What is a Number? - Math Philosophers' Views

[matt grime]

but a bag is an object.

It is also an analogy.

 

[matt grime]

yeah that's the part of set theory i never got...because I didn't understand why {{}}!={}

because one is a set that contains no elements and the other is a set that contains one element, and that element is itself a set, the empty set. It is important you distinguish between a subset of a set and an element of a set.

Always thought that set theory started with one element/singleton {x}

Nope, set theories do not say what the sets or the elements of the sets are, it just tells you the rules that the sets obey. The empty set, and the sets whose existence are implied by the existence of the empty set and the other axioms (such as constructibility) are the only ones that must be in the model.

 

[Mickey]

Jimmy, I don't think so!

We need the axiom of extensionality to tell us what it means for things to be equal. Then we are able to look at empty sets and see that, since they all have the same property, they are one and the same.

In your pseudo-axiom, you seem to already define your empty set as identity, taking for granted the set theory steps.

Also, you only have that one pseudo-axiom, so your pseudo-proof is more like an interpretation of that pseudo-axiom. It's like it's attempting to be an extension of that pseudo-axiom, but you forgot to have another axiom for what it means to "extend" something.

 

[neurocomp2003]

matt grime: so your saying a set that contains a set of no elements? wouldn't that be no elements? What is a set of no elements? just a pair of brace brackets?

Micky: doesn't teh axiom of extensionality states there exists something?

 

[honestrosewater]

matt grime: so your saying a set that contains a set of no elements? wouldn't that be no elements? What is a set of no elements? just a pair of brace brackets?

A pair of brackets is something. As you guys are using them, a pair of brackets denotes a set. The empty set is a set. It seems like you're looking through the brackets, as if you can just delete them if there is nothing inside of them. Is that how you're looking at things? That isn't how it works.

Micky: doesn't teh axiom of extensionality states there exists something?

No, it defines (or extends) equality on sets in terms of the membership relation: two sets are equal iff they contain the same members. Perhaps you are thinking of the Axiom of Infinity. That and the Axiom of the Empty Set are the only two axioms that I have ever seen included in any of the ZF axiomatizations that actually give you a set. At most, the other axioms give you sets if you already have other sets.

Also, ZF is not the only axiomatization of set theory. You don't have to take membership as primitive and define the subset relation, union operation, intersection operation, difference operation, etc. in terms of membership. You could simply take them all as primitive, or, for example, the equivalence that defines subsets in terms of membership

(1) $$\forall x, y, z \in D \ [(y \subseteq z) \Leftrightarrow (x \in y \rightarrow x \in z)]$$

where D is your domain of sets, can just as well be used to define membership in terms of susbets, making any necessary changes of course.As for what a number is, if you think that a number is defined by its internal structure, you can let a number be any structure that satisfies, or models, some theory of numbers, a theory of numbers being some set of formulas that you think anything called a number should satisfy. Or if you think that numbers are defined by their relationships with other things, you can let a number be a member of the domain of any structure that satisfies your theory of numbers. That is how I would answer similar questions:

Q: What is a set?
A: A set is a member of the domain of a structure that satisfies the theory of sets.

Q: What is a group?
A: A group is a structure that satisfies the theory of groups.

Q: What is an equality relation?
A: An equality relation is a structure that satisfies the theory of equality relations.

 

[matt grime]

matt grime: so your saying a set that contains a set of no elements? wouldn't that be no elements?

You have just asserted that the empty set does not exist. In the notation used {{}} is a set that contains a set. It therefore has an element in it, namely the set {}, but it would not matter if it were some other set: you're still making the same fallacious step*. You have also asserted that the empty set is a set that contains itself (as an element), so that is two times where you contradict the axioms of ZF (at least), three if we throw in the axiom of extension discussion you're having independently of me.

* the fallacious leap is that you think that the following two things are equal:

A set S, and the set {S} which is a set that contains S.

{S} has one element irrespectiveof what S is, be it the empty set (you agree the empty set is something) or the integers, or some large cardinal.

