What is a Number? - Math Philosophers' Views

[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.