What is the role of the empty set in defining the number 1 in mathematics?

[NeutronStar]

Hurkyl wrote:
We are "supposed" to think of a set as a container.

This is absolutely incorrect. If I were permitted to think of as set as a container there would be no conceptual problem. However there then would be tons of problems with the formalism.

Generically speaking:

A set is a collection of things

Number is a property of a set

If what you say is true then why don't mathematics books just say so:

A set is a container

Number is a collective property of the contents of the container

You won't find that in any mathematics book in the world. Why? Because it is a totally incorrect picture of the underlying formalism. So if you are intuitively thinking of Cantor's set theory in this way then you have totally ignored the details of the formalism.

Mathematics books cannot teach it this way because to do so would be incorrect.

In fact, if the modern mathematical community has accepted this way of thinking of set theory then they have pulled a sly trick on the scientific community and I would love to start my book off with that story in the very first chapter! It would be a very scandalous story historically speaking.

 

[NeutronStar]

Joy Division wrote:
You can make arguments from a conceptual stance but it doesn't prove anything. I've seen many conceptual arguments from a physics standpoint. They often make sense and bring a certain understanding. Most don't fall into the trap of thinking that they are actually rigorous proofs however.

Well this could be the ultimate answer right here.

Pure mathematicians are out to prove things.

Physicists are more interested in understanding things.

I am definitely a physicist and not a pure mathematician. I am more interested in building formalisms that explain things in a comprehensible way than I am in proving that they must be true based on some superfluous axioms that I pulled out of a hat earlier.

I actually came to this discussion board because I am interested in talking about the physical nature of the universe (including its property of quantity and the correct communication of that idea).

But I supposed that because of the fact that physicists have leaned heavily on mathematical formalism to convey their conceptual ideas this has made physics attractive to pure mathematicians.

They day when pure mathematicians take over physics will be the day that science dies.

Mathematics is definitely not a science. It does not use the scientific method and therefore has no right calling itself a science. And the day that physics becomes a purely axiomatic formalism will be the day that it can no longer be called a science.

It is actually quite disturbing to me that physics is becoming more and more reliant on abstract axiomatic mathematics. Especially when I know that mathematical formalism has incorrectly modeled the idea of quantity as the universe exhibits it.

I actually came here in the hope of pointing this out to my fellow physicists. Instead I seem to have run into some pure mathematicians who are hell bent on defending their axiomatic formalism to their deaths.

I didn't come here to prove anything. I came here to offer my insight into the problems associated with these various formalisms and sciences.

I seem to be beating dead horses (or lively pure mathematicians) in this thread so perhaps it's time to call it quits.

It really isn't my intention to argue my case to those who are not interested. I was merely attempting to share my ideas and explain why I believe that they are valid.

I'm not out to prove anything to anyone.

So believe what you will.

 

[Hurkyl]

If what you say is true then why don't mathematics books just say so:

A set is a container

Number is a collective property of the contents of the container

Because those statements are useless, from a practical point of view. If I want to prove something about sets, telling me "A set is a container" is perfectly useless unless I already know how to prove things about containers.

The axioms of a theory cannot tell us what the objects are; they can only tell us how to use the objects. Thus the axioms tell us things like (to paraphrase a few ZF axioms):

(format is name of axiom, formal statement, "english" translation)

Axiom of extensionality:

For any A and B:
A = B if and only if (for all x: x is in A iff x is in B)

Two sets are the equal iff they contain precisely the same elements

Axiom of the unordered pair:

For any A and B:
There exists an x such that for all y: (y is in x iff (y = A or y = B))

For any (possibly equal) A and B, {A, B} is a set.

Axiom of the sum set:

For any A:
There exists an x such that for all y: (y is in x iff (there is a z in A such that y is in z))

The union of all of the sets in a set is a set.

Axiom of the power set:

For all x there is a y such that for all z: (z is in y iff z is a subset of x)

P(x) (the power set of x) is the set of all subsets of x

Axiom of the empty set:

There exists an x such that for all y: (y is not in x)

There is an empty set

Axiom of subsets:

For all A and propositions P:
There exists an x such that for all y: (y is in x iff y is in A and P(y))

{y in A | P(y)} is a set


Incidentally, note that if you have the existence of at least one set, the axiom of subsets can prove the existence of the empty set. The only reason the axiom of the empty set exists is so that you can prove there exists at least one set.


Oh, and the fatal flaw in Cantor's set theory lies in the axiom of subsets; Cantor's set theory implicitly used the axiom:

For any proposition P:
There exists an x such that for all y: (y is in x iff P(y))

{y | P(y) }

which allowed one then to construct terrible things like the set of sets that don't contain themselves.


Anyways, all of those sound like things I can do to collections. (A collection is simply a container whose boundaries are conceptual, not physical!)

 

[drnihili]

It sounds to me as if you want to consider objects plurally rather than considering their collection as a unity. If that's more or less right, you might have a look at some of the work on the logic of plurals. I don't know all the work in that field, but you might try Byeong-Uk Yi's Understanding the Many as a start. I know that he argues that plurality provides a better basis for number theory than do sets.

 

[NeutronStar]

Hurkyl wrote:
to paraphrase a few ZF axioms.

So those are the ZF axioms! Hmmmmm. I just took a refresher course this spring called Conjecture & Proof. In that course we were given all those axioms. However, neither the professor nor the book mentioned anywhere that they were the ZF axioms. They were simply presented as definitions. Having taken the course year ago I also don't remember them being called the ZF axioms. It could be that it was mentioned somewhere in passing and I just didn't catch it.

It is a shame that educational institutions can't do better job at creating a bit of consistency when teaching mathematics!