 

[3trQN]

yeah that's the part of set theory i never got...because I didn't understand why {{}}!={}

Was this adopted in axiomatic set theory after such problems as Russell's Paradox? Was naive set theory approach such that {{}} = {}?

 

[neurocomp2003]

ok i sort of get what you all are saying now...thanks for explaining itto me.

Is there a "Set Existence Axiom" ( thereexists x = x)

matt grimes: As pertaining to the 2 things are equal S= {S}...only for the empty set because i never really understood the empty set.

 

[honestrosewater]

ok i sort of get what you all are saying now...thanks for explaining itto me.

Is there a "Set Existence Axiom" ( thereexists x = x)

There certainly could be. As things are usually done, I think you're bordering on the logical matters, which are usually left in the background in math. Assume your background theory includes first-order logic, which I think is the most used logic in math. For any theory that includes an equality symbol, that symbol gets interpreted, by convention, as the identity relation on the domain of your structure. So for any such theory, the formula

(2) $$\forall x \in D \ [x = x]$$

will be satisfied by every structure, i.e., it will be true in every structure. If you also include as part of your background theory the assumption that all structure domains are non-empty or that

(3) $$\forall x \ [\phi] \rightarrow \exists x \ [\phi]$$

is satisfied by every structure for any formula $\phi$, it follows that

(4) $$\exists x \in D \ [x = x]$$

will also be satisifed by every structure that satisfies (2). So your background, logical theory and interpretation conventions might give you one or more sets, but you don't know anything else about them except that each is equal to itself. I think it's more of a technicality anyway, as I think you can change this stuff without it having any non-boring effects.

i never really understood the empty set.

The empty set arises naturally from several places. One rather intuitive place, I guess, is the connection between properties and sets. A property that no object has, or that is satisfied by no object, corresponds to the empty set. If no object has the property of being a square circle (or not being equal to itself or being a penguin on my lap), the set of all objects that have the property of being square circles (or not being equal to themselves or being penguins on my lap) would be empty. What objects would such a set contain?

Relating back to your other question, given that (2) is satisfied by your structure, you could define the empty set as the set of all members of your domain that are not equal to themselves:

$$\emptyset =_{\mbox{def}} \{x : x \not= x\}$$.

 

[honestrosewater]

Was this adopted in axiomatic set theory after such problems as Russell's Paradox? Was naive set theory approach such that {{}} = {}?

What does that equation mean, exactly?

Russell's Paradox showed that you can't just make up any property whatsoever and turn it into a set (or that not every property defines a set, or that the extension of every property is not a set, or however you want to think of it). You have to put some restrictions on what kinds of things are allowed to be sets. The axioms from which the antinomy is derived are too loose, too inclusive -- they allow you to derive both some formula and its negation ($(x \in y)$ and $\neg(x \in y)$), so you need to change them to rule out one or both of those formulas. I think that's the basic idea, at least.

 

[neurocomp2003]

honestrosewater: so then the empty set arises because one assumes that there exists objects and these objects are not members of this set [\phi]. Is that correct?

 

[matt grime]

The empty set 'arises' because it is the set of all real roots of x^2+1=0, because it is the intersection of the set of even integers with the set of odd integers, because we find it so much easier to have an empty set so we can talk about aribtrary interesections and sets that possibly might have no element satisfying its defining properties. To do without it would be like trying to do addition without 0, or multiplication without 1 (and that is more than just an unfounded analogy: the empty set is the identity element for the operation of symmetric difference).

 

[honestrosewater]

honestrosewater: so then the empty set arises because one assumes that there exists objects and these objects are not members of this set [\phi]. Is that correct?

No, not really. I meant phi to be a formula, a well-formed string of the language in which your theory is expressed, but I don't think explaining that is going to help anyway. The empty set can arise in many ways. The most straightforward is to just say that it exists.

(1) There exists a set that contains no members.

