0, 1, infinite model, question

[matt grime]

If you are now going to declare the "infinite environment is 2 through infinity and 0" then I don'et know what you're remotely getting at. Why is 1 not in this set? Because it is "the finite"? This is a departure from the usual ideas, and almost certainly wouldn't agree with whatever notion of numbers the bloke from princeton had.

I don't see whty you need to make a distinction between different symbols for the same object: 3=1+1+1.

You are reading more into it than I do (what I termed "personifiying", such as assigning terms like mysterious to 0).

If however, all this princetonian was doing was explaining to you that there is a initial ordinal, successor finite ordinals, and that the set of finite ordinals is infinite, then I can see what he was getting at.


Oh, and 1 is also a "neutral element" for the operation of mutliplication. Though I don't like those labels for objects - naming them the real and imaginary numbers is one of mathematics' worse choices.

A reason why some people do and some do not take 0 to be natural is that some wish for the naturals to be the"counting numbers" as, say, going back to the romans. Some wish to use 0. The point is that we can view the set of natural numbers, in either version, as a model of an "inductive set". Loosely a set with an initial term, and for every other term there is a "next" element.

Some start coutnign with 0 soem with 1. 0 comes about becuase of the use of the empty set to create the model in peano's axioms, I would suggest.

We could equally start with -1,0,1,...

All that matters is the ordering. It is only if we also wish to define addition to these that we must make more specifications. But the choice of how we add things is what makes 0 the identity.

There is a perfectly valid form of addition on N that says 1+1=3, and 2+2=5.

To see this, hold up two fingers and then two more fingers and you have "1+1=3" by coutnign the spaces between them. So 0 is only special with respect to your common notion of addition.

 

[arildno]

One, that read beautifully to me. Thank you for that thoughtful explanation.

If 0 is a neutral element, that clearly delineates it from 1, no? the fact that you can distinguish one binary operation from another binary operation is the same distinction I am referencing regarding the distinction between 0 and 1.

perhaps, and maybe I am being a bit bold here, but let me say that we can define 'relationships' as 'distinctions'. Things that have 'distinctions' have clearly defined borders, boundaries. Indeed, a distinction 'draws' the finite conceptual object that we model it to be.

So when I say the relationship to 0 to 1 to 2, I simply mean the observable basic distinctions to how we perceive those conceptual mathematical environments that exist with their own natural set of rules, and then I am saying that now that since I CAN make the distinctions, and the distinctions are observable to all p.o.v. in the objective sense, I want to define those distinctions mathematically and rationally.

I see what you mean above about 0 and 1. Do you understand what I mean here now, and do you see how your analogy does not apply to this particular environment?

MR

I think I'll need to clarify a few points here (hopefully, I won't be obfuscating..)

1. Concerning the distinction between "addition" and "multiplication"
Now, in order to get to grips with this, we need to focus on those rules mathematicians CHOOSE to regard as the defining characteristics of those operations.
That is, the axioms for these operations:
a) Commutative&Associative rules:
a11) The commutative law for addition
Pick any two numbers "a" and "b" ("a" and "b" need not be different numbers):
Then we say that the number called the SUM of a and b, that is a+b, is equal to the number called the SUM of b and a.
That is, for any choices of a,b the equality a+b=b+a is assigned the truth-value "true".
a12) The commutative law for multiplication
Pick any two numbers "a" and "b" ("a" and "b" need not be different numbers):
Then we say that the number called the PRODUCT of a and b, that is a+b, is equal to the number called the PRODUCT of b and a.
That is, for any choices of a,b the equality a*b=b*a is assigned the truth-value "true".
a21) The associative law for addition:
Given any numbers a,b,c, we state that the number called the sum of (the Sum of a and b) and c, is the same number as the sum of a and (the sum of b and c).
That is:
(a+b)+c=a+(b+c) is fundamentally "true"
a22) The associative law for multiplication:
Given any numbers a,b,c, we state that the number called the product of (the product of a and b) and c, is the same number as the product of a and (the product of b and c).
That is:
(a*b)*c=a*(b*c) is fundamentally "true"

b) The distinguishing rule: The distributive law.
Note that apart from different fancy words (addition, multiplication, sum, product) and fancier symbols (+,*) the laws as specified under a) does not at all make it clear that the "+"-operation is any different than the "*"-operation.
Thus, if we are to be justified in regarding these as DISTINCT operations (rather than being just different names for the same "thing") we need to specify a rule which breaks the operational symmetry under a)
We do so by specifying how addition and multiplication act together:
The distibutive law says that given any numbers a,b,c, the product of a with (the sum of b and c) equals the sum of (the product of a and b) and (the product of a and c).
That is:
a*(b+c)=a*b+a*c

Note that now we are in possession of a distinguishing tool, because, at the outset, it does not follow from the other rules that we also must have:
a+b*c=(a+b)*(a+c) for all numbers a,b,c


That is, we may distinguish "addition" and "multiplication" by choosing a particular set of axioms.

