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

[Joy Division]

Well I've read some excerpts from Cantor's original papers. It's very hard to see what he means most of the time. His style is really that of the time and mathematics has made leaps an bounds in clearing up his original ideas and putting them into a more clear concise form.

He laid the ground work for modern set theory. He's it's pioneer. But to say that we base all of mathematics on his works directly, is well, wrong. At least your little conditional statement is vaccuously satisfied.

Cantor didn't set out to build a foundation of mathematics what he did was try to put rigor to the notion of what a "set" really is.

With his work and that of others mathematicians were able to axiomatize the foundations of mathematics. It was what everyone wanted to do at the time and his set theory worked.

I don't think many people are going to buy your whole math=numbers argument. Abstraction is one of the most powerful tools mathematics has. I know in physics we really like to wonder what things really represent and how the math really connects with the real world. In mathematics however, generalisation and abstraction lead to connections between things that would be difficult if studying them at face value.

 

[NeutronStar]

Hurkyl wrote:
The "feeble quantitative idea" of a set containing the empty set is a model, not a definition.

I find this rather humorous because mathematicians usually try to convince me of just the opposite. They usually tell me that it's just a definition, and it's not intended to be a model.

Hurkyl wrote:
A typical formal definition of 1 is "the identity element of a multiplicative group".

This is totally absurd. I realize that it is taught this way, but that doesn't give it any merit.

In reality they are doing just the opposite by defining the identity of a multiplicative group using the number one. To claim that it is the other way around is totally absurd. They didn't have a choice! Would it really make any sense to use any other number? No. Why not? Because 1 is already defined as the cardinal property of a set with a particular property (i.e. the set containing the empty set).

Multiplication is ultimately just a formal shorthand for addition. And addition is most certainly defined on the cardinal property of sets. To perform 1 operation of addition has meaning based on the cardinal meaning of 1. And this is really where the multiplicative identity comes from. To pretend that we re-invented the concept of 1 by defining it as the identity element of multiplication totally absurd. It has precisely the same quantitative meaning. Why define it twice? It simply isn't necessary. Much better to stick with one definition and use that to define multiplication, which is really what they actually did!

There is simply no need for all of these superfluous definitions. We can build a single formalism based on a single concept of number and just stick with it the whole way through. Why make things unnecessarily complicated?

Hurkyl wrote:
"Endless" sets do have cardinality. Some cardinal numbers are finite, some are infinite. Some infinite cardinal numbers are bigger than others.

Yes I know that current mathematical formalism recognizes these concepts. However, I now understand that all of these concepts stem from our misunderstanding of infinity. And our misunderstanding of infinity stems from the fact that the Natural Numbers are defined on the empty set.

Hurkyl wrote:
Actually, off the cuff, I can't remember anywhere in set theory that the term "infinity" appears; just the adjective "infinite", whose meaning is just what you describe in your formalism.

Well, if the meaning of an infinite cardinal number is that it represents a set that contains an endless quantity of elements, then how can one infinite cardinal number be bigger than another one? There must be something more to the idea of the infinite than simply the idea of being endless.

How can one infinite cardinal number being bigger than another infinite cardinal number not imply that one set's elements are more endless than another set's elements?

Hurkyl wrote:
Which step doesn't follow?

I'll have to get back to you on that power set thing. I'm too tired to think about that right now.

 

[NeutronStar]

Joy Division wrote:

Abstraction is one of the most powerful tools mathematics has.

I hear this all the time when I voice my concerns about set theory. The formalism that I am proposing is not in any way any less abstract that current mathematics. It is simply more correct.

It is not necessary to be incorrect just to facilitate abstraction.

By the way, you never did answer the orginal question of the thread. What do you think makes mathematics?

What should be the criteria for something to be called a mathematical idea, and not just merely a logical idea?

I hold that the idea of number should be the criteria. If it's not about number then it's not mathematical in my opinion.

And of course, I believe that there should only be one definition and meaning for the idea of number. I hold that it would be an ambiguous idea otherwise.

 

[Joy Division]

Originally posted by NeutronStar
I hear this all the time when I voice my concerns about set theory. The formalism that I am proposing is not in any way any less abstract that current mathematics. It is simply more correct.

