draw a straight line from any point to any other point.
- Postulate 2: It is possible to produce a finite straight line continuously in a straight line.
- Postulate 3: It is possible to describe a circle with any center and distance.
- Postulate 4: It is true that all right angles are equal to one another.
- Postulate 5. It is true that, if a straight line falling on two straight lines make the interior angles on the same side less than two right angles, the two straight lines, if produced indefinitely, intersect on that side on which are the angles less than the two right angles.

A lesson learned by mathematics in the last 150 years is that it is useful to strip the meaning away from the mathematical assertions (axioms, postulates, propositions. theorems) and definitions. This abstraction, one might even say formalization, makes mathematical knowledge more general, capable of multiple different meanings, and therefore useful in multiple contexts.

Structuralist mathematics goes farther, and develops theories and axioms (e.g. field theory. group theory. topology. vector spaces ) without *any* particular application in mind. The distinction between an “axiom” and a “postulate” disappears. The postulates of Euclid are profitably motivated by saying that they lead to a great wealth of geometric facts. The truth of these complicated facts rests on the acceptance of the basic hypotheses. However by throwing out the Euclid's fifth postulate, we get theories that have meaning in wider contexts, hyperbolic geometry for example. We must simply be prepared to use labels like “line” and “parallel” with greater flexibility. The development of hyperbolic geometry taught mathematicians that postulates should be regarded as purely formal statements, and not as facts based on experience.

When mathematicians employ the axioms of a field. the intentions are even more abstract. The propositions of field theory do not concern any one particular application; the mathematician now works in complete abstraction. There are many examples of fields; field theory gives correct knowledge about them all.

It is not correct to say that the axioms of field theory are “propositions that are regarded as true without proof.” Rather, the field axioms are a set of constraints. If any given system of addition and multiplication satisfies these constraints, then one is in a position to instantly know a great deal of extra information about this system.

Modern mathematics formalizes its foundations to such an extent that mathematical theories can be regarded as mathematical objects, and logic itself can be regarded as a branch of mathematics. Frege. Russell. Poincaré. Hilbert. and Gödel are some of the key figures in this development.

In the modern understanding, a set of axioms is any collection of formally stated assertions from which other formally stated assertions follow by the application of certain well-defined rules. In this view, logic becomes just another formal system. A set of axioms should be consistent ; it should be impossible to derive a contradiction from the axiom. A set of axioms should also be non-redundant; an assertion that can be deduced from other axioms need not be regarded as an axiom.

It was the early hope of modern logicians that various branches of mathematics, perhaps all of mathematics, could be derived from a consistent collection of basic axioms. An early success of the formalist program was Hilbert's formalization of Euclidean geometry. and the related demonstration of the consistency of those axioms.

In a wider context, there was an attempt to base all of mathematics on Cantor's set theory. Here the emergence of Russell's paradox. and similar antinomies of naive set theory raised the possibility that any such system could turn out to be inconsistent.

The formalist project suffered a decisive setback, when in 1931 Gödel showed that it is possible, for any sufficiently large set of axioms (Peano's axioms. for example) to construct a statement whose truth is independent of that set of axioms. As a corollary. Gödel proved that the consistency of a theory like Peano arithmetic is an unprovable assertion within the scope of that theory.

It is reasonable to believe in the consistency of Peano arithmetic because it is satisfied by the system of natural numbers. an infinite but intuitively accessible formal system. However, at present, there is no known way of demonstrating the consistency of the modern Zermelo-Frankel axioms for set theory. The axiom of choice. a key hypothesis of this theory, remains a very controversial assumption. Furthermore, using techniques of forcing (Cohen ) one can show that the continuum hypothesis (Cantor) is independent of the Zermelo-Frankel axioms. Thus, even this very general set of axioms cannot be regarded as the definitive foundation for mathematics.

In the field of mathematical logic. a clear distinction is made between two notions of axioms: *logical axioms* and *non-logical axioms* (somewhat similar to the ancient distinction between "axioms" and "postulates" respectively)

These are certain formulas in a language that are universally valid. that is, formulas that are satisfied by every structure under every variable assignment function. In colloquial terms, these are statements that are *true* in any possible universe, under any possible interpretation and with any assignment of values. Usually one takes as logical axioms *at least* some minimal set of tautologies that is sufficient for proving all tautologies in the language; in the case of predicate logic more logical axioms than that are required, in order to prove logical truths that are not tautologies in the strict sense.

These axiom schemata are also used in the predicate calculus. but additional logical axioms are needed.

is universally valid.