Why does (1) bother you while

(2) There exists a set that contains one member.

doesn't? People have already said everything else I can think of to say. Can you try to explain why it bothers you? That might help us figure out where the problem is.

 

[loseyourname]

Russell's Paradox showed that you can't just make up any property whatsoever and turn it into a set (or that not every property defines a set, or that the extension of every property is not a set, or however you want to think of it). You have to put some restrictions on what kinds of things are allowed to be sets. The axioms from which the antinomy is derived are too loose, too inclusive -- they allow you to derive both some formula and its negation ($(x \in y)$ and $\neg(x \in y)$), so you need to change them to rule out one or both of those formulas. I think that's the basic idea, at least.

Russell's paradox is the most famous of the logical or set-theoretical paradoxes. The paradox arises within naive set theory by considering the set of all sets that are not members of themselves. Such a set appears to be a member of itself if and only if it is not a member of itself, hence the paradox.

Some sets, such as the set of all teacups, are not members of themselves. Other sets, such as the set of all non-teacups, are members of themselves. Call the set of all sets that are not members of themselves "R." If R is a member of itself, then by definition it must not be a member of itself. Similarly, if R is not a member of itself, then by definition it must be a member of itself.

...

The significance of Russell's paradox can be seen once it is realized that, using classical logic, all sentences follow from a contradiction. For example, assuming both P and ~P, any arbitrary proposition, Q, can be proved as follows: from P we obtain P or Q by the rule of Addition; then from P or Q and ~P we obtain Q by the rule of Disjunctive Syllogism. Because of this, and because set theory underlies all branches of mathematics, many people began to worry that, if set theory was inconsistent, no mathematical proof could be trusted completely.

Russell's paradox ultimately stems from the idea that any coherent condition may be used to determine a set. As a result, most attempts at resolving the paradox have concentrated on various ways of restricting the principles governing set existence found within naive set theory, particularly the so-called Comprehension (or Abstraction) axiom. This axiom in effect states that any propositional function, P(x), containing x as a free variable can be used to determine a set. In other words, corresponding to every propositional function, P(x), there will exist a set whose members are exactly those things, x, that have property P. It is now generally, although not universally, agreed that such an axiom must either be abandoned or modified.

link

Don't you just love the Stanford Encyclopedia? I was getting ready to pull out my copy of The Principles of Mathematics, then I realized I didn't even have to. Notably, in that volume, Russell does define the empty set as the set of all objects that are not equal to themselves, which I believe is one of the definitions Matt gave above, the set of all x's such that x does not equal x.

 

[matt grime]

That wasn't one of the definitions I gave, at least not knowingly. I believe the best definition to give is that the empty set is the (unique) set X for which the statement x in X is always false. In particular this is the kind of thing we have to bear in mind when proving "x in X implies something": this is always true if X is the empty set.

 

[neurocomp2003]

honestrosewater: I think the real reason why it bothers me is because the thread is about counting/numbers and the way i view counting i guess comes from a psychology standpoint...in that as humans we like to label/quantify things(particularly as objects). Thus the #1(singleton/entity/identity) IMO should exist before #0(empty set/null) in set theory. With 1+1=2(add 1/succ op.)...1-1=0(remove 1,pred op.) You have created 2 and 0. However 0+1=1(but how do you add one if 0 exists first and one does.).

Then Someone brought up something from penrose's book which defined a number by the empty set rather than a singleton. And this confused me.
Since i thought the singleton would have came first(but i guess its because of the axioms, though i recall having the thereexists x=x taught in my set theory class)

Lastly I thought the foundations of mathematicsc would have come from
{ 1, {x}-singleton, thereexists,=,succ(), pred(), Union,Intersection}. But clearly I am wrong.
But for counting {1, succ() operator,pred() operator}

 

[loseyourname]

That wasn't one of the definitions I gave, at least not knowingly.

Sorry, it was rosewater.

I believe the best definition to give is that the empty set is the (unique) set X for which the statement x in X is always false. In particular this is the kind of thing we have to bear in mind when proving "x in X implies something": this is always true if X is the empty set.

