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[Antone]

Is [0] an exception to the reciprocal fraction rule?

For example, [1/2] is the inverse of [2/1]. But [0/1] is said to be zero, while [1/0] is said to be undefined.

.
In my opinion, the answer is NO, [0] does not violate the mentioned reciprocal rule. Here's why:

First, I would say that (except for the ordinal numbers) it is impossible to have a [number] without having a [unit of measure] attached to it. For instance, it is meaningless to say, "I have one," unless the sentence refers back to some previously established object. If someone came up to you and without preamble and said, "I have one," the first question that would spring to mind (after wondering why they were talking to you at all) would be, "One What?"

Even in abstract mathematics (such as the equation 1+1=2) each of these numbers are assumed to be identical units. The size (and type) of that unit may be arbitrary, but it must be the same. We can't add apples and oranges. [1 apple] plus [1 car] does not equal [2] of any specific, physical thing. It is [2 physical objects], but a [physical object] is a vague concept. So [2 physical objects] doesn't really refer to [something specific] or [something physical].

On any given number line, [1] is the unit of measure[/quote]. When not specified, the number [1] represents an arbitrary unit of measure; [2] only has meaning in terms of being two of that arbitrary unit--whatever [1] was.

Second, let's look at a normal case of the reciprocal rule:
When we multiply [5/1] by [1/5], traditional theories say the [5]s should cancel out leaving us with [1/1 = 1]. Canceling out the [5]s, however, is an arbitrary choice. We could just as easily have removed the [1]s, leaving us with [5/5]. When we proceed in this direction we can interpret the math as follows:

• [5/1] means there are [5 wholes] that are [1 unit in measure] in size.
• [1/5] means there is [1 unit of measure] that has [5 equal parts].
• [5/5] can be interpreted to mean that we have [5 parts of a whole that is divided into 5 parts]. Since we have [all five parts of the whole] we have (in essence) the [whole], which is why [5/5] reduces to [1].

Now, If we look at [0/1] and [1/0] in the same way, we might say that

• [0/1] means there are [0 wholes] that are [1 unit in measure] in size.
• [1/0] means there is [1 unit of measure] that has [0 parts](or 0 size).
• [0/0] can be interpreted to mean that we have [0 parts of a whole that is divided into 0 parts].

Once again, however, which number we choose to cancel out is an arbitrary choice. We could just as easily have canceled out the [0]s, leaving [1/1 = 1]. This, of course, implies that we have [1 of something]--which is why mathematics suggests that [0] in the denominator is undefined. For it suggests that [1 something] is equal to [0 parts of 0 wholes].

The resolution to this apparent dilemma is simple, however, if you realize that the [something] we have [1] of is [0]:

What may be less apparent is why we are forced to invert our thinking from [actual reality] to [conceptual reality] when dealing with [0/0], but not when dealing with [5/5].

To understand this, it should be noted that [0] is always reciprocal in nature with respect to the rest of the numbers. Exactly how [0] is reciprocal can vary, depending on what type of number we’re dealing with, but it is always reciprocal in some way. For example, consider the following set:

{2, 5, 9, 6, 2, 4, 7, 3, 9}

This set contains [2 nines] and [2 twos], among other things, but it doesn’t contain a [1], so it contains [0 ones].

Notice that [0] describes [what is not there], while the other numbers describe [what is there]. Clearly, since [0] is describing what is not there, it must be describing something that is not physical—(i.e something that is conceptual).

It is this [reciprocal nature of 0] that forces us to invert our thinking when we are dealing with numbers that have [0] in the [denominator]. So for instance, one way of thinking about what we have when we have [5/0], is to assume that we have [five different ways of thinking about what nothing means].
For example:

• Nothing
• Not anything
• Not some thing
• The absence of something
• the idea of nothing

This means that [5/0] is essentially the [cardinal number 5] where the [unit of measure is 0]. Or, in other words, it is the [ordinal number 5], which is just the [cardinal number 5 when it has been stripped of its [unit of measure], or its magnitude aspect]. Since it has no magnitude aspect (and thus, no unit of measure, which is required for all cardinal numbers) it necessarily has a cardinal value of [0]... Thus, we can interpret [5/0] as [cardinal 0] or [ordinal 5].

Because there are two equally valid ways to interpret this number, it should not be surprising that it can lead to paradox (or problems) when encountered.

Similarly, [0/5] is the [cardinal number 0] where the [unit of measure is an object divided into five equal parts], and we have [0 of those parts]. Or you get the [ordinal number 5], which is basically the [concept zero] subdivided into [five equal parts].

In the foundations of mathematics, the ordinal numbers are said to be constructed from nothing by starting with the [empty set] or [nothing] and then placing that set inside another empty set, and that set inside another and so on.

So if we let {...} equal the empty set we can list the ordinal numbers as follows:

• 1 = {...}
• 2 = {{...}}
• 3 = {{{...}}}
• 4 = {{{{...}}}}
• 5 = {{{{{...}}}}}
• and so on

In other words,

• 1 = nothing
• 2 = the idea of nothing
• 3 = the idea of the idea of nothing
• 4 = the idea of the idea of the idea of nothing
• 5 = the idea of the idea of the idea of the idea of nothing
• and so on

In physical terms, each of these ordinal numbers are physically identical, because they are all various permutations of the idea of nothing. None of them has any more magnitude than any of the others. Yet we have divided the [concept of nothing] into 5 physically equal and conceptually distinct parts.

Now, when you put these together:
[5/0] x [0/5] = you get either the [Cardinal number 0] or the [ordinal number 1]. Or in other words, you get [one 0]... or [1 zero].

In truth, [0] in the denominator IS defined; and it does not violate the reciprocal rule in any way. It is simply not understood that [what you have 1 of] is literally the [0 unit of measure]. And even if it was understood, it would probably be a good idea to avoid [0 in the denominator] anyway, to avoid confusion and mathematical errors.

[Ungomma]

I received a degree in mathematics and nothing I had been learning for N years resembles Antone's musings. It is philosophy, not mathematics.

If you want a mathematical answer - I have never heard of a reciprocal fracton rule, so I don't really know what would constitute a violation.

Long answer:

Peano's axioms define multiplication as:

Code: Select all

a*0 = 0
a*S(b) = a+a*b

Then, x is said to be a result of division of a and b (a/b) if a = b*x.

From this definition:
x = a/b if a = b*x
y = b/a if b = a*y
We multiply those and get:
a*b = b*x * a*y, thus
a*b = a*b * x*y
As we can see, x*y = 1 otherwise the equality wouldn't hold.

But x*y = (a/b)*(b/a)!
QED.

Now, what about zero?

0/a for all a can be defined by above process:
c= 0/a if c*a = 0, therefore c = 0

Not so easy with a/0:
0 = a*0 for all a.

c=a/0 should, according to definition, be defined as
a=0*c, but 0*c=0 therefore a=0, but a is any number, so that's a contradiction. So, we can't define division by zero this way.