2. The distinction between 0 and 1
Again, this is not something we can deduce unaided by specified axioms, in fact, we CHOOSE to regard the statement 0\neq1 AS AN AXIOM IN ITSELF.
This emphasis on choice is in fact a rather subtle, but crucial, point, in that it shows that from a logical point of view, there is nothing inherently contradictory in regarding the multuplicative and additive identities as the same number; effectively, what you'd get then is a perfectly consistent mathematics which only deals (in a consistent manner..) with a single element..(all numbers can be shown to be equal)
It is not a particularly fascinating mathematics, but it is not internally contradictory.


EDIT:
I have tried to emphasize the "CHOOSE"-word throughout this post, because it indicates that there may well exist other internally consistent systems&operations which bear very little resemblance to our "ordinary" way of thinking about numbers, and the operations we may perform upon them.

 

[arildno]

I think I'll post a few comments on what ought to be crucial if we are to perform a philosophical analysis on mathematics.
Then, I hope you agree with me, we need to represent "how mathematicians think about maths" in a correct manner!
In particular, we should pay special attention to where a mathematician's perspective differs from the layman's perspective, i.e, where we can pinpoint a conceptual shift, if you like.

Now, I assert, the ordinary way of thinking about numbers, is first and foremost to think about what numbers ARE.
This is NOT the way a mathematician thinks about it; instead of being interested in what numbers ARE, he is interested in what we may DO with them!

That is, the question of what a particular number is, unrelated to whatever numerical relations we might choose to put it in, is of little interest to the mathematician.
Rather, he focuses his attention precisely onto these relations, and couldn't care less if numbers have independent existence apart from these relations.
Indeed, that tends to be a rather meaningless issue, since we basically "bring into existence" numbers by specifying the defining relation of that number.
For example, we may introduce the symbol "2" to represent that number we gain by adding 1 to itself; i.e, the equality 2=1+1 might be regarded as the definition of the number 2
(Note: There are valid quibbles to this way of thinking, in particular, it may be convenient to introduce definitions more intimately linked to the fundamental level maths, i.e, set theory&logic).

What "2" inherently is, is an issue faded into insignificance, i.e, we may say with Hegelian terminology, "2" is sublated/"aufgehoben".

What remains therefore as mathematics proper, is the study of self-stated axiomatic systems, chosen rules of inference and operations, rather than the study of inherent properties of quantities.


This subtle shift in focus, that is to shift from speculating what numbers are, to study what numbers will "do" under a specified set of axioms is therefore, in my opinion, of the utmost importance to grasp if we are to perform a good philosophical analysis of maths.

The ROLES of "no thing", "finiteness ("one")/infinity) might be assigned to some of the players (numbers) in the logical games mathematicians devise&study, or these roles might not appear, if the mathematician chooses differently.
Thus, they are, essentially "aufgehobene Grössen", rather than the main object of study, i.e, the game itself.

 

[honestrosewater]

