Is Addition Really a Basic Skill?

[arildno]

You should start with proving proof with no idea about what must be present in a valid proof. Or perhaps not..

 

[NeutronStar]

Erm, no, that isn't quite what prove means in mathematics, is it? To prove something one needs a hypothesis from which to make a deduction.

Ok, so prove minus, then, prove 3, prove composition of functions.

IF $$a=1+1$$ THEN $$a=2$$

There is a hypothesis and conclusion to prove.

Attemping to prove minus, or 3 make no sense because there is no hypothesis and conclusion. With 1+1=2 there is.

I'm not sure about the composition of functions, that could probably be put into a conditional statement and proven. But to do so we would probably need to refer intensely to the definitions of functions and composition. To prove the conditional statement above we would likewise need to refer intensely to the definitions of number (specifically the numbers 1 and 2) and the definition of operation of addition. While that may not actually "prove addition" it would at least show that addition is a valid logical concept. And in a very real sense that is proving it.

The bottom line that will be the death of this whole thing is that the operation of addition is dependent on the definition of numbers and vice versa. So the whole thing become circular. That's because of the empty set definition of numbers. If the numbers where defined on a concept of Unity then they wouldn't depend on the operation of addition for their definition and the process would no longer be circular. In other words, it would be "externally" provable via other forms of logic.

I might add that if Peano's Unity had been accepted as the foundation of the definition of number, then Kurt Gödel's incompleteness theorem would no longer apply to mathematics because mathematics would no longer be a self-contained logical system.

Drcrabs is either going to have to accept the axiomatic methods of mathematics or prove to the mathematical community why it is flawed. The former is way easier.

 

[arildno]

Nonsense!
You INTRODUCE the mathematical symbol "2" by defining
2=1+1
You've got some silly, unmathematical preconception about what "2" is; get rid of that.

 

[NeutronStar]

Nonsense!
You INTRODUCE the mathematical symbol "2" by defining
2=1+1
You've got some silly, unmathematical preconception about what "2" is; get rid of that.

You may very well be correct. But if I have an unmathematical preconception about what "2" is I'm afraid that you'll have to blame that on the educational institutions that I've been taught by.

I was introduced to the idea of number in both kindergarten and in early grade school using a concept of collections of things (sets). The idea of "2" is ingrained in my intellect as the quantity than after having removed a quantity of 1 there is only 1 remaining. (that's actually backwards from the way it is taught, but I think it makes for a more rigorous definition).

I don't think of "2" as a printed numeral that we are using here to communitate the idea of two. That's just a symbol no different the the English word "T-W-O". The actual concept of "2" is a collection of individual elements such that after removing an individual element all that remains is enough "stuff" to define precisely the definition of whatever it is that it being quantified.

This may sound a bit complicated, but in reality I believe that this is everyone's everyday intuitive experience of the idea of quantity that we have come to formalise as numbers.

So based on this concept of unions of sets I have no problem at all comprehending addition. Unfortunately the most rigorous axiomatic definitions for addition do not permit this intuitive view. We must resort to Cantor's idea of collections of nothing.

 

[jcsd]

That isn't a proof neutron star, but even then proving 1 + 1 = 2 (which you'll see was done earlier if something so trivial can be called a proof) is not known as 'proving addition'.

We don't have to start out with the empty set in order to build Peanos axioms (I listed them above notice no mention of empty set), though it is possible to construct Peano's axioms starting with the empty set

0 = {}

0^* = 1 = {0} = {{}}

(0^*)^* = 1^* = 2 = {0,1} = {{{}},{}}

etc.

This was done by Von Neumann after Peano and there are other ways of constructing Peano's axioms.

 

[drcrabs]

So basically after reading these pages, you are trying tell me that you actually can't prove addition?

 

[NeutronStar]

We don't have to start out with the empty set in order to build Peanos axioms

Technically I disagree with that. Peano relies on the idea of 1 which has been formally accepted by the mathematical community to be defined by Cantor's definition. Therefore any reference to the number 1 is automatically a reference to the empty set by default. (In other words, Peano doesn't actually define the number 1 in his axioms, he merely uses the preexisting concept)

0 = {}

0^* = 1 = {0} = {{}}

(0^*)^* = 1^* = 2 = {0,1} = {{{}},{}}

Now this has always been of interest to me. Since 1 = {0} = {{}}
and 2 = {0,1} = {{{}},{}} then the whole intuitive idea of addition as the union of sets gets blown out of the water because {{}} U {{}} does not equal {{{}},{}}. It would be {{},{}}. This is actually the way we perceive number as a property of the real universe by the way. We don't perceive 2 as {{{}},{}} we percieve it as {{},{}}. So why the need to define it in such an unnatural way?

If we define 2 as {{},{}} then it would be easy to prove addition as the union of sets. We can't prove that 1+1=2 using our current mathematical formalism because it makes no sense.

Instead we need to rely on axioms that merely state that it is true by the rule of formalisms rather than being able to prove it by logical deduction.

 

[drcrabs]

The OP was aksed to "prove red" and he didn't (though he appears to think he did). He gave a "definition" of red. That in itself doesn't "prove" red.

Yea waddup Grimey. I've noticed that you realized that i gave a definition. The purpose of the 'definition of red post' was to point out that the person who was asking me to prove red was getting off the topic

 

[jcsd]

Your not quite making sense, Peano did not make any refernce to the number 1 in his axioms, the only number explcitily mentioned is zero and the only properties it has are those that are defined by the axioms (though you can start with the number one in order to construct the non-negtaive integers). In Peano's system of the natural numbers 1 is just another name for 0^* and the only properties it has are those that are defioned by Peano's axioms. Generically 1 is the mulpilcative identity in a semiring (I guess, as N is certainly not a ring under normal additon and multiplication), but this is something that emerges out of the defiuntion of mulpilication and additon in this set and we need not assume any preconceived notion of the number '1', other than the one given to us by our defintions.