Short answer:

Division by zero is undefined because there can be no inverse operation for multiplication by zero.
(a/b)*b = a always. But if (a/0) is meaningful then (a/0)*0 = a, but also (a/0)*0 = 0 regardless of a, and therefore all numbers are zeroes.

So yes, there is no inverse to 0/6 because 6/0 cannot be defined. And if was defined somehow, (6/0)*(0/6) = (6/0)*0 = 0 when it should be 1 because (a/b)*(b/a) = 1.

[Antone]

I received a degree in mathematics and nothing I had been learning for N years resembles Antone's musings. It is philosophy, not mathematics.

This comment doesn't really surprise me. I have a couple of "mathematicians" in my family, and when I tried to talk to them about this they were completely and utterly clueless--and highly resistant to learning anything more than what they already think they know about math.

I assume, however, that you realize there are many branches (or fields) of mathematics. And just because you have a degree in mathematics doesn't mean that you are familiar with all of them. Most mathematics degrees study the practical applications of math. What I was discussing is part of a field that I believe has been called the Foundations of Mathematics. This might also be thought of as the "philosophy of math". So yes, it is at least as much philosophy as it is math.

A good book that discusses this type of thinking is John D Barrow's The Book of Nothing. I believe Charles Seife also discusses similar things in his book Zero, but I couldn't find the specific reference I was thinking of. In any case, I have seen similar thoughts expressed in several different books. So know I'm not the only one who has expressed similar ideas.

I have never heard of a reciprocal fracton rule...

I've never heard of the rules of walking either, but if you don't follow them, you'll fall on your face.

Keep in mind that I AM NOT a mathematician. Everything I've written about I either read in a book or came up with on my own, from my own musings. The so-called reciprocal fraction rule is a term I made up to indicate a readily observable fact, which is that the reciprocal of [1/n] (i.e. 1/(1/n)) is [n/1]. So, for instance, 1/(1/2) = [2/1].

This is an observable fact, that I have called the reciprocal fraction rule. Whether or not anyone else calls it that is completely immaterial.

Peano's axioms define multiplication as:

Code: Select all

a*0 = 0
a*S(b) = a+a*b

Peano's axioms introduce the same basic ideas that I did, just worded in a slightly different way. Peano creates the ordinal numbers using the term successor. So the successor of [0] is [1] and so on.

It should be obvious that these numbers have no magnitude, because [0] CANNOT refer to a physical thing. With [0], we must refer to the things that aren't present. So, if I say there are [0 apples on my desk] that means apples aren't present in the area of my desktop. It doesn't mean there aren't things on my desk top. Similarly, when I say, "I have nothing in my pocket." that is (strictly speaking) false. If nothing else, there is air in my pocket. Even the strongest vacuum is teaming with a great many subatomic particles. Physically speaking then, "What is the successor of that which does not exist," is a meaningless question. And so if Peano were referring to anything physical, his axioms would necessarily be wrong--at least philosophically speaking.

This is why we have to create the ordinal numbers first. We construct these numbers out of [nothingness] and they do not indicate magnitude--only order. Order is not a physical characteristic, it is primarily conceptual in nature. Once we have the names for this order, we can then place the physical objects (of a set) into a one-to-one correlation with the ordinal numbers, and the last ordinal number to be paired up is the cardinal number of the set.

This is not entirely my own thinking--although I think I introduce a number of very important variations. For instance, Barrow (and the others I've seen) suggest that the empty set should correspond to [0] while I believe that it should correspond to [1], since it is the first idea. This is a rather simple change, but I believe it leads to profound consequences in the later development of my theory.

By the way, if you're interested in seeing all of my theory, presented in a detailed and logical manner--instead of jumping in at the middle--drop me a private message, and I'll be happy to let you see it.

Short answer:
Division by zero is undefined because there can be no inverse operation for multiplication by zero.
(a/b)*b = a always. But if (a/0) is meaningful then (a/0)*0 = a, but also (a/0)*0 = 0 regardless of a, and therefore all numbers are zeroes.

So yes, there is no inverse to 0/6 because 6/0 cannot be defined. And if was defined somehow, (6/0)*(0/6) = (6/0)*0 = 0 when it should be 1 because (a/b)*(b/a) = 1.

I'm not going to get into your long answer now, because I don't have time and I'm not a mathematician. But let us look at your short answer.

The long-version of the equation (in your short answer) is:
a/b * b/1 = a/1
The s cancel out leaving a/1 = a/1
So, for instance,
1/2 * 2/1 = 1/1,
In other words, [1 part] of [2 units] * [2 parts] of [1 unit] equals [1 part] of [1 unit].
However, we can also cancel out the [1]s to get
1/2 * 2/1 = 2/2,
In other words, [1 part] of [2 units] * [2 parts] of [1 unit] equals [2 part] of [2 unit], which reduces to 1/1.
But why does it reduce to 1/1 = 1?
Because we must change the units from [wholes] to [halves]--thus, [2 parts] of [2 halves] equals [1 whole].
Why do we have to change the unit to [halves]? Because the denominator of 1/2 can also be understood as [1 whole that is divided into 2 parts]--or 2 halves.

Now, let us consider the specific case of (a/0) where [a=1]:
1/0 * 0/1 = 1/1,
The traditional analysis would produce: [1 part] of [0 units] * [0 parts] of [1 unit] equals [1 part] of [1 unit], because we cancel the [0]s.

Keep in mind, however, that unlike the other numbers, [0] is what isn't there. This means that we can not analyze this equation the same way we would the others. Instead I believe it should go something like this:
[1 part] of [what isn't there] * [no parts] of [1 unit] equals [1 part] of [1 unit with zero magnitude], which reduces to [1 nothings].

And, as we saw with the other formula, we can also cancel out the [1]s to get
1/0 * 0/1 = 0/0,
Which we can analyze as,
[1 part] of [what isn't there] * [no parts] of [1 unit] equals [0 part] of [0 unit], which reduces to [1 nothings].

Think about it, since
1/a * a/1 = 1/1,
then
0/0 = 1/1 = 1.
But what do we have [1] of?
The denominator of 1/0 can be understood as [1 whole that has 0 parts]--or simply [no parts]. So we literally have [0 somethings].
This inversion is possible because [0] is unique among all the numbers, being the [absence of something], instead of the [presence of something]. This gives it the ability to invert the rule:
1/a * a/1 = 1/1,
Instead of 1/1 reducing to [1 something] it must reduce to [1 something that isn't there] or [1 nothing].

So again, I would suggest that it isn't that 1/0 is undefined... it's just that mathematicians do not adequately understand the definition of [0] to begin with. And trying to use 1/0 in an equation is guaranteed to create confusion--and an increased likelihood of error. So it is generally best to be avoided, even though it is not all that difficult to make sense of it.