Is there a better way to state that? Is that another way of saying the statement "x is a member of X" is always false? The way you've written it now, it just sounds like the set of all false statements.

 

[loseyourname]

honestrosewater: I think the real reason why it bothers me is because the thread is about counting/numbers and the way i view counting i guess comes from a psychology standpoint...in that as humans we like to label/quantify things(particularly as objects). Thus the #1(singleton/entity/identity) IMO should exist before #0(empty set/null) in set theory. With 1+1=2(add 1/succ op.)...1-1=0(remove 1,pred op.) You have created 2 and 0. However 0+1=1(but how do you add one if 0 exists first and one does.).

There are two ways to frame the question we are asking here. The way that has been taken in this thread is of defining numbers the way they are defined in modern day mathematical logic. We can always frame the question historically, however, to ask how the word "number" ever came to be and what it meant. If I had to guess, I would say the word's extension originally included only small counting numbers which themselves had an empirical reference to sets of physical objects or to the passage of time according to whatever measure was used (days, moons, years). The problem of numbers that could not possibly have an empirical reference did not come about until those numbers were invented. I believe there are even languages today that have only one word (akin to "indefinitely many") to refer to amounts greater than the first few counting numbers in English.

 

[matt grime]

Is there a better way to state that? Is that another way of saying the statement "x is a member of X" is always false? The way you've written it now, it just sounds like the set of all false statements.

I don't believe what I wrote does sound like that, I don't see how one gets that I said the empty set is the set of all false statements.

All I did was say in words not symbols that the empty set is the (unique) set X such that $\forall x, x \notin X$ (i.e. there is no x that is an element of X).

 

[loseyourname]

I think you meant to write 'x is in X' is always a false statement. You just left out the "is." No problem. The symbolic version makes it pretty obvious what you're saying.

 

[matt grime]

No, I meant to write x in X, just like I would write 'for x in $\mathbb{Z}$'. Better perhaps would have been to put it in brackets thus: (x in X). It is perfectly reasonable literal writing of the symbolic version you prefer, in my opinion. I could have written it in pseudo-tex as x \in X, or even itexed it here.

 

[0rthodontist]

Church numerals deserve a mention. http://en.wikipedia.org/wiki/Church_numeral

 

[loseyourname]

No, I meant to write x in X, just like I would write 'for x in $\mathbb{Z}$'. Better perhaps would have been to put it in brackets thus: (x in X). It is perfectly reasonable literal writing of the symbolic version you prefer, in my opinion. I could have written it in pseudo-tex as x \in X, or even itexed it here.

Yes, putting it in brackets does clear it up. The problem before was that it seemed "the statement 'x' in X is always false" was also a possible interpretation.

Nitpicky, I know. Frankly, I think the term "empty set" is pretty self-explanatory with or without a definition.

 

[neurocomp2003]

loseyourname: ""empty set" is pretty self-explanatory" would you need to define an object or symbol and {} before defining the empty set?

 

[matt grime]

Yes, putting it in brackets does clear it up. The problem before was that it seemed "the statement 'x' in X is always false" was also a possible interpretation.

That doesn't make sense as an interpretation, at least it does not correspond to your initial interpretaton. It would imply that the set X contained exactly one statement, x, (the statement 'x'), whereas you said I called it the set of all false statements.

 

[matt grime]

loseyourname: ""empty set" is pretty self-explanatory" would you need to define an object or symbol and {} before defining the empty set?

no. the empty set is a set X for which (x in X) is false for all x. beyond defining the concept of 'in' or 'for all' there is no need do define any symbols like {} or what an object is. Set theories do not say what objects are, or even say what a set 'is'.

 

[neurocomp2003]

but how do you use the terms false and x,X to define it?

 

[loseyourname]

loseyourname: ""empty set" is pretty self-explanatory" would you need to define an object or symbol and {} before defining the empty set?