You probably hear this because naivly equating mathematics to numbers guts abstraction from mathematics. If everything mathematical is just a way to count then you could be a graduate "mathematician" by watching Sesame Street. I believe the field deserves a bit more respect than that.

And of course, I believe that there should only be one definition and meaning for the idea of number. I hold that it would be an ambiguous idea otherwise.

You'll have trouble with this one then. Every time something has an alternate definition which can be proven equivalent the it is no longer well defined in your "mathematics"? You'll have frightfully few concepts left to work with. I suppose you won't care too much anyways since you're only dealing with natural numbers.

I suppose if you wanted to be rigorous about what would be part of mathematics you could say that, "Anything that can be derived from set theory by the rules of logic and axiomatic structure could fall under the heading of mathematics." Or we could adopt the slightly more encompassing, "Anything which is an axiomatic system or the logical properties derived from those axioms is mathematical."

Really I don't care but there are definitely things that are abstract enough to not be numbers or quantities but still be mathematical.

Should logic fall under the umbrella of mathematics as you put it? Maybe. Symbolic logic certainly can and should. Could logic fall under philosophy? It could. Does the destinction mean anything? Not really.

 

[NeutronStar]

Hurkyl wrote:
Proof that for any set S, |S| < |P(S)|: (the cardinality of any set is less than the cardinality of its power set)

Yes, I agree that the proof you gave is perfectly correct in current mathematical formalism.

But this proof and conclusion would not be applicable in my formalism for infinite sets. For finite sets it would be virtually identical in my formalism.

I keep calling this my formalism for lack of a better title. My formalism is actually based a set theory, and retains almost identical operations on sets. The only difference is that my formalism is based on an idea of some thing while current mathematical formalism is based on an idea of nothing (the empty set).

While much of the formalism remains identical there are some significant differences. The most obvious one is the way with which my formalism handles the idea of infinity.

I do not view infinity as a cardinal property of a set. To do so would be to recognize infinity as a number.

In both current mathematics, and in my formalism the generic idea for the concept of number is the idea of the cardinality of a set. Therefore, in the most rigorous logical fashion, if infinity is considered to be a cardinal property of a set, then it must be considered to be a number.

In my formalism infinity is not a number. On the contrary it represents the idea of a collection of things who's cardinality cannot be finitely determined. In my formalism infinity is a purely qualitative idea rather than a quantitative idea.

Like On Radioactive Waves suggested at the onset of this thread:

On Radioactive Waves wrote:
It seems to me the entire basis of mathematics is the number 1, and its relation to itself.

This is really the foundation of my formalism.

When a set cannot be quantitatively described based on the quantitative idea of 1 and its relationship to itself, then such a set is said to be endless, or infinite.

Remember! In my formalism there is no such thing as a empty set. Zero is the concept of the absences of a any thing, and therefore the absence of a set. So Zero and infinity are totally different concepts. One represents an idea of the absence of quantity, the other represents a collection of things who's cardinality is endless and therefore cannot be finitely determined.

So the quantitative idea of a power set in my formalism would necessarily be restricted to infinite sets.

Having said that we can still use the idea of a power set to say that different infinite sets have different qualities because they necessarily must contain elements that can't be matched up via a functional approach. However, it is clear that these qualitative differences are not an idea of quantity based on our definition of that concept. Instead they are purely qualitative differences.

From my point of view we haven't lost anything. Instead we have gained a better understanding of what we are attempting to communicated.

So anyhow, that's how I would handle the Power set when applied to infinite sets. To claim that one has a larger cardinality is not only incorrect in my formalism, but it's not even a useful idea!

This whole explanation here should only serve to show yet another flaw in Cantor's idea of cardinality. Once a set becomes infinite it is absurd to talk about it's cardinality in a quantitative manner. We must necessarily move it into a qualitative category. And this is precisely how my formalism deals with the concept of infinity.

But remember again. In my formalism a finite line may only contain an finite number of points!

So there are many other consequences to accepting my formalism.

 

[Joy Division]

Well I hope that you understand that when you keep talking about your formalism while we don't know what it is it's hard to understand exactly what you're on about.

Really from what I can gather you just seem to want to cut all the things you find pathelogical from math and the things that you don't feel are right. Then rewrite a formalism encompassing the things you do like.

Please try to remember when talking about things like infinity and other mathematical concepts which are beyond the ken of human experience, waxing lyrical and navel gazing will get you nowhere.