[Ungomma]

I am leaning heavily towards Wittgensteinian formalism and I really don't think that [0],[1],&c refer to parts, objects or whatever. I hope you understand that from my perspective your ideas are not particulary interesting to me.

Also, it is of no consequence how you designate the empty set, [0] or [1]. It is still the empty set and it fulfills the same function in Peano's system. It is a convention to name empty set zero, but you can call it anything you want. The resultant structures will be isomorphic and that is the usual criteria for identity in mathematics.

[Antone]

I really don't think that [0],[1],etc refer to parts, objects or whatever.

If not, then what do they refer to. Does it make any sense to walk up to someone out of the blue and say, "I have 1"? With no context they'd think you were a loon and what you were saying made no sense. An appropriate response might be, "1 what?"

On the other hand, if you walked up and said, "I have 1 apple," they might think you were strange talking to them, but they would understand what you were saying, even if not why.

Even in abstract math, where there are no explicitly stated units, it is still necessarily assumed that the numbers refer to a set unit. Is [1+1=2] true or is it false. It is only TRUE if you are adding identical units. [1 atom] plus [1 universe] doesn't equal two of anything--except maybe something like [2 ideas].

Without a specific reference to something else, numbers have no meaning except order. And it is meaningless to try to add order. The [third man in the line] PLUS the [fifth man in the line] does not EQUAL the [eighth man in the line]... Nor does it equal [8 men], or anything else related to cardinal number math.

I hope you understand that from my perspective your ideas are not particularly interesting to me.

So why bother responding? Sounds a bit like you're just being contrary and trying to waste both of our time.

Also, it is of no consequence how you designate the empty set, [0] or [1]. It is still the empty set ... It is a convention to name empty set zero, but you can call it anything you want. The resultant structures will be isomorphic...

Actually this is just plainly false. Just because you call two different things by the same name doesn't make them the same thing.

And I think it clearly is important which way you decide to define things. If you define the empty set as [0] then there isn't any correlation between [1] and [0] and so there is no philosophical or mathematical justification for making the suggested transition when dealing with [n/0]. But because the [ordinal number 1] is identical to the empty set (or the cardinal number 0) we can make this transition by simply changing our perspective from the cardinal numbers to the ordinal numbers.

In addition, there are a number of important ways in which this understanding affects mathematics (and set theory) at higher levels. For example,

POWER SETS
the [power set of S] is understood to be the [set of all subsets of S]. Traditionally the power set is held to be larger than the set--but this only makes sense if you ignore the unit of measure. Consider a set of two apples. {A1, A2}... according to traditional theory the power set would be {0, A1, A2, [A1, A2]}

My theory doesn't consider the empty set to be an enumeration set at all, I refer to this idea as the non set. The true empty set can only be an abstraction set. One way to represent it would be:

{x: x is nothing}.

So in my DS theory, the power set would be:

{[A1], [A2], [A1, A2]}

Now, the set contains two apples: [A1] and [A2]. But the power set contains [0 apples]. What it contains are [GROUPINGS of these two apples]. These are ideas, not objects, and as such, they have no physical magnitude. Thus, in terms of magnitude the power set is actually smaller than the set. The power set is only larger than the set in terms of separate and distinct ideas.

Now, my theory has a rule which I call the Principle of Elemental Redundancy which states that set can only have one instance of a given element. So {A1} and {A1, A1} are the same exact set.

Thus, when we turn the power set back into a set that refers to physical objects which can be meaningfully compared to one another we get:

{A1, A2, A1, A2}

The redundant elements drop out leaving {A1, A2} which is the original set.
Thus, the set and the power set are literally the same size. We might say that they are just two different 'names' for the same set--one name focuses on the physical aspect of the set; the other name focuses on a conceptual aspect of the set, which had nothing to do with magnitude of physical objects.

This bit of logic is a fairly radical departure from traditional set theory, and it requires that the numbers be constructed in the manner I prescribe, with the [empty set] being [ordinal number 1]. Otherwise, the logical equivalence of the set and power set doesn't hold.

One reason creating this equivalence is important is because it allows us to easily resolve a number of paradoxes similar to Cantor's and Russell's Paradoxes.

CANTOR'S INFINITE SETS
Another way that it radically changes things is that it sets up an understanding for why Cantor was 180 degrees wrong on every single thing that he said about infinite sets. This is a rather involved discussion (and you've indicated you're not really interested anyway) so I won't go into my ideas on Cantor's fallacies here...

But again, if my ideas are correct, this would seem to be a fairly big deal, for it has a profound impact on how we look at the Continuum Hypothesis and the Axiom of Choice, for example. These ideas have fairly important implications for mathematics, set theory, etc.

[Keen]

There are lot of things you are discussing, but I rather fail to see your point. Anyway I studied maths for three years now and I take a particular interest in maths foundations, so here is how I understand numbers.
Representation of numbers
In mathematics what usually imports are not really how you represent the objects, but the relations between them. You talked about the representation of natural numbers with sets. Yes it has to be done so that mathematicians can have a consistent theory reductible to axioms of the set theory, but once natural numbers constructed you can forget its construction with no serious consequences as long as you know that there is a set that satisfies the Peano's axioms, so to mi mind you shouldn't try to give much meaning to the representation of natural numbers as 0=empty set 1={empty set} 2={{{empty set}}}, or any other representation you can find.
Inversion of 0:
I'm open to any ideas and I personally wouldn't mind dividing by 0, but if so we must realise what it does mean to divide. To divide means to multiply by an inverse number. For instance to divide by two means to multiplyh by 1/2. So we could divide by 0 only if we can give a sense to an inverse of 0. Such inverse would have to satisfy: 0*1/0=1
The problem is, that the multiplication of numbers is made the way that if you multiply ANY number by 0, you obtain 0. You can define another rules I don't care, but if so you wouldn't obtain the same multiplication as we know it. So let's get back to our equality 0*1/0=1. The problem is that when you multiply the number that happens to be the inverse of 0: 1/0 you obtain 0, therefore 0=1. From this you easily deduce that all numbers are equal to 0, now such theory is not interesting at all, so that's why we avoid dividing by zero.

[Keen]

Representation of numbers I agree... which is why I would NEVER, EVER intentionally do such an extremely silly thing.

I use this method to construct the ORDINAL NUMBERS... not the NATURAL NUMBERS.

Well this method is used to construct natural numbers as well. In fact Natural numbers are just a particular case of Ordinal numbers, but the Ordinal numbers are more general so I'll avoid dealing with them at least for now.
What really matters for Natural numbers is that it's a totally ordered infinite set in which every non empty subset has a minimal element.
With only this property you can define what is a successor and with this notion of successor you can define addition.