In Von Neumann's constuction the succesor of some number n is simply n U {n}. Defining 2 as {{},{}} indeed makes no sense.

 

[jcsd]

So basically after reading these pages, you are trying tell me that you actually can't prove addition?

It's not a case of 'cna't' it's just that the question makes no sense.

 

[StatusX]

addition may be the most basic mathematical concept. in fact, i think its safe to say that addition came first, and the numbers were defined in terms of it. is there any other way to define 2 besides 1+1? if so, then maybe there would be a way to prove addition, but i don't think there is.

also, all this talk about set theory is, in my opinion, off topic. set theory is not the only way to define addition, and it implies there is only one way it could work. its possible to imagine a universe where 1+3 is not equal to 2+2. addition is, whether you like it or not, an empirical law, from which its set-theoretic defintion was derived.

when you gave the definition of red, you were actually proving my point. addition, like the color red, is not something up for debate. it is defined, plain and simple. from a definiton, we can derive all sorts of things about the things properties, like that its commutative, or that its wavelength is 650 nm. but the defintions themselves are, in a sense, handed down by god.
(red may have been a confusing example, because it is subjective, where as we all believe addition to be objective, and based on logic. my point was that they both are so basic and innate, to prove them makes no sense.)

 

[NeutronStar]

also, all this talk about set theory is, in my opinion, off topic. set theory is not the only way to define addition, and it implies there is only one way it could work. its possible to imagine a universe where 1+3 is not equal to 2+2. addition is, whether you like it or not, an empirical law, from which its set-theoretic defintion was derived.

I totally agree that addition is an empirical law of our universe. Heck, I think anyone who's been paying an attention at all to human history can clearly see that our mathematics arose from observing the quantitative property of our universe.

I totally disagree, however, that Set Theory was derived from this emperical observation. I wish it were, then addition would be provable.

Finally, concerning the idea that set theory is off topic in any discussion of addition is absurd in my humble opinion. And my reason for feeling this way is because it is through the very concept of sets (or collections of things) that the universe exhibits it quantitative nature. So even though we can disguise the idea of quantity (or addition of quantities) behind the guise of other types of logic doesn't change the fact that it can always be reduced to ideas of collections of the fundmental property of individuality for this is the nature of the universe from which the idea came.

There's just no getting around it. If we are talking about number we are talking about quantity. And if we are taking about quantity, we are talking about collections of individual things. It's a basic truth of the universe. Any concept of number that is not the concept of a collection of things is simply an incorrect concept of number. Such a concept is something other than the concept of number. So to just mention mathematics or number we necessarily have to think in terms of sets. There's just no other way to comprehend it.

True, we can write up a bunch of rules and axioms and follow them. But is that truly comprehension? Especially when we can't even prove them?

 

[jcsd]

Anyone who starts to talk about 'empirical laws' when talking about math proofs is most defintely on the wrong track. Maths does not claim to describe reality, the axioms of mathethamtical systems do not come from the observation of reality, they are true for the system because we define them to be true. It is pewrfectly possible to have a statement that is axiomatic in one mathematical syetm but false in another.

 

[StatusX]

all I am saying is that addition is so basic, it is impossible to prove within math(ie, using set theory). all you can do is define it to match empirical observations. to prove the physical law of addition is impossible, just like any other physical law. in general, id agree that math and reality are completely separate (although its naive to take this too far, since math is only useful when it helps us in the real world), but i think addition may be the point at the bottom where they meet. if anything, this is a philosopical question.

 

[NeutronStar]

Anyone who starts to talk about 'empirical laws' when talking about math proofs is most defintely on the wrong track. Maths does not claim to describe reality, the axioms of mathethamtical systems do not come from the observation of reality, they are true for the system because we define them to be true. It is pewrfectly possible to have a statement that is axiomatic in one mathematical syetm but false in another.

Had you been a student of Pythagoras he would have had you thrown overboard into the Aegean sea for making such a remark.

Actually I agree with you that modern day mathematicians view mathematics in a totally different way than many historical mathematicians did. And while I don't share this modern view I can't say that it is incorrect in this day and age. But what I can say is that modern day mathematics is not a correct model of the quantitative nature of our universe.

I usually put my view in a conditional statement and claim that the statement it true.

IF "mathematics is supposed to be a good model of the quantitative nature of our universe" THEN "mathematics is logically flawed".

I claim that this statement is true, although I certainly don't intend to prove it on an Internet board to a hostile audience. I'll save it for a personal lecture to people who are genuinely interested in hearing it.

I believe that most mathematicians wouldn't even be interested in hearing the proof because they would deny the hypothesis right off the bat. (This appears to be jcsd's stance, he simply doesn't believe that mathematics is supposed to be a good model of the quantitative nature of the universe and therefore any proof to show that it isn't a good model is trivial and uninteresting)

As a scientist I don't share his view. I believe that it is extremely important that mathematics properly model the quantitative nature of the universe. Therefore I am concerned with any flaws in mathematics that might make the above conditional statement true.

 

[Hurkyl]

Since we're talking about the philosophy of mathematics, I'll move this thread here.


Things have changed slightly since Pythagoras's time. Science/philosophy decides what axioms you should be using, and mathematics tells you what those axioms imply.

 

[NeutronStar]

Science/philosophy decides what axioms you should be using, and mathematics tells you what those axioms imply.

Actually I wish that was true.

 

[matt grime]

Yea waddup Grimey. I've noticed that you realized that i gave a definition. The purpose of the 'definition of red post' was to point out that the person who was asking me to prove red was getting off the topic

that post you refer to wasn't off topic, it perfectly illustrated that just because you can ask for something to be proven, doesn't mean the question makes sense. And when do you get to start using a familiar version of my surname?