People often have trouble with the way math descibes certain things and have trouble accepting them. They very often don't notice that these things are very much not part of their experience and should not be expected to obey their ideas.

It's like disregarding quantum mechanics because you don't like it and the world you know doesn't act that way.

I don't mean to tell you that you can't try to rewrite all of mathematics. By all means go ahead. Just don't expect me to accept what you have to say when try marginalize achievement that you haven't shown a very good understanding of.

 

[Hurkyl]

It has sounded to me more like Neutron Star just doesn't like the names of things, at least at the fundamental level; he has no beef with the study of collections and shapes... he just doesn't like the term "mathematics" applied to it. He has no problem with infinite cardinals and one infinite cardinal being bigger than another; he just doesn't think they should be called cardinal numbers. He has no problem with the empty set, it just shouldn't be called a set. He doesn't think the unqualified term "line" should apply to what one calls a line in infinitely divisible spaces; he thinks it should apply to what one calls a line in discrete spaces.

IOW the concepts are fine, but the names are wrong.


Incidentally, why object to the term "cardinality" as it is traditionally used? The term does not exist outside of mathematics, I can't possibly imagine what confusion (real or imagined) could arise from using the term as is.



Oh, and if you're willing to consider other possible formalisms, you could try ZF with the axiom of infinity excluded. Without the ability to say "the set of natural numbers", ZF loses its ability to create infinite cardinals or model things like the real numbers.

You have to be very careful about the distinction between a "class" and a "set" in this theory, but you can describe many of the same ideas. (though you might have to wait until your abstract algebra course to get an idea about how to do this)

Of course, this theory still has the empty set in it.

 

[NeutronStar]

hurkyl wrote:
He has no problem with the empty set, it just shouldn't be called a set.

You are almost right here.

In a pure category theory where a set is almost thought of as a container of a particular class of objects I have no problem with the idea of an empty set. It is indeed just the idea of an empty container. And that is a very valid concept.

But that isn't the way that the idea of set is being used in Cantor's set theory. We can't think of the set as a container. That causes all sorts of problems not the least of which is that number is defined as the property of a set. Well, technically if the set is the container then number is a property of the container, not the collection of things within the container.

So you may say, just think of number as a property of the contents of the set. Well, that won't work either because there is no formal definition for the contents of the set. All we have is the definition of the set or container itself. Not to mention the fact that if we push number through to the container onto the contents then we lose that all-important distinction between an element and a set containing an element.

Trust me, there's some pretty heavy logic going on here. I've been thinking about this for over 40 years and while I may come across as a complete idiot I can assure you that most of that is only an illusion.

There really are some serious logical flaws in Cantor's empty set theory (even if it isn't suppose to properly represent the idea of quantity.

Joy Division wrote:
Really from what I can gather you just seem to want to cut all the things you find pathological from math and the things that you don't feel are right. Then rewrite a formalism encompassing the things you do like.

Well, I totally agree with this if we define logical contradictions to be pathological.

However, I think it would be incorrect to suggest that I am solely looking for things that please me. Unless, or course, we recognize that logical stability pleases me.

I didn’t set out to create a formalism that ends up only allowing a finite number of points in a line. That just fell out of the logic after correcting Cantor's errors. Define the concept of quantity correctly and this is where the logic leads. I'll grant you that it also happens to make more sense to me, but then logical things usually do make more sense to me.

I actually think it is amazing that the concept of number would lead to such a revelation. I mean, via other sciences (physics obviously) we have discovered that distance can indeed only be divided up finitely. (I'll post more about this in the Zeno's Paradox thread.) And here mathematics would have told us this abstractly when we formally defined the Natural Number if we had only done it correctly. Descartes would have been proud. Wasn't he the one who believe that abstract logic could ultimately be used to correctly describe reality? To think that pure abstract mathematics could have told use that distance can only be divided up finitely before we actually discover this fact would have been phenomenal. Unfortunately it's too late for that now. All we can do is correct the math and hope that maybe then it can be used to properly predict some more fundamental truths about our universe.

Hurkyl wrote
Oh, and if you're willing to consider other possible formalisms, you could try ZF with the axiom of infinity excluded. Without the ability to say "the set of natural numbers", ZF loses its ability to create infinite cardinals or model things like the real numbers.

