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

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.

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.

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]

Originally Posted by loarevalo

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]

Originally Posted by loarevalo

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.

Originally Posted by Qfwfq

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]

Originally Posted by loarevalo

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