Moonrat,
The main point of my questions was just to understand what you were saying. You're using familiar terms in unfamiliar ways and introducing new terms without defining them, so it's very hard to figure out what you mean. I still don't understand most of what you're saying, but here's a relationship to consider (bearing in mind that I don't really understand your terms or what you're trying to do...):
In classical logic, a proposition is tautological if its truth-value is always "Truth" (T); a proposition is contradictory if its truth-value is always "Falsehood" (F); a proposition is contingent if its truth-value can be T or F (but not both). In this way, contingent propositions seem to agree with your "mystery" concept, leaving tautologies and contradictions to be "true" or "false". However, all atomic (the smallest "unit") propositions are contingent, and all tautologies and contradictions are "combinations" of contingent propositions and connectives (i.e. truth-functions). In this way, contingent propositions seem to agree with your "true" concept, while tautologies and contradictions would both seem to agree with "false". This seems to show that "mystery", "true", and "false" can apply to the same object. I know you want to apply these concepts to numbers, but you seem also to want an object to have only one of these properties for each "POV", and this doesn't seem to hold in general.
I don't understand why you dismissed other's points about 0 and 1 having the same property with respect to addition and multiplication, respectively. If 0 has a "non-value" because x + 0 = x, then 1 has a "non-value" because x*1 = x. If "non-value" only applies to addition, why? You want a logical interpretation, so you need a logical reason for the restriction (a definition would work but still seems ad hoc). This is important for your identification of 0 with mystery and 1 with true.
Also, why do you include addition, multiplication, and divison but not subtraction? Because 0 = 1 - 1, making 0 seem to be "false"?

It seems to mean "0 is the value of x if and only if x has no value", but that's a blatant contradiction.

hmm, no more so than 0 in general, I would imagine.

The definition is self-contradictory, period. The problem has nothing to do with 0 but with x both having and not having a value. If that is your definition, then there is no object whose value is 0.

Things that have 'distinctions' have clearly defined borders, boundaries. Indeed, a distinction 'draws' the finite conceptual object that we model it to be.

Have you never watched the sun rise or set? I can distinguish between night and day, but the transitions between them are not clearly defined, but perhaps this isn't what you meant.
Also, if you want to be liberal in your interpretation, you get 1 and 2 simultaneously. The very concept of distinction implies 2 objects: x and not x, each of which is 1 object.
Anyway, you want to consider a certain relationship between 0, 1, and 2. As distinct numbers, they are indistinguishable (in that they are all distinct numbers). The properties you want them to have seem to rest on order and addition, does that seem right?

 

[Moonrat]

Hehe, me and my big mouth! okay, it is probably going to take me a bit for me to be clear about where I follow what the three of your are saying, and where I get lost in confusion. However, I do like the switch around, usually, I am used to understanding people, but them not understanding me..now I get my dream come true, you can understand now me, but I am not so sure I understand you!

Really, it makes me wish the world was run by mathematicians, you all are so neat and orderly and elegant.

So, give me a bit to digest this, and let me respond to this one little point to keep this moving a bit forward...

To be honest, this is a bit of a dream come true, you all are providing me with extraordinary feedback...

The properties you want them to have seem to rest on order and addition, does that seem right?

Yes, that does 'seem' right. And I of course am referencing the 'common' way numbers are referenced. But what I am saying is that there is a pattern to the ‘common way’ and this must be able to be parsed.

I am really only dealing with two themes in ONE set, Finite, and infinite. Naturally, this is an infinite set. This is merely describing the players on an infinite number line. Now, keep in mind that how I define finite and infinite may NOT be how it is defined in mathematical principles, however, I do want to define or parse my definitions into mathematical elements. I define 'infinity' not necessarily as an 'eternal' or anything metaphysical, but rather as something that may have a beginning point, just no ending. It is merely 'continuous'. Thus, 0 and 2 represent the 'continuum' in which the only thing continuing is the 1. it is almost like verb to noun. the noun is the finite 1, and the verb is the 'continuing' or 'continuum', expressed as 0 and 2.

I don’t know how to define 'infinite' other than 'continuous'.

At first, finite and infinite appears as a bivalent set, however, what I then define is that there is finite (1), and finite is expressed or perceived via two modes or basic perceptions, both infinite and not finite, 0 and 2. this makes for 'continuous' combination of 1. It is not that the finite 1 does not exist in the set of 2 through infinity and 0, it is that in the environment that is continuous inside of the set 2 through infinite, '1' is the only 'real' thing that is continuing, and '2 through infinity' are merely 'abstractions of the 1 abstraction' that we use for convenience sake. Like you said

Have you never watched the sun rise or set? I can distinguish between night and day, but the transitions between them are not clearly defined, but perhaps this isn't what you meant.

YES! However, the signifiers ‘night’ and ‘day’ clearly distinguish them! We do this for simplicity’s sake. Thus, the bivalency of night and day create a new distinction of ‘sunset and sunrise’. Sunset and sunrise, as you can ‘see’, is both ‘night and day at once’.