You focus on the fact that 0=empty set 1={empty set} 2={empty set,{empty set}} 3={empty set,{empty set},{empty set,{empty set}}} and try to give it some meaning, but at least for the construction of natural numbers the choice of empty set really doesn't matter and you can instead of empty set chose a set like {cat}: a set containing a single element which is a cat, then you could define numbers like this:
0={cat} 1={cat,{cat}} 2={cat,{cat},{cat,{cat}}} and you would get absolutely the same construction.

[Antone]

Representation of numbers
In mathematics what usually imports are not really how you represent the objects, but the relations between them.
... once natural numbers are constructed you can forget its construction with no serious consequences...

I don't entirely disagree. That's what allows us to do abstract math. But the ONLY reason you can disregard the construction is because [1] is the unit of measure. Whatever arbitrary unit [1] is, [2] is another one of those units added to the first.

The importance of the unit isn't explicitly stated, but it is still EXTREMELY important. Everyone knows you can't add apples and oranges--not unless you change your arbitrary unit to FRUIT. As fruit, [apples and oranges] are conceptually identical--just as [one apple] is conceptually identical to [another apple].

Because part of the definition of a natural numbers is that [1] is the [unit of measure], you can work the math without considering the units--but that doesn't mean the idea of units isn't critically important. In fact, without the idea of units, numbers would be completely meaningless.

Even Peano's axioms provide a unitary measure, which is succession.

... to my mind you shouldn't try to give much meaning to the representation of natural numbers as 0=empty set 1={empty set} 2={{{empty set}}}, or any other representation you can find.

I agree... which is why I would NEVER, EVER intentionally do such an extremely silly thing.

I use this method to construct the ORDINAL NUMBERS... not the NATURAL NUMBERS. Peano bypasses the ordinals by using the idea of succession, without making any effort to define succession or what succession implies. That's because his system is a formal system, and his construction is not intended to explain anything about what [1+1=2] means. It is merely a collection of symbols and rules that. After creating the formal system, then we can interpret the system by overlay what we understand onto the formal system. But the meaning is not part of the formal system proper.

When explaining the foundations of math this is not a luxury that we are allowed--because our purpose is to explain [what the numbers are], [where they come from], [their relation to each other] and so forth.

When dealing with the foundations of mathematics, we are not allowed to construct the numbers using anything that isn't carefully introduce and fully explained.

This is why we begin our construction with the [idea of nothing]... or the absence of everything. This is essentially the empty set, which can be written as an abstract set as follows:

{x: x is nothing}

We can abreviate as

{...}

This is the ORDINAL number [1], so we can construct the rest of the ordinal numbers by (using my terminology) power setting the empty set.

According to my theoryt: A powerset is a reconceptualization of the original set. So the set
{1,2} can be reconceputalized as the the possible groupings of the numbers in {1,2} or
{[1],[2],[1,2]}. This set contains three elements, but the set defines [groupings], not [numbers]. When we think in terms of numbers, {[1],[2],[1,2] still only contains [2 numbers].
Similarly, the empty set can be changed by making it not an [idea], but the [idea of an idea]. Again, we see that the essential meaning of the power set hasn't changed. It still denotes [nothing], but it is also different because the [idea of something] is not the same thing as the [thing itself].

Using the idea of the power set, we can construct the ORDINAL NUMBERS as follows:
{...} = 1
{{...}} = 2
{{{...}}} = 3
and so on.

[Ordinal Numbers] have no MAGNITUDE, just as the sets that are used to construct them have no magnitude. All they denote is order. The order of the successive power sets.

We create the [natural numbers] by placing the [ordinal numbers] into a one-to-one correspondence with a set of conceptually identical elements.

Again, this is why the unit is critical. The set, {Apple, Orange} does not equal [2] when using the [Apple] or the [Orange] unit of measure. But it does equal [2] when using the [Fruit] unit of measure.

Inversion of 0:
I'm open to any ideas and I personally wouldn't mind dividing by 0, but if so we must realise what it does mean to divide. To divide means to multiply by an inverse number. For instance to divide by two means to multiplyh by 1/2. So we could divide by 0 only if we can give a sense to an inverse of 0. Such inverse would have to satisfy: 0*1/0=1

Which it does, as I previously explained.

We have [1 unit]... and that unit is [nothing].
So we literally have [1 nothing].
This inversion of the unit is necessary, for the reasons I previously explained--so I won't repeat the whole thing again.

The problem is, that the multiplication of numbers is made the way that if you multiply ANY number by 0, you obtain 0.

Exactly... and my analysis DOES NOT violate this fundamental rule, since [1 nothing] is the same as [nothing] or [0 somethings].

In other words, at least with respect to magnitued,
[1 ordinal unit] is the same as [0 cardinal units]

Notice also that it doesn't matter what the other number is. 0*7/0=7 is also true, because [7 nothings] is still [nothing].

Moreover, this is not all that different from:
0*7=0. Here again, we have [7 units of nothing]= which has a magnitude of [0].
With 0*7/0=7. 7/0 explicitly states that the unit is [0 magnitude], because the denominator is [0].
The numerator tells you how many parts you have.
The denominator tells you how many parts [1], the [unit of measure] is divided into.

1/2 means that [1] is divided into [2 parts] and we have [1] of those two parts.
1/0, means that [1] has [0 parts].
[1] has [1 part], so [0 parts] is the same as [nothing]; and we have [1 of those non-existent parts].
It doesn't matter how many times you add [nothing] you still have [nothing].

let's get back to our equality 0*1/0=1. The problem is that when you multiply the number that happens to be the inverse of 0: 1/0 you obtain 0, therefore 0=1. From this you easily deduce that all numbers are equal to 0, now such theory is not interesting at all, so that's why we avoid dividing by zero.

Obviously!
As I have just shown,
[ordinal number 1] = [cardinal number 0].
[ordinal number 2] = [cardinal number 0].
[ordinal number 3] = [cardinal number 0].
[ordinal number 4] = [cardinal number 0].
...
[ordinal number n] = [cardinal number 0].

All [Cardinal numbers] have some magnitude,
All [Ordinal numbers] have zero magnitude

Conversely, [Cardinal numbers] do not imply order.
If you have ten apples, none of those apples is designated as the [first apple], etc.
So if we have two apples, [A1] and [A2],
the following sets

{A1, A2} and
{A2, A1}

have the exact same cardinal value.

See new thread for discussion of the nature of multiplication

[Keen]

I have studied a bit the ordinals (which isn't the easiest notion to grasp in the set theory) and if you don't mind I'll point out a few mathematical mistakes you made, before commenting on your ideas.
You defined ordinals as:

# 1 = {...}
# 2 = {{...}}
# 3 = {{{...}}}
# 4 = {{{{...}}}}
# 5 = {{{{{...}}}}}

The usual definition of ordinals is rather that any of their member is also their subset, that means
0=empty set
1={0}={empty set}
2={1,2}={empty set,{empty set}}
3={0,1,2}={empty set,{empty set},{empty set,{empty set}}}
and so on...
the reason is simply technical: it is much easier to order such sets rather the ones you have written, but it's more or less the same(up to isomorphism),so not really important.