 

[matt grime]

Actually I wish that was true.

That is true.

Plus you're logical proposition that you won't prove here is trivially true since the conditional is false.

 

[CrankFan]

IF "mathematics is supposed to be a good model of the quantitative nature of our universe" THEN "mathematics is logically flawed".

Mathematics isn't intended to be a good model of the quantitative nature of the universe; whatever that means. I think maybe you're confusing the roles of mathematics and physics.

You've not given any justification for your belief that the foundation of mathematics, as it's currently understood, is insufficient or inconsistent. Instead, you've insisted that some facet of the current approach is wrong (like say, defining 0 first) and fail to show how it leads to either a contradiction or limitation in characterizing some useful area of mathematics. Essentially, you don't like construction of N starting from 0, but can't articulate why this is problematic; it's just a (religious) belief.

Regarding Peano, the first publication of the Peano postulates appeared in "Arithmetices Principia", which was published in 1889. In that document 1 is defined as first natural number.

Later, in 1892(?), "Formulario Mathematico" is published, which shows 0 to be the first natural number. Peano changed his approach between those two publications.

But, more to the point: The selection of 0 or 1 as the first natural number is probably only a matter of convenience.

 

[jcsd]

Thanks, I've always wondered why some versions of Peano's axioms start with 0 and others start with 1.

 

[NeutronStar]

You've not given any justification for your belief that the foundation of mathematics, as it's currently understood, is insufficient or inconsistent. Instead, you've insisted that some facet of the current approach is wrong (like say, defining 0 first) and fail to show how it leads to either a contradiction or limitation in characterizing some useful area of mathematics. Essentially, you don't like construction of N starting from 0, but can't articulate why this is problematic; it's just a (religious) belief.

With all due respect it wasn't my intent to prove or show anything. I was merely responding to the original poster's concern with proving addition. I simply providing information concerning that topic.

I think that there's been a general consensus among the responders to this thread that addition cannot be proven (or as some claim it is even meaningless to ask such a question). I agree that it cannot be proven using current mathematical axioms. And I also agree that within that axiomatic framework it is a nonsensical question.

I was merely pointing out that the particular axiomatic framework that is preventing the question from being meaningful was introduced rather recently (with respect to the entire history of mathematics). And that the theory that prevents it is Georg Cantor's empty set theory.

I believe that everything that I've said here is true. I believe this on solid logical grounds. I actually don't have a religious bone in my body. (maybe spiritual, but that's another topic)

I have no intent to attempt to convince people who aren't interested in this topic. The mathematical community as a whole is well aware that there are logical problems associated with set theory. This is no secret and it has been a philosophical debate for many years (not the specific empty set concept that I am referring to, but set theory as a whole logical system).

Just for a quickie I will give you concrete example of the problem,…

Originally (that is to say that before the time of Georg Cantor) a set was intuitively understood as a collection of things. Cantor then introduced the idea of an empty set. A set containing nothing.

Well, this is actually a logical contradiction right here. We have a collection of things that is not the collection of a thing. This is a logical contradiction to the very definition of what it means to be a set. Similar objections actually came up by other mathematicians at the time that Cantor proposed his empty set idea. These objections were never (and I mean NEVER) fully resolved. In fact, the significance of their implications wasn't even fully understood at the time. Nor does it seem to be fully understood today.

In any case, there are only two ways to get around this logical contradiction. One is to claim that nothing is a thing in its own right. Therefore the empty set does indeed contain a thing and there is no logical contradiction. However, that logic leads to further contradictions by the simple fact that they empty set is then no longer empty. It contains this thing called nothing. In fact, this solution was pretty much tossed out as being far too problematic. The so-called "genius" of Cantor's idea was to remove the idea of number from any connection to the idea of a thing thus making it a "pure" concept. Even Cantor did not like the idea of treating nothing as a thing.

This leaves us with the second choice,… simply change our intuitive idea of the notion of a set. A set is no longer considered to be a "collection of things". That is merely an intuitive notion that is not needed for an axiomatic system to work. Instead, Cantor suggested, let's just ignore definitions, and forget about trying to comprehend the idea intuitively and make an axiom that simply states, "There exists an empty set". He somehow sold this idea to the mathematical community and they bought into it.

However, later on (possibly even after Cantor's death, I'm really not sure on that) the mathematical community started seeing other problems cropping up. For example, there must be a distinction between a set, and an element within a set. If this distinction isn't made then 1 would equal 0.

In other words, by Cantor's empty set theory

0 = {}
1={{}}

Now if there is no distinction between an element, and a set containing a single element then there is no difference between Cantor's definition of zero and One.

I'll grant you that this may appear quite trivial but I assure you that it is not.

The problem does not exist for Cantor's higher numbers because they are more complicated combinations of sets and elements.

2 = {{{}},{}} for example. Even if the elements were to stand alone there would be not be equal with any other number. In other words, you can remove the outermost braces (which merely convey the idea of a set) without reducing the contents (the actual collection of things) to configuration that represents a different number.

By the way this is much easier to see if you actually use the symbol err to represent the empty set. Or I think modern mathematicians use the Greek letter phi.

The bottom line to all of this is that Cantor's empty set theory is logically inconsistent. It's based on a logical contradiction of an idea of a collection of things that is not a collection of a thing.

So big deal you might say. It's a trivial thing just let it go and get on with using the axioms. Well if you think like than then you truly are a modern mathematician.

This logical contradiction does exists none the less, and it really does have an effect on the logic that follows from using these logically flawed axioms.

Two things should be apparent right way. First off, there is no formally comprehensible idea of a set. If you think of a set as a collection of things you are wrong. That is an incorrect idea in Cantor's set theory. It simply doesn't hold water in the case of the empty set which is the foundation of the whole theory.