Well if the word axiom and definition are interchaneable then my formalism certianly contains axioms. They just aren't self-contained axioms. The rely on concepts external to the formalism.

In any case, here's how I view the difference between a mathematician and a physicist.


********************

A mathematician runs down the street screaming, "I proved it! I proved it!"

A layman stops him and asks him what it is that he proved.

The mathematician says, "I have no idea, I can't translate it into English!"

The layman asks, "Then how do you know that you proved it?"

The mathematician replies, "I have axioms!"

The layman frantically runs off in the fear that he'll also contract the disease.


*********************

A physicist runs down the street screaming, "I figured it out! I figured it out!"

A layman stops him and asks him what it is that he figured out.

The physicist tells him, "The faster I run, the slower my watch will go!"

The layman says, "I don’t believe you."

The physicist replies, "It doesn't matter!"

The layman then scratches his head as he watches the physicist run off.

**********************

 

[Hurkyl]

So those are the ZF axioms!

Some of them; there are 9 in all. (10 if you consider ZFC)

Well if the word axiom and definition are interchaneable then my formalism certianly contains axioms.

They are indeed very similar, but not quite the same. An axiom is a statement you can use in a proof (converse not true), a definition is notational shorthand.

For example {a, b} is defined to be the set that contains a and b, but to prove such a set exists we need to appeal to an axiom.

Definitions can be replaced with axioms though; one could add as axioms to ZF that {a, b} is a set and that {a, b} contains a and that {a, b} contains b. The reverse is not true in general.

 

[NeutronStar]

Hurkyl wrote:
They are indeed very similar, but not quite the same. An axiom is a statement you can use in a proof (converse not true), a definition is notational shorthand.

I do agree that they aren't the same. But it does seem to be open to interpretation.

I mean, I see adefinition as an explanation that defines a concept. How well a definition works can be tested by how well it defines a concept in a comprehensible way. For this reason different people may like or dislike a particular definition. Or arguments can be made as to the logical validity or completeness of a particular definition.

An axiom, on the other hand, is a statement of truth within a formalism. An axiom can simply state that something is true without any need to actually define it or prove it in a comprehensible way.

Therefore almost any definition can be used as an axiom in a formalism. If people don't like the definition that's too bad. If they want to accept the formalism then they must accept that particular definition as an axiom of truth within that formalism.

One problem that I have with axioms is that they aren't even really required to define a concept at all. They can merely state conditions that must be met without actually givein a comprehensible explanation of what that means. (In other words, without really defining it)

This is how I see the differences between a definition and an axiom.

However, having said that, any formalism that claims to be logical must only contain axioms that do not logical contradict each other. And if we are attempting to comprehend a formalism this begs that axioms be understood as definitions. Otherwise how can we know whether or not they contradict each other.

So there is a catch-22 associated with the differences between an axiom and a definition.

I have no problems with axioms that make sense.

 

[NeutronStar]

By the way, I think I should mention here that my real concern is with the way that Cantor used his idea of an empty set to define the number 1. This is what Cantor originally set out to do. Once we accept his set containing an empty set as the definition of 1 I really don't have any problem with the following ZF axioms.

Also, I would like to point out that from an engineering or applied point of view it really doesn't make much difference. I mean, airplanes fly (most of the time), bridges don't fall down (too often), and applied mathematics works pretty good from an engineering view point. This is because most engineering problems don't rely on lofty abstract concepts of infinities within infinities, etc.

The only place where Cantor's incorrect definition for the number 1 is going to show up is when we start formulating extremely abstract mathematical concepts to describe the nature of the universe. As in the cutting edges of QM, GR, and notions like String Theory. This is where the consequences of Cantor's empty set definition for the number 1 is going to really start showing up.

And since physics is heading off into the realm of this purely abstract mathematics I feel that it might be interesting to take note that our formal mathematical definition for the number 1 may have subtle differences in comparison with the actual property of the universe that we call quanta.

As I've stated several times. Changing (or correcting as I like to say) the definition of 1 as the foundation of the formalism does indeed have consequences that reach to the penthouse of pure mathematical abstraction. Precisely what all of these consequences will be I cannot say as I am not well-versed enough in those penthouse concepts. But I can see where it would force mathematicians to make a distinction between the idea of a quantitative cardinal property of a set, and other qualitative properties. By forcing them to make this distinction this would necessarily affect they way that the formulate future lofty axioms. Change the axioms, you change what they can prove.

As it is now, they just simply accept that one infinite set is cardinally larger than other. But that is an erroneous concept caused by the ill-defined number 1.

That's my story, and um stick'in to it!

 

[Hurkyl]

One problem that I have with axioms is that they aren't even really required to define a concept at all. They can merely state conditions that must be met without actually givein a comprehensible explanation of what that means. (In other words, without really defining it)

I don't think it's a problem. Concepts only really have any meaning in their relationship with itself and other concepts, and that's what the axioms specify.



One reason that physics has gone off in the direction of abstract mathematics is because the techniques physicists need have not been invented yet.

Another is that it is an efficient way of searching for a better theory of the universe. The abstractions created by mathematical physicists help them strike out vast classes of potential theories that can't describe reality, and helps them discover how to bend theories so that they still look like the theories we have now that work on the scales where they do work.

 

[HallsofIvy]

neutronstar wrote:
By the way, I think I should mention here that my real concern is with the way that Cantor used his idea of an empty set to define the number 1. This is what Cantor originally set out to do. Once we accept his set containing an empty set as the definition of 1 I really don't have any problem with the following ZF axioms.

Why in the world would you have a problem with that? Cantor effectively defined "the number 1" as a set containing exactly 1 thing. Of course, he couldn't phrase it that way since he hadn't yet defined "1"! So he used the one thing he had already at hand: the empty set.