The basic patterning of distinction is; this, that, and that which is both this and that (the meta signifier)

Also, if you want to be liberal in your interpretation, you get 1 and 2 simultaneously. The very concept of distinction implies 2 objects: x and not x, each of which is 1 object.

YES! It is implied…hmmm..perhaps that is what I ‘mean’, it is the ‘implication’, built in, to the numbers 0, 1, and 2. 0 implies ‘no 1 or combination of 1 here’. 1 implies only itself, ‘ I am here’. 2 through infinity are always just implying ‘some combination of 1 here’.

You are most certainly correct, The very second we have ‘one’, we get two distinctions, which we can call ‘subject and object’ if you wish. Now, that is the ‘basic’ of distinguishing perception. The next step is to see how when we get 1, we really get 3 distinctions. 0, nothing, or mystery existed before the 1st distinction, however we could not perceive it or distinguish it yet! And not only that, but we don’t perceive it even though it is ‘here’ before we make our next distinction!

The 0 is always ‘implied’ by the 1, whether we recognize that or not.


And I wish to only suggest that what I am observing or distinguishing is essentially, 'common sense'. It's not rocket science, I don’t expect you to go 'wow, that is some pretty advanced formulae!' however, it is profound...and rational, and very very simple.

Essentially, to give the three of you a bit more idea of where this leads to, is I have been developing this dialectic that is viable for internet discussion. It has a crude method of creating a non zero sum 'environment' in conflict to arrive at win win in the ‘conflict of idea’. So it's like a game, a game of perception, and how we classify all ideas we encounter into the discussion (o, 1, and 2). All players are competing for the 1 how 1 is defined in the dialectic.

Although it is logical and rational, it is not logical in the sense that 'tractucus' is logical and only appeals to those with such a philosophical bent, but rather in the same manner that 'checkers' is a logical game with defined moves, kings, etc etc. So it has 'common sense' appeal.

It is still in crude form, but the bottom line that it's 'crude form' at the foundation is very sound and rational, and very novel and unique. the trick of the dialectic is once you have the simple principle, then you apply it to your perception of idea, even the dialectic, and 'game' conflict in a profound new way. It has strategy, moves, all kinds of things. I have a strong feel for applying it, but describing many of it's principles into purely rational language (math or symbolic) eludes me.


this play on perception made my professor friend comment both in the beginning of our chat and at the end of our chat that what I was describing was, in his opinion, touching upon principles used in quantum computing. To me, that sounds, sure, really 'cool' sounding, however, I don’t know what that really means, or if that helps any of you or not.

I can accept if this is what matt was refrencing me 'personalizing', however, what I am saying is that there is a pattern in the 'personalization' and the 'human element', from the p.o.v. of how we ‘naturally’ perceive.

Okay, you three got my head spinning, let me go back and reread those posts of yours about a hundred more times…

MR

 

[Moonrat]

If you are now going to declare the "infinite environment is 2 through infinity and 0" then I don'et know what you're remotely getting at. .

I am getting the funny feeling that I may be overcomplicating this in an attempt to match all of your expertise, so please bear with me a bit...


Let's say all number sets can be divided into two distinctions. Finite sets, and infinite sets. An infinite set, according to how I am defining this in the dialectic, by default, has two distinctions inside of it that cannot be escaped or excluded. I know that there are many different sets of infinite in mathematics, but this is not how I am defining it here, this is about how infinity is or can be 'percieved' naturally. Infinity, by it's nature, is not 1, but '2'.

First, there would be what we would call the 'ding an sich' of infinity, which can find no 'true' expression. the philosophical question to this would be 'What is the thing in and of itself of infinity? what can infinity be deconstructed down into? How can we define that which is continuous?'

In math, I would assume this is the same principle that there is no 'infinite number' as there is no 'total' in an infinity, right?

So what is this number then? what is the number that can define that which has no final or finite value other than 0? For this, I assign 0. 0 has no final value, no finite ( at least in the environment of order and addition as Honestrosewater suggested) just as infinity has no final value. Indeed, all infinity can be said to be is merely the 'collection' of finites, which are defined as 1's.

