If there is no BIGGEST number, is there a SMALLEST?

[Guest loarevalo]

The problem is this:

 

Set Theory, and the whole of Mathematics, requires that there be no Absolute Infinity defined, no number that could be called "the last number" or the biggest. Yet we use an "smallest number" - the number zero.

 

If there is no largest number, no absolute infinity, no last number (and the consistency of mathematics rests on this premise), it is only intuitive for symmetry's sake to suggest that, likewise, there must not be a smallest number, zero.

 

Philosophically speaking, the ideas "one-half" and "two" are equivalent, though we differentiate them mathematically by the particular meaning attached to them, like we differentiate between "having two" and "not having two." Both "negative two" and "two" allude to the number "two," and both are equivalent in the sense that they are both that same "two," only in different contexts, thus acquiring their different meanings. Likewise "one-half" and "two" allude to the same number and are equivalent ideas in that sense - "one-half" could not exist without "two." It follows that if such an absolute infinity is not subject to determination or definition, so does not an absolute infinitesimal, which we have named zero.

 

Nonetheless, zero does exist, in the sense that infinitesimals do exist because infinite numbers do also. One minus one is still zero, if by zero we mean "some infinitesimal," if zero is defined as the number (or any number) that added to a finite number it is equivalent to the same finite quantity. Likewise, a characteristic of a transfinite number is that such a number when added to a finite number, results in another transfinite quantity (this suffices though we may also say 'it is the same transfinite number').

 

I have not found, and there may not be, a rigorous proof that zero does not exist in mathematics. But I can see how we would do so much better if we did not appeal to an absolute zero. Because we do, I suppose, is the current hinder of all versions of Analysis, because we have not yet defined within the system a true infinitesimal. Nonstandard Analysis's so called infinitesimals are not infinitesimals, but numbers smaller than the smallest reals - "thinner" than reals. A comprehensive system for infinitesimals would disallow the use of an absolute zero, as classic set theory disallows absolute collections.

 

Besides metaphysics, besides the nature or definition of zero, one point I want to make clear: the infinitesimal is defined under the notion and definition of infinity if we are to define the term 'infinitesimal' as 'infinitely small.' The true infinitesimal, added to a finite, results in that same finite - not an approximate finite. Thus far I have not learned of any system that defines true infinitesimals.

 

If for an infinite N: N + 1 = N

Then for an infinitesimal n: 1 + n = 1 or 1 + 1/N = 1

 

NSA's infinitesimal n: 1 + n ~ 1 or std(1+n) = 1

One plus n is infinitely close to one.

 

This insight may bring us to greater understanding of many mischeavous behaviours like division by zero, and the called indeterminate forms. If the absolute zero is not defined, division by the same is not defined (as it currently is); for poet's sake we may say that division by zero is the absolute infinity, but that also is undefined. Because by using zero we usually mean something other than the absolute zero (or not necesarily the absolute zero), zero divided by zero is undetermined not due to the operation or form, but to the undetermined nature of zero. As we mean many things (allude to many numbers) by the generic "infinity" so we mean various things by a simple "zero" - that could be a specific infinitesimal, an undetermined class of infinitesimals, or even the absolute zero.

 

Thank you for your attention. If you know of any reasoning that deals with these issues, please inform me of such.

[Guest loarevalo]

Forgot to tell:

 

These are ideas that I have thought over for many years. Most of the initial entry is an actual argument that I have sent to many colleagues - It didn't come up from the blue in five minutes!

[Turtle]

___Welcome to the Forum. Off the cuff philosphically, I reason 1 is the smallest number (say quantity) & also the largest. I also say philosphically off the cuff, the number 1 is the first prime number & the first perfect number. It is all you need to contruct all numbers which is the 1 complete set of numbers. 1 is the smallest number, not zero. :)

[Guest loarevalo]

Thanks for the welcoming.

 

I am so glad you think that 1 is the first number - I didn't mean that it wasn't. Actually, the number system that I would propose builds from 1 and not zero. (Well why? Because zero doesn't exist). Set Theory (and mathematics officially) constructs the number system starting at zero.

 

Are you saying that an Absolute Zero does exist?

[UncleAl]

If there is no largest number, no absolute infinity, no last number (and the consistency of mathematics rests on this premise), it is only intuitive for symmetry's sake to suggest that, likewise, there must not be a smallest number, zero.

10^(-1)+10^(-2)+10^(-3)+10^(-4)+10^(-5)+10^(-6)+10^(-7)+... = 1/9 *exactly*

 

What is the trend limit when taking reciprocals of the largest allowed numbers? Zero is a perfectly fine and exact limit. It's also the identity element for addition and therefore necessary not merely convenient. Hell man, you only need the smallest infinity , the number of itnegers, and that is exactly countable.

 

http://en.wikipedia.org/wiki/Natural_number

"History of natural numbers and the status of zero"

 

You can play your word games with the number of real numbers or the number of functions drawable through a point.