[ordinal number 1] = [cardinal number 0].
[ordinal number 2] = [cardinal number 0].
[ordinal number 3] = [cardinal number 0].
[ordinal number 4] = [cardinal number 0].

I'm not sure what exactly you mean by these equalities, but cardinality of your ordinals except for the empty set is 1, because all of you sets you used to define ordinals have one element: in fact the only set that has 0 cardinality is the empty set.

I agree... which is why I would NEVER, EVER intentionally do such an extremely silly thing.

I use this method to construct the ORDINAL NUMBERS... not the NATURAL NUMBERS.

As a matter of fact even though ordinal numbers is a much more general class then natural numbers, there is no difference between them when you consider only the ordinals you mentioned, because the definition of natural numbers is that they are finite ordinal numbers.

We create the [natural numbers] by placing the [ordinal numbers] into a one-to-one correspondence with a set of conceptually identical elements.

If you want to place two sets into one-to-one correspondence with another set, you first need to prove that such bijection exists. And as I don't know what you mean by conceptually identical elements, I can hardly tell whether what you are doing is right or wrong.

Now that I pointed out the problems in your construction I'm going to comment on your ideas.

The beauty of mathematics is that they don't care too much about the objects, but more about how these objects are related.
Mathematics are about defining rules and proving what these rules imply. Mathematicians usually start from a common notion, like distance volume space and generalize them by defining abstract rules.
When you require a unit of measure you already make an interpretation of these rules, so that they can fit better into our world, but nevertheless these rules do not require anything like a unit of measure.
+: can be defined as an operation on two elements a and b of a set that is associative, commutative and has a zero. Yes a and b have to be in the same set, nevertheless you can put any objects in the same sets and define + on them.

You are upset about the fact that Peano's Axioms are a formal system and that they don't really explain what numbers are. Well honestly all I can tell you is that Natural numbers is any set that obeys to the Peano's axioms and is unique up to isomorphism, i.e no matter how you call your objects, no matter what meaning you try to give them, the only thing important about them from mathematical point of view are the rules they obey to.
When you did the construction of natural numbers, you just exchanged one formal system (Peano's arithmetic) for another (Set theory). And again the only thing I can tell you about a set is that it's anything that obeys to Zermelo's Axioms (You can include the axiom of choice, but it's not necessary for construction of natural numbers).

[Antone]

The usual definition of ordinals is rather that any of their member is also their subset, that means
0=empty set
1={0}={empty set}
2={1,2}={empty set,{empty set}}
3={0,1,2}={empty set,{empty set},{empty set,{empty set}}}
and so on...

If you read my post carefully, then you surely noticed that I pointed this fact out myself.

The point of my post is to show a new way of understanding the [ordinal numbers] and some of the advantages that this new way presents for us to understand other difficult mathematical scenarios.

...it is much easier to order such sets rather the ones you have written, but it's more or less the same(up to isomorphism),so not really important.

I disagree.
First, it is not easier. Using the traditional strategy you must count the number of elements in the set to determine the [ordinal number]. Using my strategy, you count the number of set brackets...
{{{...}}}
Equals the cardinal number 3.
Simple as can be. And logical.

Second, as I've said before, it is not (in my opinion) UNimportant. It provides a valid logic for resolving a number of very important mathematical situations. As I've stated several times.

I'm not sure what exactly you mean by these equalities, but cardinality of your ordinals except for the empty set is 1, because all of you sets you used to define ordinals have one element: in fact the only set that has 0 cardinality is the empty set.

True, according to traditional definitions. And I apologize for using terms in unconventional ways. But the traditional way of defining things doesn't really make any sense, to me. So I define cardinal sets somewhat differently.

As I see it, a cardinal number determines [magnitude], or how much of something we have. Thus, the cardinal number of the enumeration set, {x} ...where [x] = [the idea of nothing]... is in fact [0] because it doesn't contain anything physical.

Similarly, the cardinal number of a set
{Apple1, orange1} depends on what the unit of measure is. If the unit is [apples] then the cardinal number of the set is [1]; same for [oranges] as the unit of measure. But with [fruit] as the unit, the cardinal number is [2].

It makes no sense to ignore the unit, because then you're adding apples and oranges and the cardinal number is rendered meaningless. To have a cardinal number [2] you have to have a second unit of the [unit of measure].

Again, I realize this is an unconventional interpretation. But I think it makes things work much better, and leads to much more reasonable conclusions with respect to many things, including Cantor's infinite sets and so forth.

As a matter of fact even though ordinal numbers is a much more general class then natural numbers, there is no difference between them when you consider only the ordinals you mentioned, because the definition of natural numbers is that they are finite ordinal numbers.

This is completely, totally and utterly false.
The dictionary defines as follows:

ordinal number any number used to indicate order. ...Distinguished from the CARDINAL NUMBERS.

cardinal numberany number used in counting or in showing how many. Distinguished from the ORDINAL NUMBERS.

No math authority believes that the ordinal numbers are essentially the same as the natural numbers--although traditionally, the Cardinal numbers and the Natural numbers are held to be very similar--and there is some variation on how certain people see these numbers.

For example, sometimes [natural numbers] are held to include [0]; other times not. I seem to recall hearing a distinction that [cardinal numbers] do not contain [0] while [natural numbers] do.

My own personal theory holds that the cardinal numbers are the ordinal numbers placed into a one-to-one correspondence with the "like" elements of a set. So, for example, a [bag that contained 10 marbles] would represent the cardinal number [10].

By contrast, the natural number 10 would be represented by an empty marble bag, into which 10 marbles are placed.

In other words, the [natural numbers] are essentially the union of the [cardinal numbers] and the [plus sign]... or the idea of being added.

On the number line, we can see the distinction between the numbers. Lets consider the number [3]:
Ordinal number 3 is like three points, with no line. {...}
Cardinal number 3 is like three unconnected line segments, with no line. {_ _ _}
Natural number 3 is like a single line segment that is three units long: {___}.

The NUMBER LINE can be said to simultaneously represents all the numbers. The hash marks represent the ordinal numbers. The cardinal numbers sees each hashmark as a break that divides the number line into infinitely many line segments--all exactly equal to the first line segment [1]. The natural numbers sees the number [3] as the [line segment from 0 to the ordinal number 3].

This way of understanding the numbers allows us to understand the irrational numbers in a completely new way. Traditionally, the irrational numbers are understood to be a set that is infinitely larger than the [set of all fractions]. But my theory understands the irrational numbers as a (relatively small) subset of the infinite numbers.

Whereas the [natural numbers] represent a line segment, the [infinite numbers] represent a single point on the number line. In a sense, they are very like the [ordinal numbers], because they have no [magnitude]. But instead of [order], they define a specific relative position on the number line.