Now, since we have defined this infinity has having no grand total, or a value equal to 0, next, we define how we arrive at all numbers that are 'finites' inside of this grand collection. The grand collection of 1's is what is referenced as the '2', false. There is no such thing as a singular '2' in this infinite set. If there was, then there would be an infinite number of '2's none the less. So in this set, we can then define all numbers as being infinite. We could say the same thing about 3, 4, 5, etc etc that we do about 2. 2 is just the first pure false number, and we refrence it for false for simplicity sake since two is the first distinction of 1. Here, there is no distinction between the numbers 2 through infinity, they are all false. there is only 1 true number, and that is 1 itself. all other numbers are abstractions of the infinite defining the finite for the sake of order and simplicity.

So when 2 is defined as false and assigned to another description of infinity, we define this in the extreme literal sense. 2 is false. there is no '2'. there is no such thing, there is merely a infinite collection of 1's, and nothing more.


You mentioned that mathematicians don’t care if the math is 'real' in the sense that if it has representation in the real world, it is only concerned with the 'rules' of how it is defined conceptually.

I accept that,however, how I am using 0, 1, and 2 has every bit to do with the real or objective world. It is the translation to how we perceive the objective world, so it must. I see that where I need the most work is to understand how the conceptual rules overlap, if at all, in mathematics or mathematical principles, but these principles are very 'real' as they function all the time.

Why is 1 not in this set?

From the pov of what I am referencing, 1 is the only number in that set in the true sense (of 0, 2 through infinity). Now, of course I am not saying that 0 is not a number, or 2 through infinity are not numbers, I am only saying that '1' is the only number with 'balls' if follow my drift. Every thing else is either mystery (like the ding an sich of infinity) or false (like every expression of the combinations of 1)

Because it is "the finite"? This is a departure from the usual ideas, and almost certainly wouldn't agree with whatever notion of numbers the bloke from Princeton had.

well, he did agree, but again, I explained it different to him...remember this is how we 'perceive' these numbers not just inside of us in the conceptual sense, that mathematicians do, but how 'we' in the collective sense perceive them outside of us, and how they apply outside of us.

I don't see why you need to make a distinction between different symbols for the same object: 3=1+1+1.

Ahh! it is not the same object from this POV. 3 is the one 'conceptual' object for THREE real or true objects in objective reality.

I have three bucks in my pocket. In my head, I make '1' distinction that I have '3'. however, when I take the money out of my pocket, I now have 'three' distinctions of 1 dollar. 1 dollar + 1 dollar + 1 dollar. There is no such thing as a three dollar bill ;-)

THAT is the only thing I am talking about, that, and no other. The basic perception of number that if it did not exist, we could not percieve very much, much less mathematics!

You are reading more into it than I do (what I termed "personifying", such as assigning terms like mysterious to 0).

that is the point, however, of this particular work, to 'humanize' these rational principles and define how we humanize them mathematically.

If however, all this princetonian was doing was explaining to you that there is a initial ordinal, successor finite ordinals, and that the set of finite ordinals is infinite, then I can see what he was getting at.

yes...that is where I am getting at too, thank you...

To see this, hold up two fingers and then two more fingers and you have "1+1=3" by counting the spaces between them. So 0 is only special with respect to your common notion of addition

great, now your going to keep me up all night again counting my fingers...hehe, 1, 1, 1,1, 1...

MR

 

[CRGreathouse]

Let's say all number sets can be divided into two distinctions. Finite sets, and infinite sets. An infinite set, according to how I am defining this in the dialectic, by default, has two distinctions inside of it that cannot be escaped or excluded. I know that there are many different sets of infinite in mathematics, but this is not how I am defining it here, this is about how infinity is or can be 'percieved' naturally. Infinity, by it's nature, is not 1, but '2'.

For communication purposes, you're saying that you're dividing sets into three categories: the null set, nunnull finite sets, and infinite sets, and that your number system is based on their respective cardinalities.

 

[Moonrat]

I can see a particular argument for 0, 1, and infinity as the building blocks for numbers (taken to be the set \{0,1,2,\ldots\} adjoin infinity): 0 is the base element, and is the successor of no other; 1 is the 'typical' number, the first successor of 0, and the additive building block of the counting numbers (1, 1+1, 1+1+1, ...). Infinity is the only 'number' that is its own successor.

Please let's eloborate on this! That seems to pinpoint this environment. Now, what are the 'distinctions' that make those what you just mentioned? how does 0 imply a base element? how does 1 imply a 'typical' number? How is infinity it's own successor? what are the percievable distinctions that can be observed not just by mathematicians, but by all in this regard?