[Turtle]

Are you saying that an Absolute Zero does exist?

I honestly don't know (both(or either) if I was saying that, and if there is an Absolute Zero (sic)).

___The Wickpedia article Al linked to played the word game of differentiating between natural numbers as either counting or ordering. My post made the philosphical proposition from the side of counting. Now as I mull it over I thought of this wordplay: 1 is the least you can have, but zero is the most you can't.

[Qfwfq]

Actually, the number system that I would propose builds from 1 and not zero. (Well why? Because zero doesn't exist). Set Theory (and mathematics officially) constructs the number system starting at zero.

 

Are you saying that an Absolute Zero does exist?

What is the solution of the following equation:

 

x = a - a

 

for a given value of a? If you like, we can say for a = 3, or for a = 11 etc...

[Tormod]

loarevalo, welcome - and great post.

 

I disagree, however. We base our entire current communications systems on the binary system - 0 and 1. "0" in binary is not the absense of a value, it is a real value. While it obviously has a positionary value (ie, it shows whether the 1, 2, 4, 8, 16, 32, 64 etc positions in a binary number are given), it also is used in logical gates. If value=0 then do this, otherwise do that.

 

It is also relevant in technologies like data storage, compression, cryptography etc.

 

In base 10, it has both a value as an absense indicator, and as a positionary value. This is actually what makes our numbering so much better than the Roman numeral system (which has no zero, IIRC). A zero indicates the beginning of a counter.

 

So, 1000 is the whole number after 999 and the number before 1,001. All those three zeroes play an important role.

 

Likewise, the value 0.1 indicates that there is a value higher than zero but less than 1. Thus our counting system starts at 0.

 

The negative numbers are simply mirrors of our counting system. They are not "real" - there is no "minus 3" in nature, but there is a 0 in nature. The planck length would constitute a zero in spacetime, for example.

 

These are not absolute positions, I am just trying to chime in.

[Guest loarevalo]

What is the solution of the following equation:

 

x = a - a

 

for a given value of a? If you like, we can say for a = 3, or for a = 11 etc...

 

Honestly, I don't disagree with what you all have said. What all have said is true - heck! it's in the books and has been taught forever.

 

You see,

 

if x = a-a, then x = 0.

 

Consider the solutions to the equation:

 

a + X = a , a is a integer

 

It is quite obvious that X = 0. But that X doesn't necesarily have to be the Absolute Zero, X could be an Infinitesimal. The problem then is this, Mathematicians haven't yet defined true infinitesimals which would behave like this:

 

a + 1/INF = a

 

Yes, I am implying that 1/INF = 0

But that is because this symbol "0" now becomes something as fuzzy as the symbol of the "eight laid down" for infinity. That symbol could stand for any infinite, aleph-null, etc... The symbol "0" would be such a notation, denotating instead of a single number, a class of numbers.

 

Such an infinitesimal as I am suggesting WAS considered 300 years ago, but dismissed for the paradox:

 

How can a number both be equal to zero (a + X = a) and yet not equal to zero (so we can distinguish it from zero).

 

Actually, there was something key that they didn't understand then: That such an infinitesimal was in reality not the Absolute Zero, but was in a sense a zero (an infinitesimal).

 

Consider then this equation:

 

X + a = X , where a is an integer

 

X then must be an infinite number. But is it that simple? Is it just Infinity? Well, we all know there are many kinds of infinity. Could such an X solve this equation and at the same time not be the BIGGEST number? Yes, there many kinds of infinity, and all of these are less than the Absolute Infinity.

 

Sorry for the mess; I tried my best to explain that confusing initial post.

[C1ay]

The problems is then, Mathematicians haven't yet defined true infinitesimals which would behave like this:

 

a + 1/INF = a

 

Yes I am implying that 1/INF = 0

But that is because this symbol "0" now becomes something as fuzzy as the symbol of the "eight laid down" for infinity. That symbol could stand for any infinite, alef-null, etc... The symbol "0" would be such a notation, denotating instead of a single number, a class of numbers.

I disagree. 0 < 1/INF/2 < 1/INF

[Tormod]

I'd like to recommend a really nice book called "The Nothing That Is - A Natural History of Zero http://www.amazon.com/exec/obidos/tg/detail/-/0195142373" by Robert Kaplan. A very well written book about the history of the concept of zero.

 

As for this "Absolute Zero" thing it seems like a nonsensical thing to me (no offense, loarevalo). The way you present it it sounds as if it is a Holy Grail that people are looking for. Whether zero exists in nature is an interesting question but hardly a mystery.

[C1ay]

I'd like to recommend a really nice book called "The Nothing That Is - A Natural History of Zero" by Robert Kaplan. A very well written book about the history of the concept of zero.