The real numbers combine the natural numbers (Plus the negative integers) with the infinite numbers... creating line segments [from 0 to the infinite number].

One interesting consequence of this construction is that the [set of infinite numbers] (which is demonstrably larger than the set of irrational numbers) is necessarily the same size as the [set that defines the natural numbers].

And yes, I realize this is a radically different understanding from current math theories... That's why it's MY THEORY!

If you want to place two sets into one-to-one correspondence with another set, you first need to prove that such bijection exists.

This is nonsensical.

To place the elements from one set into a one-to-one correspondence with another set, all you need to do is pair the elements. If both sets run out of elements at the same time, they have a one-to-one correspondence.
{1, 2, 3}
{2, 6, 9}
have a one-to-one correspondence because the elements can be paired as follows:
1...2
2...6
3...9
Neither set has any more elements, so they are in a one-to-one correspondence. That and that alone is what a one-to-one correspondence means. There is no need to worry about bijunctions or any other such nonsense.

And as I don't know what you mean by conceptually identical elements, I can hardly tell whether what you are doing is right or wrong.

Any [given apple] is different from any [other apple]. However close to the same they are, they will almost certainly have slightly different weights. They may have different stem lengths, number of seeds, different coloration, shape, skin thickness and so forth. Thus, no [second apple] is ever an exact second of the first apple. However, the two apples are conceptually identical, and so we can add them, even though they aren't physically identical.

The beauty of mathematics is that they don't care too much about the objects, but more about how these objects are related.

That is one opinion.

I believe the beauty of mathematics should be that it makes sense, and explains something about the real world. Which, is (in fact) "how these objects are related". If math isn't about objects, then it can't be about how those objects are related--because the [relation of the objects] is necessarily ABOUT the objects.

When you require a unit of measure you already make an interpretation of these rules, so that they can fit better into our world, but nevertheless these rules do not require anything like a unit of measure.

The rules of mathematics demand a "unit of measure". This is not a term that I made up.

Formal systems attempt to present a system using only a topographical structure (symbols and rule with no interpretation)--and some attempts have been made to create a formal system for mathematics, but Godel's incompleteness theorems (if you accept them) clearly seem to indicate that this isn't really possible for mathematics.

Besides, I believe that my theory of mathematics could be reduced to a formal language--with at least as much success as traditional versions have had. But frankly, turning math into meaningless symbols and rules isn't beautiful--in fact, it strips math of the beauty that it does contain.

+: can be defined as an operation on two elements a and b of a set that is associative, commutative and has a zero. Yes a and b have to be in the same set, nevertheless you can put any objects in the same sets and define + on them.

I have no idea what your point is here. Doesn't seem to address anything I said, other than perhaps to agree with what I said in a oblique sort of way.

You are upset about the fact that Peano's Axioms are a formal system and that they don't really explain what numbers are.

LOL... you're talking nonsense again.

First, I'm not upset.
Second, Peano's axioms are entirely compatible with everything that I've said. It's just another perspective--another way of saying the same thing. Peano's axioms are a more formal (and limited) presentation, but they certainly doesn't conflict.

[Keen]

If your theory is consistent, then it's fine to me. Let me just point out few last things:

First, it is not easier. Using the traditional strategy you must count the number of elements in the set to determine the [ordinal number]. Using my strategy, you count the number of set brackets...

Yes of course with these brackets comes the idea of succession, which behaves exactly like the succession in Peano's axioms, so you can define the order relation just like in Peano's theory by induction, but then you'd have to prove that it's really an order relation. In traditional construction you have such an order relation for free: it's the set inclusion, because each set not only belongs to it's successor but is also a subset of it's successor, but as I said it's purely a technical problem, so not really worth debating.

As for the cardinals ordinals and naturals: the traditional definitions, that can be found in the books of the set theories is:
-cardinal is an equivalence class of sets that can be put into one-one correspondence.
-ordinal is a set such that each of it's elements is also it's subset.
(or it can also be defined as a member of a well ordered set)
-natural number is an element of N, which is the smallest set that contains the empty set and that has the following property:
if x belongs to N then x U {x} belongs to N.
You can then easily prove that all elements of N are ordinals, that they are finite and that there are no other finite ordinals.
As long as you work with countable sets these three notions are almost the same:
Order and Naturals:
You simply identify the first element with the number 0, the second with 1, the third with 2 and so on.

Cardinality and Naturals
Given any finite set you simply how many elements it has and you associate a natural number to it.

Things get different when dealing with uncountable sets, but then these notions become rather complicated.

To place the elements from one set into a one-to-one correspondence with another set, all you need to do is pair the elements. If both sets run out of elements at the same time, they have a one-to-one correspondence.
{1, 2, 3}
{2, 6, 9}
have a one-to-one correspondence because the elements can be paired as follows:
1...2
2...6
3...9
Neither set has any more elements, so they are in a one-to-one correspondence. That and that alone is what a one-to-one correspondence means. There is no need to worry about bijunctions or any other such nonsense.

When you did this you have actually proven that there is a bijection between sets {1,2,3} and {2,6,9}.
But this method doesn't work for infinite sets and natural numbers is one of them. You could hardly place the ordinals in one to one correspondence like you did with the finite sets, because you would never run out of elements. You would have to prove there is a one-one correspondence between your set of ordinals and the set of what you call conceptually equal elements: whatever this is.

Formal systems attempt to present a system using only a topographical structure (symbols and rule with no interpretation)--and some attempts have been made to create a formal system for mathematics, but Godel's incompleteness theorems (if you accept them) clearly seem to indicate that this isn't really possible for mathematics.

What Godel's incompleteness theorems state is that given any mathematical language that is rich enough to contain arithmetic do have theorems that are true, but not provable. The second one states that given any axiomatic system rich enough to contain arithmetic you can't prove the consistency of this system. Yet you still can have a purely formal system. What you can't have is an algorithm that would decide given any statement of this system, whether it is true or not and also you can't be sure that your formalization is without contradictions.

Anyway I respect the point of view that mathematics should explain something in the real world. It would just be a pity to limit them only to reality as there are many interesting things to discover in the formal language of maths and that are purely abstract notions not linked(at least not directly) with reality: like infinite dimensions, integral domains, sigma algebras and so on. Many of these abstract notions are useful in informatics, cryptology and for solving problems more directly linked with the real world.

[Antone]

In traditional construction you have such an order relation for free: it's the set inclusion, because each set not only belongs to it's successor but is also a subset of it's successor, but as I said it's purely a technical problem, so not really worth debating.

As I understand it, Peano's axioms are the rules of a formal system. As such, there are no sets or subsets involved. In fact, there aren't even any numbers involved. These are interpretations that we give to the formal system. The formal system is a good one if it allows for a one-to-one correspondence between itself and what we wish it to model--i.e. our interpretation.