Actually I have no problem at all with the idea of infinity. I believe that it is quite possible that the universe itself is infinite. I mean, I have no reason to believe one way or the other, but it really doesn't matter to me whether the universe is infinite or finite. Either way it's mere existence is a complete logical impossibility as far as I'm concerned.

Ironically, I can accept that the mere existence of the universe is illogical, while at the same time standing in awe at just how logically it behaves!

As far as the ZF axioms go I really should sit down here an go over them to see how they can be used to prove that a finite line contains an infinite number of points. I never really looked into the ZF axioms very deeply because they are based on the idea of the empty set. And since I see a logical flaw in the foundation of that idea what's the point in climbing up the stairs to see what's in the penthouse? It can't be anymore logical than the foundation that it is resting on can it?

Joy Division wrote:
It's like disregarding quantum mechanics because you don't like it and the world you know doesn't act that way.

I don’t see the comparison here. Quantum mechanics is backed up by experimental evidence so to deny it is to deny experience. I mean, we can certainly argue about the interpretations of it, but we can't very well argue the results.

There is no experimental evidence to back up the number of points in a line, or the cardinality of infinite sets. Do these things even have meaning in our universe? Maybe they do and maybe they don't. Only time will tell.

In the meantime I'm going to go post my new thoughts on the infinite divisibility of distance on the Zeno's Paradox thread.

 

[Joy Division]

Ok so you want to instead of taking the empty set to be a given, (by the empty set axiom) being a set which has no members. To instead define some other object which is an atom in the theory (an object that is not a set) and make that be zero. This zero has all the properties of the normal set theoretic empty set but is just not the normal set theoretic empty set.

Now where we would normally say zero is the cardinality of the empty set. Since the most naive notion of cardinality is the number of objects in a set. You instead just say that there is this other object. Is this zero a number itself? If it's not that has far ranging implications for fields, additive groups and other useful algebraic structures.

As far as I can tell that's what you're doing. Now the you haven't made any logical changes, and you haven't "fixed" set theory any. You've just renamed something so that you like it more.

It's just like some mathematicians say that 0 is an element of the natural numbers while others prefer that the natural numbers begin at 1. You seem to make a big deal out of this when in reality it's just a matter of convention. I'm sure you're going to jump on me for this but it really doesn't matter. It works both ways and the only confusion is if you change definitions in the middle of your work.

 

[NeutronStar]

Joy Division wrote:
To instead define some other object which is an atom in the theory (an object that is not a set) and make that be zero.

I don't define zero as an object at all. In my formalism the concept of zero is simply the concept of the absences of a set, the absences of a collection of things, the absence of quantity. It simply means that we have nothing to quantify.

So I guess in my formalism zero represent the idea of nothing (not the collection of nothing). For if you have a collection of nothing then you do indeed have something and zero becomes a thing. In my formalism zero is the concept of the absence of a thing.

So in this sense zero is not a number. It is still useful as a symbol to convey an idea of quantity (or I should say the absence of quantity). And this is in fact how we use zero. The number 503 for example represents 5 hundred units, no ten units and 3 one units. We actually use zero to represent the absence of quantity in our number system, but we have it defined as a quantity by definition. Albeit the quantity it no quantity at all! It's a superfluous definition. Not to mention the fact that it is a logical contradiction to the idea of a set as a collection of things.

Joy Division wrote:
Is this zero a number itself?

Not in my formalism. But in Cantor's it must necessarily be a number, because the whole idea of a number is that it is the cardinal property of a set. In Cantor's formalism zero is the cardinal property of the empty set, so technically (or logically) it must necessarily be a number in Cantor's formalism.

Joy Division wrote:
It's just like some mathematicians say that 0 is an element of the natural numbers while others prefer that the natural numbers begin at 1.

Well actually mathematician don't really have a choice. By Cantor's set theory zero is defined as a part of the natural numbers and is necessarily a part of the set. To ignore that formal definition is to simply ignore the formalism of mathematics. So any mathematicians who claim that the natural numbers being at 1 are just making up their own rules and ignoring mathematical formalism.

Having said that, mathematicians are most certainly free to make up a set of the positive integers where they can toss zero out because it is neither positive or negative. But it would be incorrect to then call that set the natural numbers.