The difficulty you're having here points to a larger problem with definition in general. Words must always be defined using other words, which forms an intricate web that is circular in a certain sense in that it has no real foundational elements. Mathematicians avoid this difficulty by taking certain primitive notions to be undefinable and defining all other symbols using these undefinables. 'x' and 'X' are defined as variables denoting an individual and a set, respectively, but individuals and sets are not themselves defined.

Note that "undefined" does not mean "has no meaning." A definition is simply the use of one string of symbols to explain another by employing the fact that both strings have equivalent meanings. "Meaning" itself is intensional and has no symbolic representation.

but how do you use the terms false and x,X to define it?

Well, 'x' and 'X' are defined, as above. 'x' is any individual and 'X' is the set of all x's. In the case of an empty set, there are no x's. A way to define this in plain English is that 'empty set' refers to the class of all objects denoted by a term with no extension. For instance, the class of all married bachelors is the empty set. It has no existing members. The proposition 'x is a married bachelor' is false for all x. In so doing, we can derive the notion of an empty set from predicate logic. A false proposition is any proposition that implies all possible propositions. That is, 'x is false,' means that 'x->y' is true for all y.

Of course, this appears circular, which is what I was talking about above, and is a great illustration of why a certain number of notions must be taken as undefinable. Otherwise, we can play this game forever, asking for the definition of every term we use to give a certain definition, until we arrive back at the terms we set out to define in the first place.

 

[honestrosewater]

but how do you use the terms false and x,X to define it?

Did you see the earlier mentions of theories, structures, and models? I think you'd be interested in this, so we can run through what these things are and how they are related. I'm still piecing some of these things together myself, but I'll try not to stray too far into those areas.

Every formal theory is connected to a language. (By some definitions, a formal theory itself is a language, but we'll stick with the more inclusive usage.) We can speak just as generally about L-theories as we can about formal theories by letting L be an arbitrary formal language. An L-theory is a set of formulas of L. We build L from a set of symbols. The symbols are undefined primitives, whose role loseyourname already described. We form strings out of our symbols by simply stringing some number of symbols together in some order. (Note that, to keep L arbitrary, we'll let strings be of any length, finite or infinite, but strings are usually restricted to finite lengths.) Formulas are special strings that have some form, some pattern or orderliness, that we want to take advantage of. They are the well-formed strings, strings that meet some well-formedness conditions.

What makes an L-theory special as a set of formulas is that it is closed under a special relation. This relation can be any of several related ones, depending on the details of the treatment, and they go by several names: implication, entailment, consequence, deducibility, syntactic entailment, semantic entailment, formal consequence, logical consequence, and so on. What these relations have in common is the idea of one formula following from other formulas. One formula might follow from the others because you can prove it from them or because its truth-value is related in a special way to their truth-values or for some other similar reason.

So an L-theory is a set of formulas of L such that if a formula f follows from any set of formulas already in the theory, then f gets put into the theory as well. And it all starts with just a set of symbols:

Set of symbols --> set of strings --> set of formulas --> set of formulas closed under entailment relation.

Is this a comfortable foundation for you? Meaning and truth haven't entered the picture yet. L-theories technically involve only meaningless symbols being manipulated mechanically according to formal rules. It's all just a bunch of operations and relations on a set of symbols, which we are describing and studying from up above in our metalanguage (English or whathaveyou) and our metatheory (first-order logic and set theory or whathaveyou).

I'm not sure how to describe the situation of, say, using set theory as both your metatheory and object theory. You could perhaps recast it as languages talking about themselves. Natural languages are rich enough to talk about themselves, to incorporate their own metalanguage, and you could perhaps look at formal languages as being a kind of refinement within the natural language. Maybe your object language is in fact a model of your metalanguage description of it. It seems a lot of what you're doing is just narrowing down the possible interpretations, making your language more precise, and I guess you might be building some new things as well, but derp, this is an area where I didn't want to stray. So anywho, back to safer ground...

A formal language L that you want to use for set theory, or, rather, an L-theory of sets, will contain some special symbols, variable symbols, which is what x and X are functioning as in the example formula that you're asking about.