*Non-logical axioms* are formulas that play the role of theory-specific assumptions. Reasoning about two different structures, for example the natural numbers and the integers. may involve the same logical axioms; the non-logical axioms aim to capture what is special about a particular structure (or set of structures, such as groups ). Thus non-logical axioms, unlike logical axioms, are not *tautologies*. Another name for a non-logical axiom is *postulate*.

Almost every modern mathematical theory starts from a given set of non-logical axioms, and it was thought that in principle every theory could be axiomatized in this way and formalized down to the bare language of logical formulas. This turned out to be impossible and proved to be quite a story (*see below* ); however recently this approach has been resurrected in the form of neo-logicism.

Non-logical axioms are often simply referred to as *axioms* in mathematical discourse. This does not mean that it is claimed that they are true in some absolute sense. For example, in some groups. the group operation is commutative. and this can be asserted with the introduction of an additional axiom, but without this axiom we can do quite well developing (the more general) group theory, and we can even take its negation as an axiom for the study of non-commutative groups.

Thus, an *axiom* is an elementary basis for a formal logic system that together with the rules of inference define a *deductive system*.

This section gives examples of mathematical theories that are developed entirely from a set of non-logical axioms (axioms, henceforth). A rigorous treatment of any of these topics begins with a specification of these axioms.

Basic theories, such as arithmetic. real analysis and complex analysis are often introduced non-axiomatically, but implicitly or explicitly there is generally an assumption that the axioms being used are the axioms of Zermelo–Fraenkel set theory with choice, abbreviated ZFC, or some very similar system of axiomatic set theory. most often Von Neumann–Bernays–Gödel set theory. abbreviated NBG. This is a conservative extension of ZFC, with identical theorems about sets, and hence very closely related. Sometimes slightly stronger theories such as Morse-Kelley set theory or set theory with a strongly inaccessible cardinal allowing the use of a Grothendieck universe are used, but in fact most mathematicians can actually prove all they need in systems weaker than ZFC, such as second-order arithmetic.

*Geometries* such as Euclidean geometry. projective geometry. symplectic geometry. Interestingly one of the results of the fifth Euclidean axiom being a non-logical axiom is that the three angles of a triangle do not by definition add to 180°. Only under the umbrella of Euclidean geometry is this always true.

The Peano axioms are the most widely used *axiomatization* of first-order arithmetic. They are a set of axioms strong enough to prove many important facts about number theory and they allowed Gödel to establish his famous second incompleteness theorem.

Probably the oldest, and most famous, list of axioms are the 4 + 1 Euclid's postulates of plane geometry. This set of axioms turns out to be incomplete, and many more postulates are necessary to rigorously characterize his geometry (Hilbert used 23).

The axioms are referred to as "4 + 1" because for nearly two millennia the fifth (parallel) postulate ("through a point outside a line there is exactly one parallel") was suspected of being derivable from the first four. Ultimately, the fifth postulate was found to be independent of the first four. Indeed, one can assume that no parallels through a point outside a line exist, that exactly one exists, or that infinitely many exist. These choices give us alternative forms of geometry in which the interior angles of a triangle add up to less than, exactly, or more than a straight line respectively and are known as elliptic. Euclidean. and hyperbolic geometries.

The object of study is the real numbers. The real numbers are uniquely picked out (up to isomorphism ) by the properties of a *Dedekind complete ordered field*. meaning that any nonempty set of real numbers with an upper bound has a least upper bound. However, expressing these properties as axioms requires use of second-order logic. The Löwenheim-Skolem theorems tell us that if we restrict ourselves to first-order logic. any axiom system for the reals admits other models, including both models that are smaller than the reals and models that are larger. Some of the latter are studied in non-standard analysis.

that is, for any statement that is a *logical consequence* of there actually exists a *deduction* of the statement from . This is sometimes expressed as "everything that is true is provable", but it must be understood that "true" here means "made true by the set of axioms", and not, for example, "true in the intended interpretation". Gödel's completeness theorem establishes the completeness of a certain commonly-used type of deductive system.

There is thus, on the one hand, the notion of *completeness of a deductive system* and on the other hand that of *completeness of a set of non-logical axioms* . The completeness theorem and the incompleteness theorem, despite their names, do not contradict one another.

Early mathematicians regarded axiomatic geometry as a model of physical space. and obviously there could only be one such model. The idea that alternative mathematical systems might exist was very troubling to mathematicians of the 19th century and the developers of systems such as Boolean algebra made elaborate efforts to derive them from traditional arithmetic. Galois showed just before his untimely death that these efforts were largely wasted. Ultimately, the abstract parallels between algebraic systems were seen to be more important than the details and modern algebra was born. In the modern view we may take as axioms any set of formulas we like, as long as they are not known to be inconsistent.

*This article incorporates material from Axiom on PlanetMath. which is licensed under the GFDL .*