There are far reaching logical consequences to this logical contradiction. And in a very real way they are almost like relativity. Just like relativity is hardly noticeable at small velocities, so the problem with Cantor's empty set theory is hardly noticeable for quantities much less than infinity.

But just as relativity comes into play and speeds that approach the speed of light, so do the logical errors of Cantor's empty set theory come into play at quantities that approach infinity, and more precisely it affects the concept of infinity immensely.

Georg Cantor is the only human ever to start with nothing and end up with more than everything. His set theory leads to ideas of infinities that are larger than infinity. In other words, it leads to the logical contradiction of some endless processes being more endless than others. If that's not a obvious logical contradiction I don’t' know what is, yet this absurd notion has been accepted and embraced by the mathematical community.

Finally, what I have typed into this post is merely the tip of the iceberg. I'm not about to write a book on an Internet forum to try to explain what most mathematicians should already be aware of. But there are other problems associated with the empty set theory as well, and they have to do with counting, or countability.

The whole reason why Cantor introduced the idea of an empty set in the first place was to avoid having to tie the concept of number to the concept of the things that are being quantified. In other words, at the that time in history it was extremely important to the mathematical community that numbers be "pure". I might add that this was much more of a religious notion than a logical one so if we want to claim that someone was being religious rather than logical we should look at the mathematical community around the time that Georg Cantor lived!

In any case, by starting with nothing, Cantor was able to avoid having to deal with the concept of the individuality of the elements (or things) being collected. In other words, by starting with nothing he basically swept the prerequisite notion of the individuality of the objects that are being quantified under the carpet. And by doing so he has removed that constraint from any elements. In other words, in Cantor's set theory anything can be counted as an "individual" element even if it has no property of individuality. This is, in fact, the very reason why he is able to have infinities larger than infinity. He is actually counting objects that have no property of individuality yet he treats them as though they do. I can actually show the error of his ways using his diagonal proof that the irrational numbers are a larger set than the rationals say. Once you understand the whole problem concerning the property of the individuality of the elements it's a fairly obvious proof.

I don’t even know why I'm bothering to type this in actually. The mathematical community simply isn't ripe for this knowledge yet. It just isn't in the "air". I think that it will be soon though as more and more mathematician begin to study group theory where is most likely to become apparent. Some clever mathematician somewhere is bound to realize what's going on and become famous for discovering the "problem".

I think that the most important thing for mathematicians to realize is that this problem was only introduced into mathematics about 200 years ago. Compare that with the age of mathematics and we can basically say that it "just happened!".

I mean, correcting modern set theory will have no affect on things like Pythagorean's theorem, or Euclid's elements, or the vast bulk of mathematics including even calculus which came before Cantor's time. It's main impact with be in group theory. But group theory is becoming extremely important in modern science so it could end up having a big impact there.

Fortunately Cantor's illogical set theory won't affect most normal algebra or other mundane calculations, so don't expect to get tax refund checks from the government when the problem is finally corrected. The government never thinks in terms of sets anyway, they think in terms of bucks.

 

[Gokul43201]

Umm...didn't read all of that, but I believe the question makes no sense not only from an axiomatic point of view, but from a basic, language point of view.

The question makes as much sense as "prove running". This is different from asking "why does running along the (B-A) vector get you to B if you start from A ?", which while self-evident (because that's how it's defined), is at least not an improper usage of language.

 

[arildno]

How do you prove proof by not proving proofs of the principles of proof?
Or something completely different..

 

[CrankFan]

I think that there's been a general consensus among the responders to this thread that addition cannot be proven (or as some claim it is even meaningless to ask such a question).

Right, they're saying that "prove addition" is no more meaningful a statement than "prove purple bonnet banana."

Some people, (like me), have made assumptions about what the OP wanted (probably a mistake ), and sketched out a process by which the function of addition is proven to exist and retain all of the properties we intuitively expect it to have. -- these proof sketches have been ignored by the OP. No one seems to know what he has in mind when he uses the phrase "prove addition", and it's doubtful that it has any sensible meaning.

I was merely pointing out that the particular axiomatic framework that is preventing the question from being meaningful was introduced rather recently (with respect to the entire history of mathematics). And that the theory that prevents it is Georg Cantor's empty set theory.

This is nonsense. Why should the development of some theory prevent a person from asking a meaningful question? Are you claiming that if someone was asked to "prove addition" in 1850 that it would be meaningful then? But now, in 2004 it's not meaningful? and Cantor is to blame for his apparent ability to control people's behavior from beyond the grave?

I believe that everything that I've said here is true. I believe this on solid logical grounds.

Actually you've made many factual errors in your posts, more than I'd bother to correct. It's surprising that you say you believe everything you've said is true, shortly after you've been presented with information about Peano's postulates which contradicts your claim that he was some kind of advocate for starting from 1 as opposed to 0 ... as if any of that mattered.

The mathematical community as a whole is well aware that there are logical problems associated with set theory.

The problems which were widely known at the time of the development of set theory do not exist in modern set theory. If you replace "set theory" in the above quote with "naive set theory" then I don't have a problem with it, but if you mean ZF by "set theory" then your statement is false.

Originally (that is to say that before the time of Georg Cantor) a set was intuitively understood as a collection of things. Cantor then introduced the idea of an empty set. A set containing nothing.

I don't know the origin of the phrase empty set, or the first use of the term set, but I'm a little bit skeptical of taking your claims at face value given the egregious errors you've made previously about the history of mathematics.

I've only studied material which is a refinement of Cantor's work, not the original -- but I don't see how Cantor or his original work is relevant to the supposed problems of set theory as it's understood today.