When we remove Peano's axioms from this formal framework, then the sets you are talking about are (if you ignore the minor differences I introduce about [empty set] = [1] instead of [0]...) the same as the sets that are used to construct the ordinal numbers.

The axioms, as they are traditionally understood--crush several steps of number building together. But this is only possible because we already understand numbers intuitively. If you want to understand how this intuitive construction process is accomplished, then you need to do each step independently.

It's like the difference between hitting the [delete] button to get rid of text in a word processor and doing the same thing in computer language. Hitting [Delete] might seem like a single step, but it actually takes several lines in the programming language. In practical applications we can ignore these many extra steps, but when we're considering the "Nuts and bolts" of what is going on, we have to consider each step in the programing.

s for the cardinals ordinals and naturals: the traditional definitions, that can be found in the books of the set theories is:

It should be noted that the traditional definition for these terms will be slightly different, depending on the discipline you're working in. In traditional math, I don't believe a [cardinal number] has anything to do with a set.

-cardinal is an equivalence class of sets that can be put into one-one correspondence.

I'm not quite sure what this even means. What is being put into one-one correspondence with what?

My logic book defines an equivalence class as: ]

If [R] is an equivalence relation in a [set A], and [a] is an element of [A], the equivalence class generated (with R) by a, written [a]R, is the set of objects to which [a] bears [R].

Using this language, the equivalence relation is something that defines elements as being similar. For example, a set of people might contain 2 people who are defined by the equivalence relation [weighs the same as].

Thus, if we have the set:
{x1, y, x2, z}
where x, y and z represent different weights (for different people) and [x1] and [x2] are two diff people who weigh the same... then, for this set, the equivalence class (with respect to x1) is:
{x1, x2}

Now, what does it mean for the cardinals to be an equivalence class of sets that can be put into one-one correspondence? I have no idea.

An equivalence class is formed when we create a subset with similar elements. My best guess is that this means we are necessarily dealing with a set that MUST contain elements that are conceptually similar (just as I stipulated). But your definition appears to to imply that we have two such sets, and that these sets must be placed in a one-one correspondence. This, however, has nothing at all to do with what it means to be a cardinal number. If I have two buckets of marbles, and I remove one marble from each bucket, one at a time until all the marbles are gone from one of the buckets, and I observe that the other bucket is empty at the same time, then the two buckets are in one-one correspondence. But I have no idea how many marbles were in either bucket. All I know is that they contain the same number of marbles.

If, however, we place the like elements of a set into a one-one correspondence with the ordinal numbers, then we have created something meaningful. If I have removed all the marbles from a bucket, placing each marble into a one-one correspondence with the ordinal numbers, and the last marble I removed corresponds to the ordinal number [5] then I know the bucket contained [5 marbles]--and that is the cardinal number of the marbles (that were) in the bucket.

-ordinal is a set such that each of it's elements is also it's subset. (or it can also be defined as a member of a well ordered set)

Again, I'm not sure how this relates to an ordinal number.

I assume you made a typo... and meant to say, "..a set such that each of it's elements is also a subset." In which case, it's saying the same thing I said, without the long explanation. Or, you could have meant to say, "..a set such that each of it's elements is also its own subset." If you define subset the way I do, then again, it says the same thing I said--but this is not true using the traditional definition of a subset.

-natural number is an element of N, which is the smallest set that contains the empty set and that has the following property:
if x belongs to N then x U {x} belongs to N.

I have no idea what this might mean since I have no idea what is. I assume that what you're trying to say is that if [x] is a natural number then [x+x] is also a natural number. However, this doesn't tell us anything at all about what x is allowed to be. Thus, it doesn't do anything to define what it means to be a natural number.

You can then easily prove that all elements of N are ordinals, that they are finite and that there are no other finite ordinals.

This makes absolutely no sense at all. As far as I can tell, nothing you've said suggests that any element of N is an ordinal--let alone all of them.

When you did this you have actually proven that there is a bijection between sets {1,2,3} and {2,6,9}.
But this method doesn't work for infinite sets and natural numbers is one of them. You could hardly place the ordinals in one to one correspondence like you did with the finite sets, because you would never run out of elements. You would have to prove there is a one-one correspondence between your set of ordinals and the set of what you call conceptually equal elements: whatever this is.

Again, this makes no sense.

The purpose of the [ordinal numbers] is essentially to give the "numbers" their names. The cardinal number [1] is called [1] because the first ordinal number is called [1]. Each set that you place into a one-one correspondence with the ordinals is called the number of that ordinal with which it last corresponds; and that is the [cardinal number] of that set. There is no need to create this "bijection" you demand because it already exists by virtue of the construction of the [cardinal numbers]. You can't name the cardinal numbers until you've named the ordinals...

Now, the question of placing an infinite set into a one-one correspondence is interesting. But I disagree with Cantor. It makes no sense to arbitrarily place elements into one-one correspondence.

To my way of thinking,
{1, 2, 3, 4, 5...}
does not have a one-one correspondence with
{2, 4, 6, 8, 10...}

We know that both sets contain [2], both sets contain [4], both sets contain [6] and so on....

Thus the reasonable way to pair the elements would be
set 1: {1, 2, 3, 4, 5, 6...}
set 2: {....2,.... 4,... 6,...}
Because now we are pairing [like elements] with [like elements]. Because each element in [set 2] has a corresponding [like element] in [set 1], we know that all the like elements will line up, and [set 1] will have left over the set,
{1, 3, 5...}
which does not have corresponding elements in [set 2].
Furthermore, because there is exactly one non-corresponding element in [set 1] for every corresponding element, we can say that [set 1] is exactly 2x as large as [set 2].

This is a much more logical conclusion than the one reached by Cantor--and it fits perfectly with the rest of my theory.

Anyway I respect the point of view that mathematics should explain something in the real world. It would just be a pity to limit them only to reality as there are many interesting things to discover in the formal language of maths and that are purely abstract notions not linked(at least not directly) with reality: like infinite dimensions, integral domains, sigma algebras and so on. Many of these abstract notions are useful in informatics, cryptology and for solving problems more directly linked with the real world.

I don't disagree.

But as I've said, I believe my theory is compatible with being formalized. And I also think that formal systems can be rather sterile. I believe it is the interpretations that we give to the formal systems that are interesting.

Also, I should note that my construction doesn't limit itself to only what is firmly tied to reality. The entire set of ordinal numbers is built from a concept that is essentially the opposite of physical reality. i.e. the [idea of nothing]. And, my theory can give rise to plenty of abstract numbers, imaginary numbers, complex numbers, and so forth.

Last post, I introduced an totally NEW type of abstract number that I called the infinite numbers. This has no counterpart in traditional interpretations of mathematics. And it, in fact, conflicts with traditional understandings of numbers...

As I said, the [infinite numbers (between 0 and 1] )are the same size as the [natural numbers]... yet my theory holds that the [irrational numbers] is a subset of the [infinite numbers]. One "proof" for this claim is quite simple.

