Why set is taken as undefined primitive?

[AdrianZ]

the title says everything. why they don't define set? is it possible to do so? if not, why?

 

[HallsofIvy]

Yes, you certainly can define "set". But then, of of course, at least some of the words that you used in defining "set" would be undefined. It's a question of how far "back" or how "primitive" you want to be.

 

[disregardthat]

There is no particular reason to define a set; set theory only uses the axioms of sets to model mathematics. In this sense you can say that sets are defined, although it is not a definition in the traditional sense.

 

[AdrianZ]

Yes, you certainly can define "set". But then, of of course, at least some of the words that you used in defining "set" would be undefined. It's a question of how far "back" or how "primitive" you want to be.

that's obvious. the question is, why set is undefinable using "mathematical" conceptions? I'm talking about defining "set" by simpler mathematical conceptions, not about giving a literal definition of set.

There is no particular reason to define a set; set theory only uses the axioms of sets to model mathematics. In this sense you can say that sets are defined, although it is not a definition in the traditional sense.

yea, but the question is, why it is so? I mean we know that concepts like point,line,plane,space are undefined primitives in geometry. and we know why. but in the case of sets the reason is not clear for me.

 

[Hurkyl]

yea, but the question is, why it is so? I mean we know that concepts like point,line,plane,space are undefined primitives in geometry. and we know why. but in the case of sets the reason is not clear for me.

What reason do you think that is? And why do you think it doesn't apply to set theory?

 

[disregardthat]

yea, but the question is, why it is so? I mean we know that concepts like point,line,plane,space are undefined primitives in geometry. and we know why. but in the case of sets the reason is not clear for me.

How do we know why points, lines and planes are undefined primitives? It is actually as HallsOfIvy says just a matter of how far back we wish to go in order to get a formal setting in which we can do mathematics. It would though be helpful to note that set theory is not mathematics defined (or "defined" as you please), but rather a formal model of much of mathematics.

 

[jambaugh]

A good reason to start with sets is that we must apply logic so deductive logic needs to be prior to any constructions and definitions. Set theory is an objectified form of symbolic logic (nearly every symbol has a 1 to 1 correspondent in logic
"and"<->intersection,
"xor"<--> set difference
"False" <--> empty set
"implies" <--> subset

So it is hard to get more primitive without questioning the very logic you use in the proofs and definitions you will use. And as we see we don't need to be less primitive since set theory is sufficient to construct the other useful mathematical disciplines.

And as already mentioned, in making formal definitions you must start with some undefined terms, i.e. you got to start somewhere.

 

[g_edgar]

There are other possibilities. Von Neumann proposed a scheme where "function" is the undefined concept, and a "set" is a special kind of function: namely a function with values 0 and 1 only.

 

[Fredrik]

It should be clear that something has to be left undefined. You could e.g. take the integers, the rational numbers, the real numbers, etc. to all be undefined, and leave the term "function" undefined as well. The thing is, if we do, we're leaving more things undefined than we have to. Why not try to leave a minimum number of terms undefined? The real numbers can be constructed from the rationals. The rationals can be constructed from the integers. The integers can be constructed from the natural numbers. This eliminates the need to leave all of the number systems undefined, but we can take this even further by choosing "set" and "is a member of" to be the terms we leave undefined, because functions and the natural numbers can both be constructed from sets. Informally, the construction of the natural numbers can be written like this:

0={} (the empty set)
1={0} (the set that contains 0 and nothing else)
2={0,1}
3={0,1,2}
.
.
.

and a "function from X into Y" can be defined as a subset f of X×Y such that

(i) For every x in X, there's a y in Y such that (x,y) is in f.
(ii) If (x,y)=(x,z) are both in f, then y=z.

A set theory is defined by a list of axioms that tell us what sets we're allowed to construct. (If they don't allow the construction of Cartesian products and the natural numbers, what I said above isn't true in that set theory, and it would be pretty much useless. If they allow us to construct any sets we might want to, the theory is inconsistent and therefore completely useless). The most popular set theory is known as ZFC, after Zermelo, Fraenkel and the axiom of Choice. It includes all the mathematics of all the established theories of physics, all the standard number systems, and a lot more. There is however no set theory (or any other kind of formal theory) that can be said to include all of mathematics, because there are always questions you might want to ask that the theory you have chosen is unable to answer. For example, ZFC is unable to answer the question of whether there's a set that's strictly larger than the set of natural numbers and strictly smaller than the set of real numbers. The continuum hypothesis is the assertion that there's no such set.