Well, this is actually a logical contradiction right here.

No it's not.

If one of the axioms asserts that sets are non-empty, then it would be a contradiction for an empty set to exist, but there is no such axiom or theorem of ZF.

We have a collection of things that is not the collection of a thing. This is a logical contradiction to the very definition of what it means to be a set.

Again, the axioms of ZF don't assert that sets are non-empty collections of objects as you are implicitly doing. Someone recently posted a version of Berry's paradox, which "proves" that the naturals are finite. It's a very amusing "proof", the problem is that it relies on the ambiguity of natural language. Your mistake above is similar, in implicitly assuming that a collection is non empty -- of course in a formal system all of these details are made explicit, and it's not a problem.

Similar objections actually came up by other mathematicians at the time that Cantor proposed his empty set idea. These objections were never (and I mean NEVER) fully resolved.

Can you cite these specific objections to his "empty set" theory? You seem to be hung up with the notion of the empty set, but I don't think others, even those skeptical of Cantorian set theory at the of its development had any problems with the empty set. I think the objections of mathematicians like Poincare were related to the concept of infinite sets and had nothing to do with the empty set.

When you talk of "problems never fully resolved", I'm not sure if you're talking about philosophical objections to set theory or the classical antinomies.

If you're talking about the later then all of the classical antinomies were addressed in the development of theories like ZF. It's inaccurate to say that the problems of Naive set theory exist today.

If you're talking about philosophical objections to the notion of infinite sets, well that's fine I suppose that you have an opinion, but these days, philosophical objections of this sort almost always belong to non-mathematicians, and are inconsequential.

However, later on (possibly even after Cantor's death, I'm really not sure on that) the mathematical community started seeing other problems cropping up. For example, there must be a distinction between a set, and an element within a set. If this distinction isn't made then 1 would equal 0.

This is absurd. Every object in ZF is a set, there are no obvious problems with the theory.

In other words, by Cantor's empty set theory

0 = {}
1={{}}

Now if there is no distinction between an element, and a set containing a single element then there is no difference between Cantor's definition of zero and One.

This is false. You're confusing the concepts of set membership and subset.

The empty set {} is a member of {{}}, it is not a member of {}. By extentionality we know that {{}} isn't the same set as {}, since {{}} contains at least 1 element not in {}.

The bottom line to all of this is that Cantor's empty set theory is logically inconsistent.

It's strange how you keep referring to Cantor's "empty set theory".

A quick search at the following site indicates that Cantor isn't known to be the first one to use the phrase empty set or null set.

http://members.aol.com/jeff570/mathword.html

From what source are you determining that Cantor was the originator of the concept of the empty set. Note that I'm not claiming that he didn't, just that I'd like to know if this is true.

I'm not sure if Cantor's set theory can be said to be inconsistent. Certainly not as a result of the nonsensical complains you've made about the empty set. Cantor seemed to be aware of the dangers of unrestricted comprehension, apparently he wrote about it in letters to other mathematicians. That aside; whatever Cantor's theory was (consistent or inconsistent) it has no bearing on the state of set theory today. If you know of a contradiction in ZF, you can state it in the language of ZF and if can't your claim need not be taken seriously.

So big deal you might say. It's a trivial thing just let it go and get on with using the axioms.

It's not that what you've said is trivial, it's that what you've said is incorrect. Your opinions are based on misconceptions. That you have difficulty in distinguishing between elements of a set, and its subsets tells me that you need to go over some introductory material before you're able to discuss these issues competently.

 

[jcsd]

I think Neutron Star one of your main misconceptions is the between naive set theory and ZF set theory. naive set theory which is the set theory of Cantor's day the empty set is not axiomatic as you are allowed to define sets by shared properties of their mebers and clealy if you define the shared properties of the members of a set in such a way that it cannot have any members you get the empty set (for example {x|x is a nonegative real number less than zero} defines the empty set). However defining sets in such a way leads to paradoxes such as Russell's paradox.

ZF set theory remedies this by saying that a set is only defined by it's members, in ZF set theory the empty set is axiomatic and as far as I am aware no paradoxes arise because of it's inclusion.

Secindly this i sgetting off track as Peano's constructio of the nautarl numbers are not depedent on whethr or not the empty set exists (it is not an orginal idea to notice that sets are defined by ther members yet the empty set has no members, it has been pointed out before, but no logical inconsistenty arises because of this state of affairs).

 

[NeutronStar]

I think Neutron Star one of your main misconceptions is the between naive set theory and ZF set theory.

I'm not actually having any misconceptions between naive set theory and ZF. In fact, I see them as totally separate things.

ZF may very well be a logically sound theory in its own right. But that is totally irrelevant.

I am only concerned with the following conditional statement:

IF "mathematics is supposed to be a good model of the quantitative nature of our universe" THEN "mathematics is logically flawed".

So even if ZF is perfectly logically correct, if it doesn't correctly model the quantitative nature of our universe then it doesn't really matter that it's logically sound. It's still incorrect within the context that should matter to a scientist.

I never mentioned a word about ZF because ZF did indeed toss out the idea of a set as a collection of things. That's how ZF got around the logical contradictions of Cantor original set theory. And yes, that is a way around it but at what price? In ZF the notion of a set is unbound, and undefined. This leads to paradoxes like infinities that are larger than infinities.

But wait! Hey, that's only a paradox if we think of infinity in a comprehensible way as an endless process or quantity. Only then is it a paradox. But if we are willing to forfeit comprehension of ideas then we can accept that abstract infinities can be larger than other infinities. Of course it no longer makes sense to comprehend infinity as a simple idea of endlessness because it makes no sense to have something that is more endless than something else.

So ZF is only a sound logical theory for those who are willing to give up comprehension of ideas and see no paradoxes in things like collections of things that contain no thing, or ideas of endlessness that are more endless than other ideas of endlessness.