Notice how both me and you can see the same distinctions, and you describe them above, and me another way, yet the distinctions remain none the less, even outside of our language or pov.

However, outside of some unusual philosophy, I can't see the relationship between 0 and mystery, 1 and true, or 2 and false. That's arbitrary at best. This is better done as abstract algebra, perhaps...

some say symbolic, someone else told me recently propositional calculus with three truth values.

I really don't know, that is where I am stuck!

T, F, U (representing true, false, unknown). Would you like to define for us how you think these should combine?

hmm, well the first thing is to say that they are already combined, the trick is to find where they are distinguished!

True and true is true, mystery and mystery is mystery, but what of the other 7 combinations? Maybe then we can better understand how these relate to numbers and give you input regarding how well 0, 1, and 2 work for these.

I am not sure if I can answer this question properly or not. this may require me to parse something in a language I do not fully understand. I am not even sure how that question would apply because of 'mystery', but let me try this...

First off, mystery is both true and false at once, and as such, we cannot distinguish that which is true, from that which is false. it's mystery, if we could distinguish it, it would not be mystery, but either true, or either false.

Secondly, remember this is a placement for perception, and ideas. So all ideas or perception 'morph' into true, and false, and mystery. Those are just three ways to 'percieve' ideas, each with a distinctive function.

What this dialectic does with these three placements is allows us to 'resolve paradox'.

I don't know if this is going to complicate things, or confuse your question, but in normal bivalency, we are given a choice between truth and falsity. All answers must be either or, and before we can tell, all answers are defaulted to false until proven true. Right?

This relationship only defines the truth, it does not define then what is false, other than false is not true. It does not define what function false serves in relationship to truth, or even outside of truth. we are only given the relationship that false is not true.

however, when we pair false with 'mystery', or unknown, and then allow that to be our dualistic set, false and mystery instead of true and false, well some interesting things begin to occur. The first of which is an objective definition of 'mystery' must take place, and then we must see how 'truth' is 'hidden' in this very elusive relationship between 0, and 2.

And then, the other opposite, mystery and truth. What is the opposition of mystery and truth?

again, these sort of relationships we begin to observe with perception, and when we apply this mathematical 'stigma' to perception, then we must also i nclude ourself into the equation percieving it, no?

So, here are the basic sets of opposition in the dialectic. Now,this is more the advanced stuff, and i am still working out the basics, so again, this is where I also could use some great help and insight.

To resolve all paradox, simple input a tertiary princaple or distinction for that which distinguishes 'both bivalencies' at once, in addition to them seperate.

true and false
false and mystery
mystery and true.

so in perception, to true we can assign objectivity, to false subjectivity. In perception we can assign to 1, order and to 2, choas. To 1 we can assign the distinctions of applying what we percieve, or process, so now we can say that 1 is science, 2 is art...

and mystery is always that which you can't tell which is true, or which is false, where the science ends and the imagination begins, or where there is order, or where there is chaos.

How is it that the word 'mystery' is used, and not 'unknown'? Because mystery, as a signifier of 0, is both true and false, art and science. it is a word that 'invokes' imagination (2, false), yet still defines that there is 'unknown' (1, true)

Now, to bring this into practical application, let's take recent history, war in iraq, and before the invasion, the public dialouge about war. This is certainly not to get political, just refrencing a common shared event to explain something simple.

In jan and feb of 2003, WMD where either true, or they were 'false' right? Note how the perception of 'true' on the idea of WMD altered history one way, and then the perception of WMD as false altered it another way.

now imagine what sort of history we may have if, in basic arguement, common arguement, WMD were allowed to be in the 'truthful' category of 'mystery' until proven one way or another?

Just like we can decontruct all numbers down into a combination of 1's, we can decontruct all ideas down into what is true (1), what is false (2), and what is mystery. And we can do this with complete and utter certainty with the dialectic.

Now i would imagine here is where you mathematicians would start having fun with us 'filosophers', so go to it!

I still have a few more responses to get too, Honestrosewater and arildno have made some powerful suggestions, and I may not get to them til tommorow..


MR

 

[Moonrat]

For communication purposes, you're saying that you're dividing sets into three categories: the null set, nunnull finite sets, and infinite sets, and that your number system is based on their respective cardinalities.

hehe, that sounds good, but which is which? 0, 1, 2?? also, is the spelling on nunnull really non-null?

where do you wish me to send the check?;-)

MR