A structure is the thing that let's you interpret the formulas of an L-theory and assign truth-values to them. It let's you give the formulas meaning. For example, suppose you have an L-theory of equivalence relations, where L is a first-order language with one nonlogical binary predicate symbol, denoted by P. One axiomatization of this L-theory could be

(A1) $$\forall x \ [Pxx]$$
(A2) $$\forall x, y \ [Pxy \ \rightarrow \ Pyx]$$
(A3) $$\forall x, y, z \ [(Pxy \ \wedge \ Pyz) \ \rightarrow \ Pxz]$$

(An axiomatization of an L-theory consists of your rules of inference and logical theorems, which you'll recall are usually left implied in the background, together with a set of formulas from which all other formulas in that L-theory follow. Also, I'm hoping that you recognize those axioms as saying that P is reflexive, symmetric, and transitive.) A set A with the identity relation R = {(x, x) : x in A} defined on it would be a structure that would let you interpret your L-theory of equivalence relations. It let's you interpret your theory because it has a binary relation, R, to match up with your binary predicate symbol, P.

The situation with structures is similar to the one with theories in that we connect structures with a language in order to use them. An L-structure is a structure that can be used to interpret all of the symbols of a language L.

The interpretation and truth-value assignment are done with functions, but you can use different definitions depending on your purposes, and the form will depend on the form of the language that you're interpreting. The most general form of an L-structure that I can think of is an ordered pair (A, I), where A is your underlying set, or domain, which contains the individuals of your structure, and I is the set of functions that use A to interpret the symbols of your L-theory and assign truth-values to your formulas. The variations on this (A, I) pair would split I up into different functions or sets of functions. For example, you might separate out the truth-assigning functions (commonly called an L-valuation) or, if L has constant symbols, you could specify that some function maps your constant symbols to individuals in your domain. For simplicity, we'll keep everything together under the umbrella of an L-structure.

If an L-structure assigns a truth-value of true, or whatever value we have chosen to correspond to truth, to a formula, we say that the structure models that formula or is a model of that formula. Similarly, if a structure interprets every formula in a set of formulas to be true, we say that it models that set of formulas. Recall that a theory is a set of formulas. So, for example, a model of an L-theory of sets is an L-structure that interprets every formula in that L-theory as being true. If we turn our earlier example structure of a set with the identity relation into a suitable L-structure, it is a model of our L-theory of equivalence relations because the identity relation does indeed satisfy the equivalence relation axioms, and due to the properties of and relations among the entailment relations of first-order logic and the axiomatization of our L-theory, any structure that is a model of our axioms is also a model of our entire theory; if it makes the axioms of our theory true, it must make all of the other formulas of our theory true as well.

So if I finally try to answer your question by saying that the empty set is any set that satisfies the Axiom of the Empty Set, I'm saying two main things:

(i) I have an L-theory that contains a formula that I've called 'the Axiom of the Empty Set'. I just wrote down a formula in my formal language L. The axiom has no meaning. It is simply, say, the following string of symbols:

(AES) $$\exists X \forall x \ [\neg(x \in X)]$$,

(ii) I have an L-structure that is a model of that L-theory, and the domain of my structure contains an individual that, when assigned to the X symbol in (AES), makes (AES) work out to be true.

For a simplified example, which isn't a model of every axiom of set theory, we could let our structure's domain A = {a, b, c} and assign to our binary membership symbol, $\in$, the binary relation R = {(a, b), (b, c), (c, c)}. We interpret the string $x \in y$ to mean that the ordered pair (x, y) is in R. An empty set is then any member of A that doesn't show up as the second argument in any pair in R. The empty set of our example structure is a.

Does that answer make sense?

By the bye, (AES) is just another way of saying what matt already said -- we wanted to state it that way to fit in with our setup. Also, I didn't want to distract you by mentioning this eariler, but there is also an empty string, whose length is 0. It is a nice thing to have. For example, it is the identity element for the binary string concatenation operation.