To get rid of the paradoxes all we have to do is forfeit our comprehension of the ideas. Seems easy enough. But for me it just isn't something that I'm willing to do.

Moreover I don't see any need to do it. I can see a logical and sound set theory that is actually based on a comprehensible idea of a set as a collection of individual things where the property of individuality has also been defined in a comprehensible way. We can still have a symbol for the absence of a set which is yet another very comprehensible idea. The set containing all positive numbers less than zero is absence, it simply doesn't exist. But we can use a symbol to denote that it doesn't exist just like we currently use a symbol for the non-comprehensible idea of an empty set.

What's of much more importance is that once we build a set theory on this foundation we will quickly see that all sets cannot be elements of other sets because all sets do not have a valid (by definition) property of individuality. Therefore we won't be bothered by Russel's paradox concerning the set of all possible sets because such an idea is an illegal idea by definition.

So far we are doing just as good as ZF.

But much more importantly there is so much more that falls out of this idea,…

To begin with it doesn't lead to infinities larger than infinity. There can only be one condition of endlessness, a set either has this property or it doesn't. No paradox there. It's also quite comprehensible as simply an idea of endlessness. So we drop that paradox off as well.

We do however pick up a lot of new interesting stuff that I'm also afraid to mention. (ha ha)

One consequence of this "new" set theory is that the number point in a finite line must necessarily be finite. At first that might seem hard to swallow, but actually it makes perfect sense after it is understood why this must be so. It become crystal clear by a very simple proof of why it must be so. At first I had a lot of problems accept this myself, but after thinking about it for many years I have come to grips with why it must be so. I also think that it is amazing that a corrected definition of the idea of number can actually lead to something that is actually true about the real universe (that it is quantized).

Another thing that comes out of this formalism is that it makes absolutely no sense at all to talk about negative numbers in the absolute sense. Once again, this may come as a shock and seem rather weird, but after thinking about it for a while it makes perfect sense.

There can be no such thing as an absolute negative number. The whole property of negativity is a relative property. This is actually the true nature of the universe we live in. It's a good model of the quantitative nature of reality. Ironically mathematicians are already aware of the absolute properties of number. They even have the absolute value function.

Well if you actually stop and think about it, negativity in any comprehensible sense must always be relative. It makes no sense to talk about an absolute negative quantity. Yet we insist on giving negative numbers a life of their own. The fact of the mater is that negativity is always a relative property between sets. It's not an absolute property of either set. The idea of something being negative must always be in context of a larger picture in order to be comprehensible. Even on a number line negative numbers are only negative relative to the origin of the line. Take away that relative reference and negativity is meaningless.

Much of this falls out of a corrected definition of number (where again, by corrected I mean a definition that genuinely reflects the true quantitative nature of our universe). Once the definition is corrected and set theory properly reflects the quantitative nature of the universe in which we live many things fall out of it including the very quantized nature of the universe. In short, had mathematicians been on the ball they could have predicted the quantitative nature of the universe before Max Planck discoverer it in quantum mechanics. Although, I don't believe that mathematics could have actually put a number on it. But just saying that it must be so would have been quite an achievement.

In any case, I seriously don't care anymore. Call me a nut. Claim that I'm full of errors. I disagree with that of course, but I really don't care if other people believe that. My rambles certainly would be full of errors if compared to ZF because I'm not talking about ZF,…. duh?

I'm talking about my conditional statement:

IF "mathematics is supposed to be a good model of the quantitative nature of our universe" THEN "mathematics is logically flawed" or at least incorrect within that context.

I really should have known better that to even mention any of this on a pure mathematics site. Forget I ever mentioned it. I'm not out to convert anyone. Honest. I really don't care what anyone believes. Have a nice life and enjoy whatever you choose to believe.

 

[Hurkyl]

NeutronStar, I'll start with a simple question: do you reject the number zero?

 

[NeutronStar]

NeutronStar, I'll start with a simple question: do you reject the number zero?

Yes. If a set is considered to be a collection of thing, and number is defined as the quantitative property of a set, then Zero, by definition, cannot be a concept of number.

It can however be a valid mathematical concept representing the absence of number (or absence of a quantity, or set). This is, in fact, how it is actually used. Take the number 1060 for example. This number means actually represents 1 thousand, no hundreds, 6 tens, and no ones. In other words, the symbol Zero is representing the asbsense of quantity. It is a valid concept of quantity even though it doesn't officially qualify as a number by definition.

In practice the symbol and concept of Zero would be used pretty much in the same way that it is used today. The main difference would be in the actual way that it is comprehended. It would be seen as an absence of quantity, or a set, rather than as a set that contains no elements. Be definition in the strictest formalism it would be understood that it cannot be thought of as a number.

Of course we always knew that Zero was a weird number anyway because of the problems associated with division by the number Zero.

So, yes, I recognize the quantiative concept. But no, technically it wouldn't satisfy the definition of a number, and therefore it would be incorrect to claim that is is a number. It can still be used as a valid symbol of communication as a quantitative idea however.

 

[dekoi]

Addition is only a way we, as human beings, articulate ideas which we discover.

Since what we discover is perfect in every respect (although not always the way we articulate it), then addition $$itself$$ should also be perfect.

Proving an idea is, like someone else said, proving a metaphysical entitity (or a color as an example).

We can of course, empirically prove addition by the use of several objects (e.g. with apples, like children learn) -- however, that is proving the way we articulate addition, not addition in its own self.

 

[Hurkyl]

I'm sure you see the analogy between the empty set and zero.

While I shudder at labelling it the "absence of blah", I'll go with it for the sake of argument... zero is used when you would normally want to express a quantity, but you have the absense of quantity... the empty set is used when you would normally want to express a "collection", but you have the absense of a collection.

