0 is not a number

[wtf]

5 * 0 = 0. Multiplying any number by zero results in zero. So, zero is said to "absorb" the multiplication. This is a characteristic of the neutral element for the addition.

Right. And a roll of paper towels absorbs spills. So I'm asking you if you are saying that 0 is like a roll of paper towels?

If there exists an element x as such that for every element a of the domain, a + x = a, then multiplying by this neutral element x will be absorbed into x, i.e. a * x = x.

Look at a roll of paper towels with one end directly facing you. What does it look like? It looks just like a thick copy of the number 0, does it not?

It works not just with numbers. It will for example also work in a ring of elliptic-curve points. In fact, it will work in a ring or a field of any type of objects.

Well sure, you can use paper towels to wipe up a mess in a field. Why are you telling me such obvious things? What point are you making?

[Atla]

0 is not a number. 0 is just a state of nothing.
Proof: Numbers can have - (negative) signs.

a = -0

5 + a = 5-0 = 5

Oh look it still works!

a= -0
It just proves that 5 + a = 5 - 0 = 5 was a waste of time.

So is multiplying by 1

[godelian]

Well sure, you can use paper towels to wipe up a mess in a field.

When you use the roll of paper towel to absorb (⨂) a fluid, there is a neutral element, emptyfluid, that you can add (⨁) to any fluid fl1 while the resuit would remain fl1:

fl1 ⨁ emptyfluid = fl1

That works. However:

roll ⨂ emptyfluid ?= emptyfluid

is problematic.

If you attempt to absorb an empty fluid with the roll, then the roll does not become an empty fluid. No, it just stays the roll:

roll ⨂ emptyfluid = roll

That is where the analogy breaks down. The neutral element for addition (⨁) emptyfluid is not the absorbing element for multiplication (⨂). It is again a neutral element, while it should absorb.

[Age]

0 is not a number. 0 is just a state of nothing.
Proof: Numbers can have - (negative) signs.
0 cannot be negative or positive.
It follows that 0 is not a number.
Because 0 is not a number, no number can be divided, multiplied subtracted or added with 0.

Okay.

But was there any point to pointing this out here, exactly?

[wtf]

That is where the analogy breaks down. The neutral element for addition (⨁) emptyfluid is not the absorbing element for multiplication (⨂). It is again a neutral element, while it should absorb.

I think I've been out-trolled.

[godelian]

0 is not a number. 0 is just a state of nothing.

There may be two operations attached to a number-like system. The notion of zero (nz) needs to be neutral for one operation (#) and absorbing for the second operation (@):

So, for each element x in the closed environment of set A:

(1) x # nz = x

(2) x @ nz = nz

In that case, nz is zero-like in the algebraic structure of (A, $, @).

So, nz is not a "state of nothing". It is a legitimate and necessary element of the number-like set A.

[Zarathustra]

0 is not a number. 0 is just a state of nothing.
Proof: Numbers can have - (negative) signs.
0 cannot be negative or positive.
It follows that 0 is not a number.
Because 0 is not a number, no number can be divided, multiplied subtracted or added with 0.

Okay.

But was there any point to pointing this out here, exactly?

Natural number starts with 0, and goes on counting 1,2,3 .... n.
0 shouldn't be included, because it is not a number.

[godelian]

Natural number starts with 0, and goes on counting 1,2,3 .... n.
0 shouldn't be included, because it is not a number.

According to axiom 8 in Peano Arithmetic theory (PA), 0 can indeed not be reached by succession:

https://en.wikipedia.org/wiki/Peano_axioms

8. For every natural number n, S(n) = 0 is false. That is, there is no natural number whose successor is 0.

This does not mean that 0 is not a natural number, because that is exactly what axiom 1 axiomatizes:

1. 0 is a natural number.

The universe of the natural numbers is the standard model that interprets PA: Without the presence of 0 in the model, no model can interpret PA.

[Age]

0 is not a number. 0 is just a state of nothing.
Proof: Numbers can have - (negative) signs.
0 cannot be negative or positive.
It follows that 0 is not a number.
Because 0 is not a number, no number can be divided, multiplied subtracted or added with 0.

Okay.