Joy Division wrote:
As far as I can tell that's what you're doing. Now the you haven't made any logical changes, and you haven't "fixed" set theory any. You've just renamed something so that you like it more.

I've changed the logic significantly. After tossing out the idea of the empty set I can no longer define the number 1 as the set containing the empty set. Therefore I am forced to redefine the number one. It is in this new definition for the number one that all of the logical changes take place. In short I go back to Peano's original proposal of defining the number 1 on the idea of unity. But I go further to define what is meant by unity in detail. Peano was unable to do this for political reason. The mathematical community at that point in time had an extreme distaste for tying the idea of number to the idea of a thing. They wanted the idea of number to be a pure abstract idea. What they didn't seem to realize that by removing the idea of number from the idea of the things that are being quantified they were actually forcing the idea of number to be defined from a purely qualitative point of view. They virtually rejected the very idea of quantity by doing this. My formalism restores the idea of number to an idea of quantity by reconnecting it to the things that are being quantified. (Or perhaps I should say that I am forcing the idea of number to be dependent on a property of existence of the thing that is being quantified).

This doesn't cause the idea of number to be any less abstract. Why? Because the things that are being quantifying may be as abstract as we wish. My formalism merely states that if we are going to quantify a thing that thing must have particular properties of existence. If it does not, then we have no business claiming that we can quantify it in the first place. This is the essence of my formalism.

I would like to add here that because my formalism is necessarily founded on the conceptual idea of a thing it can no longer be thought of as purely axiomatic. It is dependent on the conceptual definition of the things that are being quantifies (no matter how abstract they might be). For this reason Gödel's inconsistency proof would not apply to my formalism. Gödel's proof is only applicable to self-contained axiomatic systems. This was another one of those things that simply fell out of having corrected Cantor's set theory. It was not my intention to fix mathematics so that Gödel's inconsistency proof does not apply to it. I just realized that this is the case after the fact! It was a pleasant surprise though!

Joy Division wrote:
You seem to make a big deal out of this when in reality it's just a matter of convention. I'm sure you're going to jump on me for this but it really doesn't matter. It works both ways and the only confusion is if you change definitions in the middle of your work.

I don't change my formalism while in the middle of working with it. I believe that I can say that I have made some significant logical changes in the formalism of mathematical, and that it's not just a matter of convention. As I mentioned above one of the consequences of my reconstructed mathematical formalism is that Gödel's inconsistency proof is no longer applicable to it. I'd say that is a whole lot more than just a change of convention.

By the way, I don't jump on anyone. I was merely trying to help newcomers to realize that these logical inconsistencies exist. I'm pointing right at the problem. The problem is the empty set. If they don't see the problem then so be it, they can just claim that I'm a nut and continue on their way. On the other hand, if they do see the problem they can know that they aren't alone.

The problem is there, it's real, and it's fixable. This is my story and I'm sticking to it.

I'm not trying to convince anyone who doesn't want to hear it. I'm not bending anyone's arm to respond to me. If you don't believe me fine. But by the same token don't try to convince me that I'm wrong because I know better.

 

[NeutronStar]

Joy Division wrote:
To instead define some other object which is an atom in the theory (an object that is not a set) and make that be zero.

Just as a further comment. Current mathematics does indeed have zero as the atom of the theory. My formalism has as an atom the number 1. My number 1 is not defined on the concept of zero. My number 1 is defined on a property of unity that can only be understood in relation to the thing that is being quantified.

This is the main difference. It's not just a matter of convention. The entire formalism is based on a completely different logical entity, using a different definition. Quite a few logical consequences fall out of this change.

 

[Joy Division]

It doesn't matter if you say your 1 is a completely different logical identity. If your 1 is provably equivalent to the set theoretically defined 1 then they are the same theory. Just the 1's have differnet flavors.

Also the fact that you don't feel like axiomatizing you concept of zero doesn't mean it doesn't have a proper axiomatic definition. You can't do any logic without falling under the blanket of Goedel's Incompleteness theorem.

If you don't have formal axioms you still have informal assumptions. But to really be doing math you need an axiomatic set of rules describing your system. Not defining your concept of zero to dodge the incompletness theorem, is like going to a swimming race and never getting into the water then saying you techinically didn't lose because you never started the race.

 

[Hurkyl]

We can't think of the set as a container.