And just like zero, virtually all of the operations applicable to sets can be extended to apply to the empty set.


I don't really think there's any argument that the empty set is any less legitimate than zero would be. The empty set certainly has practical value, just look at any programming language: there would be an absolutely nightmarirish semantics problem if collections were required to have at least one element!


I want to reemphasize an earlier point: just like zero, virtually all of the operations applicable to sets can be extended to apply to the empty set.

There's the old adage, if it looks like a duck, and it sounds like a duck...

Your argument sounds like it's entirely an issue of semantics. To you, a collection must have at least one element, so you have the separate concepts of "collection" and "absense of collection". Just what is the problem of having one term that includes both of these concepts?

For example, you don't reject the term "fruit" just because there are "apples" and "oranges", do you?

You are fixated on the "definition": "a set is a collection of elements". That definition is geared for those who consider the empty collection a collection. For those like you who reject the idea of an empty collection, you should take the "definition" of set to be "a set is either a collection of elements, or the absense of a collection".


Again, just like zero, this unified concept has proven its usefulness: it's silly to reject it, especially over an issue of semantics.


Oh, and a disclaimer: everything in this post is on the philosophy of mathematics -- don't try to take it as the actual thing.

I'll touch on the infinite (notice I did not say "infinity"!) in the next post.

 

[Hurkyl]

Actually, there are several different aspects of your argument, each of which I'd like to address separately. The next is axiomization.

The whole reason why Cantor introduced the idea of an empty set in the first place was to avoid having to tie the concept of number to the concept of the things that are being quantified. In other words, at the that time in history it was extremely important to the mathematical community that numbers be "pure". I might add that this was much more of a religious notion than a logical one so if we want to claim that someone was being religious rather than logical we should look at the mathematical community around the time that Georg Cantor lived!

I offer a third possibility -- it was a pragmatic notion.

You say we should look at the mathematical community of the time. Well, let's look at the early 1800s: the field of analysis was blossoming at a rapid pace, directed mostly by intuition, rather than rigor. Abel had this to say in 1826:

There are very few theorems in advanced analysis which have been demonstrated in a logically tenable manner. Everywhere one finds this miserable way of concluding from the special to the general, and it is extremely peculiar that such a procedure has lead to so few of the so-called paradoxes.

One particularly interesting fact is that there was a great reluctance to accept Fourier's method of expanding functions as a trigonometric series (today a very important tool), because it was just too weird. Because there was no axiomization of analysis, Fourier could not prove his method worked.


As you can see, it was a practical necessity that led to the search for a "pure" way to define the numbers, so that results could be rigorously justified, in the spirit of Euclid.

 

[NeutronStar]

As you can see, it was a practical necessity that led to the search for a "pure" way to define the numbers, so that results could be rigorously justified, in the spirit of Euclid.

Well, I'm not sold on the idea that it was a practical necessity. I particularly believe this for the case of Euclid's Elements. While I admire Euclid's genius and totally agree with his logical structure of geometry, I genuinely don't see the need for the axiomatic method. He could have done the same thing by defining intuitively comprehensible concepts. I believe that because I can deconstruct Euclid's axioms and reconstruct them into intuitively comprehensible concepts without having to change any of his rules whatsoever. So from my point of view the mere fact that he chose to present it as an axiomatic system is totally irrelevant to me.

I do understand why he chose to use axioms rather than definitions. It's simply easier to state an axiom than to have to explain a definition to the point where everyone will comprehend it in the same way and agree on its meaning. So there is a practical aspect to axiomatic methods.

I'm actually not against axiomatic methods. I could actually state my intuitive set theory as a set of axioms also. There would be absolutely no problem with doing that. In fact, as far as I can tell any comprehensible logical system should be able to be stated in an axiomatic format.

What I do have a problem with is when a logical system is stated in axiomatic format and no one can explain it in terms of intuitively compressible ideas. That's when I step off the boat. I like things that I can comprehend. I don't like to have to learn a bunch of rules that I don't fully understand, and that lead to paradoxes that I can't even comprehend. That makes no sense to me. My first intuition is to reduce the axioms to comprehensible ideas, understand them, then put them back into axiomatic form and continue. With set theory I wasn't able to do that because the axioms don't make intuitive sense to me.

This really bothered the hell out of me for years. I struggled to make logical sense of them for literally decades. I finally concluded that they simply are nonsensical and this is why I can't make any sense of them. Then I decided to see if I could build a system that does make sense and I could. All I have to do is start with Peano's original idea of Unity (although I like to call it the property of individuality), and then define the number one on that. In essence I am defining an element first, and then I'm going to go on and define a set based on the concept of collections of elements. There can be no empty set in my theory because the whole concept of the set is based on the concept of a collection of an element. If we have no element, we can't have a set.

Ok, so it's an intuitively comprehensible theory. So what? Does that make it dangerous?

It can be stated axiomatically. Actually I never really thought about stating it axiomatically, but what would stop me from doing so. It's based on totally comprehensible ideas so all I need to do is state those idea as axioms.

I would start with the axiom that there exists an element. Then I would go on to state the axioms that define its necessary property of individuality. The property of individuality is the crucial difference between my set theory and Cantor's. Those properties. I actually have a workable axiom for that is based on the element's definition of existence. In a very real sense it is based on an operation of subtraction. Or better yet, I should say that it actually defines an operation I call subtraction. Subtraction within this primitive context cannot produce negative sets. However, as the logical system builds the concept of a negative set does come into play. That concept is explained in a relative context. In my system it is clear that there are two entirely different meanings to the negative symbol. One meaning is the relative negativity of a set with respect to a larger picture. I think of that negative sign as an adjective in the language of mathematics. The other meaning of the negative sign is to perform the operation of subtraction, in that context the symbol represents a verb in the language of mathematics.