I enjoyed Zero: The Biography of a Dangerous Idea http://www.amazon.com/exec/obidos/tg/detail/-/0140296476/qid=1122764441/sr=8-1/ref=pd_bbs_1/104-7820831-1475128?v=glance&s=books&n=507846 by Charles Seife & Matt Zimet as well. It is a similar coverage of the history of zero.

[Guest loarevalo]

I disagree. 0 < 1/INF/2 < 1/INF

 

Thanks for responding. I know this assertion does touch the core belief of some.

 

If it is true that

 

0< 1/INF

 

then what you mean by "0" is what I call the Absolute Zero.

 

All I am saying is that BOTH an infinitesimal (1/INF) and the Absolute Zero are solutions to the equation:

 

a + X = a , a is an integer

 

With that understood, we can go on to hypothesize that maybe, we do NOT need to define the Absolute Zero, as we do not need to define an Absolute Infinity.

 

(This a weaker response.

Actually, we need to leave the Absolute Infinity undefined for consistency).

[C1ay]

All I am saying is that BOTH an infinitesimal (1/INF) and the Absolute Zero are solutions to the equation:

 

a + X = a , a is an integer

I recommend you study limits in calculus and then return to this. With a better understanding of the epsilon-delta proof you will abandon this absurdity. The only number X that satisfies the equation a + X = a is 0, just plain 0, it doesn't need any special name like absolute zero.

[Guest loarevalo]

 

The negative numbers are simply mirrors of our counting system. They are not "real" - there is no "minus 3" in nature, but there is a 0 in nature. The planck length would constitute a zero in spacetime, for example.

 

 

Hey, Thanks for that analogy. I am asserting exactly the same:

 

As there is no real "minus 3," likewise there is no real "1/3." The negative numbers are mirrors of the naturals, and so are the rationals: 1/2, 1/3, 1/4 ... though this is a bit harder to understand. As there is no real "minus 3" only 3, there is no real "1/3" only 3 - which seems perfectly reasonable if you remember how children think initially of 1/3.

 

As you say, there is no real "minus 3" only 3, there is no real "infinitesimal" only infinity. Since the real Absolute Infinity, we can directly infer that there is no Absolute Zero. That's it.

 

Is it so terribly complex to grasp? Too weird and yet too simple to accept?

 

...

 

Thank you Tormod for that reference "Zero: The Nothing That Is." I picked that book some months ago and read some chapters, not all - it was very philosophical and poetic. I would recommend it - but the book didn't confront the problem in Analysis this day:

 

Mathematicians just don't know how to define Infinitesimals.

 

As you know, there are many versions: Non-standard Analysis, Internal Set Theory, Nill-square approaches (far more controversial than what I say), Synthetic Differential Geometry, etc...

 

I don't mean to declare myself the next Cantor, or Russell. I am just pointing out what I think to be obvious and logical that stem from that fundamental idea of Set Theory: that there is no Absolute Infinity, no greatest number.

[Tormod]

I don't get what you are trying to achieve. First you say "There is no Absolute infinity", then you set out to squash Absolute zero, whereas it seems to me zero is a very real value whereas infinity is a very real concept. I do not believe in "Absolutes" and I think you need to find some other argument than "it is what we were taught". I was not taught so...but then again I am neither a scientist nor a mathematician.

 

As for 1/3 it is indeed a real value even if it is not a whole number. It is rather fruitless in my opinion to weed out "real" vs "unreal" numbers - even if this is done in guise of philosophy. 1/3 marks the distance where 2/3rds of a path remains, for example. It is a real value. But if I were to mark the same spot "-2" I would simply be saying that there are "2 more distances like this to go".

 

It is easy to confuse the term "number" with "value". Any number specifies a value and thus represents something, real or not. The smallest value will depend on which system you use (which I think you pointed out yourself), thus there either are many "Absolute zeroes" or there are none - depending on how one wishes to interpret it. I fail to see that there are dragons to slay here.

[Guest loarevalo]

I recommend you study limits in calculus and then return to this. With a better understanding of the epsilon-delta proof you will abandon this absurdity. The only number X that satisfies the equation a + X = a is 0, just plain 0, it doesn't need any special name like absolute zero.

 

I agree. The solution is plain 0. It's just that I am saying that plain 0 is like plain infinity - a bit too vague and general. I just like to differentiate that plain 0, from an absolutely smallest quantity (I know, it's confusing).

 

I did study Calculus; I learned it from reading my sister's college Calc book when I was in ninth-grade. Sorry, I didn't mean to brag, just to disipate some false ideas.

 

The epsilo-delta proof and notation of limit is perfectly fine. Traditional Calculus does NOT use nor define infinitesimals (nor does it use Infinity); Calculus just works with the Real Numbers. Other versions of Analysis do attempt to define infinitesimals. So, Calculus has little to contribute to this discussion, because Calculus (and limits) only deal with the finite.