But was there any point to pointing this out here, exactly?

Natural number starts with 0, and goes on counting 1,2,3 .... n.
0 shouldn't be included, because it is not a number.

So, just start from 1 then if you like.

And, what is the difference between 'natural numbers' and 'numbers'?

[godelian]

And, what is the difference between 'natural numbers' and 'numbers'?

Natural numbers can be reached through consecutive succession: 1,2,3,... , except for 0 which can only be axiomatized to be a natural number.

The natural numbers are closed under addition and multiplication. So, adding two natural numbers or multiplying them results again in a natural number.

Natural numbers are not closed under substraction. For example, 3 - 5 = -2, is not a natural number but an integer.

Natural numbers are also not closed under division. For example, 3/10 = 0.3, which is not a natural number but a rational.

With additional operations, you can reach new kinds of numbers, until you reach the set of computable numbers, which is a subset of the real numbers, most of which can however not be reached in any computable way.

[Skepdick]

That is where the analogy breaks down. The neutral element for addition (⨁) emptyfluid is not the absorbing element for multiplication (⨂). It is again a neutral element, while it should absorb.

I think I've been out-trolled.

Quod troglodytam ipse est troglodytam

[Impenitent]

crosses be damned, 0 is nought a number...

-Imp

[Age]

And, what is the difference between 'natural numbers' and 'numbers'?

Natural numbers can be reached through consecutive succession: 1,2,3,... , except for 0 which can only be axiomatized to be a natural number.

The natural numbers are closed under addition and multiplication. So, adding two natural numbers or multiplying them results again in a natural number.

Natural numbers are not closed under substraction. For example, 3 - 5 = -2, is not a natural number but an integer.

Natural numbers are also not closed under division. For example, 3/10 = 0.3, which is not a natural number but a rational.

With additional operations, you can reach new kinds of numbers, until you reach the set of computable numbers, which is a subset of the real numbers, most of which can however not be reached in any computable way.

Since you did not actually answer the actual clarifying question I posed, and asked here, and since you added a new one I will ask again;

What is the difference, now, between 'natural numbers', 'numbers', and, now, 'real numbers'?

[godelian]

What is the difference, now, between 'natural numbers', 'numbers', and, now, 'real numbers'?

Natural numbers: can be reached by successive succession, starting from zero: 0, 1,:2, 3, ...

Regular expression: [0-9]+

For a different base, appropriately replace the digit matching pattern [0-9].

Integers: allow for negatively-valued natural numbers.

Regular expression: [+-]?[0-9]+

Real numbers: numbers that can be represented with infinite (decimal) expansion:

Regular expression: [+-]?[0-9]+([.][0-9]*)?

These definitions reflect the formalist ontology in which mathematical objects are symbol strings.

https://en.m.wikipedia.org/wiki/Formali ... thematics)

In the philosophy of mathematics, formalism is the view that holds that statements of mathematics and logic can be considered to be statements about the consequences of the manipulation of strings (alphanumeric sequences of symbols, usually as equations) using established manipulation rules.

In that sense, a number type is a set populated with a particular string pattern. Some number types can be matched by a simple regular expression.

[Age]

What is the difference, now, between 'natural numbers', 'numbers', and, now, 'real numbers'?

Natural numbers: can be reached by successive succession, starting from zero: 0, 1,:2, 3, ...

Regular expression: [0-9]+

For a different base, appropriately replace the digit matching pattern [0-9].

Integers: allow for negatively-valued natural numbers.

Regular expression: [+-]?[0-9]+

Real numbers: numbers that can be represented with infinite (decimal) expansion:

Regular expression: [+-]?[0-9]+([.][0-9]*)?

These definitions reflect the formalist ontology in which mathematical objects are symbol strings.

https://en.m.wikipedia.org/wiki/Formali ... thematics)

In the philosophy of mathematics, formalism is the view that holds that statements of mathematics and logic can be considered to be statements about the consequences of the manipulation of strings (alphanumeric sequences of symbols, usually as equations) using established manipulation rules.

In that sense, a number type is a set populated with a particular string pattern. Some number types can be matched by a simple regular expression.

So, is 0 a number or not?