We actually already recognize these two different meanings of the negative sign in mathematics, but few people actually think of one as an adjective describing a relative situation and the other as a verb describing the action of an operation.

In any case, I won't bore you with any more. The whole idea for me is that it works, and it works in a fully comprehensible fashion that leaves nothing undefined. Moreover, it's actually a more accurate description of the true quantitative nature of our universe. It doesn't lead to logical contradictions such as collections of things that are not collections of things. Nor does it end up leading to infinities larger than infinity, or the logical contradiction of some endless processes being more endless than other endless processes. To me that whole idea is totally absurd. Fortunately my system does not allow that. It turned out to be an unexpected consequence that I had not planned on at all.

There are other things about my system that many people might have trouble with at first. Like the idea that my system leads to the inevitable conclusion that a finite line must only contain a finite number of points. I found that hard to buy into myself at first. But after studying it for years I see now why it must be the case. It turns out that there is a very beautiful eloquent proof using this set theory and the concept of functions (which my set theory does not affect at all) that not only proves that this must be the case, but the proof itself makes it crystal clear why it must be the case. It also makes the idea very intuitively comprehensible.

Oh well, I'm spending far too much time typing this stuff in. I have other work to do. But my point is that I have no problem with axiomatic systems that make sense. I just have a problem with ones that don't make intuitive sense. Oh, by the way, using my set theory one can "prove addition". Or at least explain it in a very logical intuitive way as to clearly show why it must be the way that it is. This comes partly from that property of unity or individuality that current set theory has refused to address. Once that property is included as part of the theory addition can be proven. In other words, it can be proven that addition is a valid operation within the overall logical system. The operation of addition "falls out" of the basic concepts (or axioms) rather than being a part of them. It was actually that fact that attracted me to this thread to begin with.

But now I'm almost sorry that I responded because I really don't have time for this right now.

I supposed I've explained my position far enough. I'm really not prepared to attempt to explain my entire formalism. If I were going to do that I'd write a book about it and get paid for my time.

I'm not a mathematician. I'm a physicist. I want to get back to studying group theory and differential calculus. Unfortunately I need to know these things in order to do physics.

By the way, while mathematicians may have dropped the ball with set theory I will give them credit for doing a wonderful job with the calculus!

 

[alexkerhead]

I reckon addition is a basic skill. But can it be proved?
How?

DO this first.
Look up the definition of "addition"
then use the definition with your fingers.
take the five on your left hand, and add it to the sum of fingers on your right hand. If you are normal, you should find a sum of 10(ten) now, put 6 in the in the first varible, you have 11 then, that is a false statement based on the definition of addition..
You really don't need to worry about addition being provable, becuase you use it in the tense that we define it as.

I hope this helps.

 

[Hurkyl]

I've argued that the empty set is a legitimate concept, and I've argued the practicality of using the empty set. Now I will argue that rejecting the empty set leads to a defective set theory.


One of the more powerful features of sets is that they are able to describe hierarchies. For example, if I'm playing the towers of hanoi puzzle (but, suppose I only have the 7 discs, I don't have the base and pegs), any particular position is a multiset of sets. For example, if I have discs 1, 3, and 5 where the first peg should be, disks 2, 4, 6 where the second peg should be, and disk 7 where the third peg should be, my position is then given by:

[ {1, 3, 5}, {2, 4, 6}, {7} ]

As an aside, this demonstrates the importance of distinguishing between a set and its elements -- if the set {1, 3, 5} meant the same thing as its elements 1, 3, 5, then I lose the entire structure: I have no way of representing the fact that my collection if discs has been partitioned into three distinct parts.


Back to the main point: suppose I then move discs 2, 4, and 6 onto disc 7. Using a definition of set that permits empty sets, I can now describe the position as:

< {1, 3, 5}, {}, {2, 4, 6, 7} >

But without the concept of the empty set, I cannot accurately express this -- I have no way of specifying that there is a third place that could hold a disc, but currently isn't. Because of this deficiency, I argue that a set theory without an empty set is a defective set theory.

(p.s. the reason I said that I need a multiset of sets is to describe the solved position: < {}, {}, {1, 2, 3, 4, 5, 6, 7} >. You need a multiset to express the idea that I have two copies of the empty set)



I don't see what's incomprehensible about the ZFC axioms, except possibly the axiom of choice. (Unless, of course, you steadfastly refuse to try and internalize the notion of an empty set). They are, in "English" rather than formalism:

Two sets are equal iff they have the same elements.

For any A and B, {A, B} is a set.

If I have a set of sets, their union is also a set.

If I have a set, there is a set containing all of its subsets.

There is an empty set. (this is actually superfluous -- the next axiom shows a set exists, and the axiom after that proves the empty set exists)

There is a set of natural numbers.

If I have a set S, and a property, then there is a set containing everything in S with that property.

If {a, b, c, ...} is a set in the domain of some function f, then {f(a), f(b), f(c), ...} is a set.

A set can't contain itself, either directly or indirectly.

And the axiom of choice.


What part of it don't you understand?



And finally, you're beginning to sound awfully crackpottish:

The whole idea for me is that it works, and it works in a fully comprehensible fashion that leaves nothing undefined.

Balderdash. You, for example, never defined the terms "collection" or "element". It's an elementary fact that, unless you're using circular "definitions" you must leave something undefined.

Moreover, it's actually a more accurate description of the true quantitative nature of our universe.

This is another typical crackpot claim -- but it's only "more accurate" in capturing the way you wish to describe things. You have not, for example, performed any measurement of anything and gotten a more accurate answer than a measurement based on a theory that accepts the empty set.


I won't complain yet about "infinity" because I haven't written my response yet on that concept.