[?]

We are "supposed" to think of a set as a container. That's why the fundamental operation of set theory is "is a member of". That's why typical set operations are union, intersection, and selection. That's why we use sets when we want a mathematical object that represents a collection of things!


Even if you are portraying Cantor's set theory accurately, then it is far removed from modern set theory.

I mean, via other sciences (physics obviously) we have discovered that distance can indeed only be divided up finitely. (I'll post more about this in the Zeno's Paradox thread.)

And as I mentioned in the other thread, other sciences have proven no such thing.

As far as the ZF axioms go I really should sit down here an go over them to see how they can be used to prove that a finite line contains an infinite number of points.

They can't. ZF is not a theory about lines. (though ZF may be capable of modelling theories that are about lines)

If you want to see the proofs that a "finite line" (by which I presume you mean a line segment) contains an infinite number of points, you should turn to the theories in which the meaning of line is defined (such as Euclidean geometry). Different theories, of course, will say different things; for example incidence geometry permits entire lines with only two points on them.

It is still useful as a symbol to convey an idea of quantity (or I should say the absence of quantity).

Then zero is an object of your theory.

Not to mention the fact that it is a logical contradiction to the idea of a set as a collection of things.

Intuitive contradiction, maybe, but you have demonstrated no logical contradiction. (incidentally, it doesn't contradict my intuition)

As I mentioned above one of the consequences of my reconstructed mathematical formalism is that Gödel's inconsistency proof is no longer applicable to it. I'd say that is a whole lot more than just a change of convention.

If you can define the natural numbers and their addition and multiplication in your theory, then Godel's proof is applicable to it. You don't even need zero based natural numbers; if you can define {1, 2, 3, ...} with addition and multiplication, then one can define zero from there and continue on to use Godel's proof. (and I'd be entirely unsurprised if it is a trivial exercise to modify Godel's proof so it never uses "0").

If you can't define the addition and multiplication of natural numbers, then you probably don't have a very useful theory.

I was merely trying to help newcomers to realize that these logical inconsistencies exist.

Proof of such an inconsistency would be nice. Basing your entire outlook on mathematics on an inaccurate portrayal of a discarded theory. (Cantor's set theory is inconsistent, but because one could make sets that were "too big", not because of the empty set)

 

[NeutronStar]

Joy Division wrote:
You can't do any logic without falling under the blanket of Goedel's Incompleteness theorem.

I don’t believe that this is correct.

It is my understanding that Gödel's Incompleteness theorem applies only to self-contained axiomatic systems. I am not an expert on Gödel's theorem to be sure. However, I have asked several mathematicians who are familiar with Gödel's work whether my understanding is correct and they gave me the thumbs up. So this is what I am going by.

My method of formalizing mathematics is actually conceptual not axiomatic. This conceptual system of logic relies on external ideas. Thus rendering Gödel's theorem inapplicable to it. But again I should add that this wasn't the reason that I took this route. It was a mere consequence of it.

Being more of a physicist than a mathematician I am used to conceptualized logic. Both General Relativity, and Quantum Mechanics are based on conceptual postulates. I see no reason why a mathematical system cannot also be built using a method of conceptual logical reasoning.

You have previously accused me of being hung up on the quantitative aspect of number.

I now accuse you of being hung up on the axiomatic system of logic.

It seems to me that in the face of Gödel's Incompleteness theorem you would want to steer away from such obviously incomplete methods of logic.

 

[Joy Division]

Incompleteness isn't such a scary thing. I just means that you can make some statements within an axiomatic system that cannot be proven based on those axioms.

Your system is self contained. Simply because you choose not to axiomatize it doesn't mean it doesn't have an equivalent axiomtization.

You can call them axioms, conceptual postulates, assumptions, but whatever you want to call them they are still the starting rules and statements from which you derive everything in your system. The only difference is that usually in a philosophical or physical argument you refer to things outside your logical system.

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.

I suppose I am hung up on axiomatic logic, that is seeing as how it is the only rigorous type of logic.

Goedel's incompleteness theorem isn't a death blow to mathematics and axiomatic logic. What it is is a limit on how powerful a proof can be. It makes sense too, there are always self referential statements you can make which can't be proven. Like "This statement is false." We should of course believe that these can't be proven.

 

[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.