Mirror Counting. Start with the infinite number [.1000...] then [.2000...], [.3000...] and so forth. But when you reach [.9000...] the next numbers are [.01000...], [.11000...], [.21000...], [.31000...], and so forth.

Now, we can place each natural number into a one-one correspondence with an infinite number; I'll separate them with a decimal and leave the [000...] off of the infinite numbers. This produces:

1.1
2.2...
3.3
...
10.01
11.11
12.21
13.31
...
20.02
21.12
22.22
23.32
...

This one-one correspondence is a process that can never actually be completed, but if it ever were completed, each [natural number] would have one and only one corresponding [infinite number] and each and every one of those [infinite numbers] would have infinitely many decimal places.

Only some of the numbers (on the infinite number side) of this list would be [irrational numbers]. Thus, the [irrational numbers] is a subset of the [infinite numbers] and yet the [infinite numbers] are in one-one correspondence with the natural numbers.

This contradicts Cantor's notions of infinite sets.

I have another, more detailed 'proof' if you're interested... let me know and I'll take the time to send it. The basic idea is relatively simple, but the proof itself is relatively long.

[Keen]

As I understand it, Peano's axioms are the rules of a formal system. As such, there are no sets or subsets involved.

You are right, but I wasn't talking about Peano's axioms, but about Natural numbers as they are constructed in set theory.

Now I'll explain what meant the terms I used to define the notions cardinal, ordinal and natural number.
Cardinal numbers.
It's a way of measuring sets. If the sets are of "the same size" you just say that they are equivalent. It is very easy to measure finite sets. You just count how many elements they have. Then two sets are equivalent if and only if they have the same number of elements, that's why you can identify cardinal number and natural number simply by counting the number of elements as long as you are dealing with finite sets. When dealing with infinite sets, things become more complicated, because you just can't count their elements. So you say that two sets are equivalent, if you can put their elements in one-one correspondence. You regroup all of these equivalent sets into one class and you call them cardinal number.
For instance: natural numbers and even numbers are in the same equivalence class, because there is a simple one-one correspondence between them: 0->0, 1->2 2->4, 3->6 etc...

Ordinal numbers
They were invented to explain order, but a very particular one. Sets like Natural numbers can be ordered in a way that each number has a successor and there is nothing between a number and it's successor. Ordinal numbers have this same property, but there is much more ordinal numbers then Natural numbers. As modern mathematics tend to describe every object as a set, you can define ordinal numbers as sets X such as every element of X is also a subset of X. Then the successor of X would be: X U {X} which contains everything that contained X, and also contains X itself (U stands for union of sets). It's clear then that X is a subset of it's successor.
Natural Numbers
Are constructed exactly like ordinals, but we just stop at finite ordinals.
We start with empty set, which is an ordinal and call it 0.
Then we create the successor of 0: 0 U {0}={empty set}=1
Then we create the successor of 1: 1 U {1}={0,1}={empty set.{empty set}}=2. As you can see every successor contains all of it's predecessors. When we collect all of finite sets of the type {0,1,2,...,n} we regroup them into one infinite set that we call Natural numbers.

Finally one of the main reasons why mathematical systems are completely formalized is because in mathematics you need to be absolutely precise about what you are saying. The common language does not provide such degree of precision and can be very ambiguous. That's why thanks to the work of mathematical logicians like Bertrand Russel, mathematicians invented a formal language in which every statement you say can be reduced to a statement about sets without any ambiguity. Then using this language, mathematicians describe first common notions like volume, counting, distance, deformation, but as the problems get more and more difficult these notions must become more and more abstract until they get to a degree where they have little in common with the vague idea one can have about them.[/u]

[Antone]

Thank you for defining your terms more clearly. There are, of course, several variations of set theory, and authors often use different terms and notation. Even when you've read the book from the beginning, their awkward language can make it difficult to follow what they're trying to say. Thus, I think it is best,to clearly define the terms we are using; and to keep our language as simple and straight forward as possible. I realize that's not always easy.

Cardinal numbers.
It's a way of measuring sets. If the sets are of "the same size" you just say that they are equivalent. It is very easy to measure finite sets. You just count how many elements they have. Then two sets are equivalent if and only if they have the same number of elements, that's why you can identify cardinal number and natural number simply by counting the number of elements as long as you are dealing with finite sets. When dealing with infinite sets, things become more complicated, because you just can't count their elements. So you say that two sets are equivalent, if you can put their elements in one-one correspondence. You regroup all of these equivalent sets into one class and you call them cardinal number.
For instance: natural numbers and even numbers are in the same equivalence class, because there is a simple one-one correspondence between them: 0->0, 1->2 2->4, 3->6 etc...

Ordinal numbers
They were invented to explain order, but a very particular one. Sets like Natural numbers can be ordered in a way that each number has a successor and there is nothing between a number and it's successor. Ordinal numbers have this same property, but there is much more ordinal numbers then Natural numbers. As modern mathematics tend to describe every object as a set, you can define ordinal numbers as sets X such as every element of X is also a subset of X. Then the successor of X would be: X U {X} which contains everything that contained X, and also contains X itself (U stands for union of sets). It's clear then that X is a subset of it's successor.
Natural Numbers
Are constructed exactly like ordinals, but we just stop at finite ordinals.
We start with empty set, which is an ordinal and call it 0.
Then we create the successor of 0: 0 U {0}={empty set}=1
Then we create the successor of 1: 1 U {1}={0,1}={empty set.{empty set}}=2. As you can see every successor contains all of it's predecessors. When we collect all of finite sets of the type {0,1,2,...,n} we regroup them into one infinite set that we call Natural numbers.

Finally one of the main reasons why mathematical systems are completely formalized is because in mathematics you need to be absolutely precise about what you are saying. The common language does not provide such degree of precision and can be very ambiguous. That's why thanks to the work of mathematical logicians like Bertrand Russel, mathematicians invented a formal language in which every statement you say can be reduced to a statement about sets without any ambiguity. Then using this language, mathematicians describe first common notions like volume, counting, distance, deformation, but as the problems get more and more difficult these notions must become more and more abstract until they get to a degree where they have little in common with the vague idea one can have about them.[/u]

[Owen]

Yes.
If we define x/y =df (the z: x=y*z) then, 1/0 = (the z: 1=0*z).

But, it is a theorem that 0*z = 0 for all z.
That is, there is no number z such that 1=0*z, ie. 1/0 is defined and it is not unique.
That is to say, 1/0 is not equal to any unique value of z...1/0 does not exist!

0/0 = (the z: 0=0*z), by the above definition.

But, 0=0*z is true for all numbers z.
That is, (the z: 0=0*z), 0/0, is also defined and it is not unique.
That is to say, 0/0 does not equal any unique value of z...0/0 does not exist!

Therefore, we can assert: x/0 is defined and does not exist, for all x.