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Real number axioms

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Could someone please explain to me this axiomisation of real numbers to me? I study further maths at A level and I have not come across this.

 

Is it basically a set of definitions for the various sets of numbers?

 

Cheers

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I split this from the Fermat thread because I don't want it going off-topic - this could easily be in a thread of its own :)

 

Axioms are assumptions that we make - after all, we have to base our initial ideas on some things, and then we can build the rest up from that. In fact, the set of real numbers is an example of something called a field; you can find the axioms at:

 

http://mathworld.wolfram.com/FieldAxioms.html

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Not a problem. If you're wondering about something that's come from a thread, just create another one if it's not on the same track :)

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The real numbers:

 

1. They are a field

 

2. They are ordered

 

3. They are complete.

 

these have subaxioms, however, you don't need to think about these, but understand that the real numbers are the set of numbers which are defined to allow us to do analysis - or if you're doing A-levels, they let us do differentiation by assumption.

 

Roughly they contain the rational numbers, they contain all the possible limits of sequences of rational numbers, these limits are unique (this means , in particular, that 1=0.99.. is a very basic consequence of the axioms, and must be true), that we can add subtract, multiply and divide (except for zero) and that + and * are commutative etc, and that we can compare elements with < and that either x<y, y<x or y=x and exactly one of these holds.

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Just as long as no one claims that 1 and 0.999... are distinct real numbers. Makes me want to scream when they do that.

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I split this from the Fermat thread because I don't want it going off-topic - this could easily be in a thread of its own :)

 

Axioms are assumptions that we make - after all' date=' we have to base our initial ideas on some things, and then we can build the rest up from that. In fact, the set of real numbers is an example of something called a field; you can find the axioms at:

 

http://mathworld.wolfram.com/FieldAxioms.html[/quote']

 

Dave is that list complete?

 

For instance I don't see this one:

 

Non-Triviality

 

not(0=1)

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Non-Triviality

 

not(0=1)

Because it doesn't have to hold in a field. In a field with one element, the additive identity equals the multiplicative identity.

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Because it doesn't have to hold in a field. In a field with one element, the additive identity equals the multiplicative identity.

 

Well what is a field supposed to be in the very first place?

 

You are turning truth into something relative if you do that.

 

not(0=1) is some kind of absolute fact.

 

To say that the additive identity equals the multiplicative identity, changes the meaning of what 0,1 denote.

 

You cannot de-stabilize what is true, and has a truth value which cannot vary in time for any reason.

 

Yes, I've made my point.

 

Regards

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Well what is a field supposed to be in the very first place?
A set that is closed under addition and multiplication (both operations are commutative and associative), where multiplication distributes over addition, has multiplicative and additive inverses, and multiplicative and additive identity elements.

 

You are turning truth into something relative if you do that.
The axioms are not 'absolute truths' as it were, for they cannot be proven to be true, you just have to 'believe' that they are. I suppose it injects a bit of faith into Mathematics.

 

not(0=1) is some kind of absolute fact.
You know that the '0' and '1' I referred to are not integers, or reals, correct? They're just names given to members of the field. If you like, you can call them 'x' and 'y', or whatever you want.

 

To say that the additive identity equals the multiplicative identity, changes the meaning of what 0,1 denote.
It doesn't, because I'm only saying that this is true for elements of the field with one element.

 

You cannot de-stabilize what is true, and has a truth value which cannot vary in time for any reason.
As what I've said earlier indicates, I'm doing no such thing.

 

Field axioms for real numbers

 

Here you see the non-triviality axiom included.

What I mentioned does not apply to the real numbers, since the reals are not a field with one element.

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http://mathworld.wolfram.com/FieldAxioms.html

 

And one more thing...

 

The above list of axioms at wolfram is not minimal.

 

There is nothing logically wrong with giving a list which isn't minimal, its just that once the observation is made, the list of axioms can be reduced. From a memory standpoint it is desirable to have the axioms minimized, so from a human standpoint such a thing is desirable, yet there is nothing inconsistent which will happen by listing too many axioms. here is what I mean though:

 

They give the following statements both as axioms:

 

Distributivity (addition)

 

a(b+c) = ab+ac

 

 

Distributivity (multiplication)

 

(a+b)c=ac+bc

 

And of course they give commutativity of multiplication:

 

ab=ba

 

Now, what they do not do is use the universal quantifier to say what they really mean.

 

And there is a way around that though, which I will use below:

 

Let a,b,c denote arbitrary real numbers.

 

Axiom:a*b=b*a

Axiom: a*(b+c)=a*b+a*c

Axiom: (a+b)*c = a*c+b*c

 

 

However, the third axiom above is a theorem of a system with the first two axioms.

 

 

Theorem: For any real numbers x,y,z: (x+y)*z=x*z+y*z

 

 

Let a,b,c denote arbitrary real numbers.

 

By the closure axiom, b+c is also a real number.

 

Therefore: a*(b+c)=(b+c)*a by commutativity of multiplication using real number a, and real number (b+c).

 

Now using the distributivity axiom it follows that the following statement is true in the axiomatic system under discussion:

 

a*(b+c)=a*b+a*c

 

Therefore, by the properties of equality the following is true:

 

(b+c)*a = a*b+a*c

 

And the numbers a,b,c were arbitrary real numbers, hence the statement above is true for any real numbers a,b,c. That is:

 

"x in R, "y in R, "z in R: (x+y)*z = z*x+z*y

 

Which was to be proven. QED

 

Also note that I used the closure axiom too.

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Originally Posted by Johnny5

Well what is a field supposed to be in the very first place?

 

A set that is closed under addition and multiplication' date=' that has multiplicative and additive inverses, and multiplicative and additive identity elements.

 

[/quote']

 

 

You left out the distributive axiom or no?

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Originally Posted by Johnny5

You are turning truth into something relative if you do that.

 

The axioms are not 'absolute truths' as it were' date=' for they cannot be proven to be true, you just have to 'believe' that they are. I suppose it injects a bit of faith into Mathematics.

[/quote']

 

Let me state my position quickly.

 

Mathematicians use binary logic with those axioms. That means they are treating the axioms as statements, assigning them truth value, and then manipulating the axioms, and statements which follow, using logic.

 

Now, if i were to prove that the axioms are meaningless, then all that would be in vain.

 

Now you aren't going to catch me arguing that the axioms are meaningless.

 

I could argue that numbers don't exist, and so on but it's pointless to do that.

 

Instead I permit them to have their intended meaning.

 

So they are statements, and they do have truth value.

By calling them axioms, we are saying that they are true.

 

And so they must be internally consistent.

 

And with so many minds using them, you figure they are.

 

Yet just the other day, I read an article on some Professor Escultera I think was his name, who reached the conclusion that the field axioms lead to a contradiction.

 

Now, there are still groups, fields, and rings, for me personally to contend with, but that can wait till later.

 

Right now I'd simply like to know if

 

not(0=1) is a field axiom.

 

I think such basic questions should be answered, before one moves onto the great mathematically advanced material.

 

Regards

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You left out the distributive axiom or no?
Yup. My mistake. I've fixed the original post.

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Yup. My mistake. I've fixed the original post.

 

Ok, any others you left out?

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Ok, any others you left out?
I believe associativity and commutativity are all that's left.

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I believe associativity and commutativity are all that's left.

 

 

Why don't you just send me all of your field axioms. Yours personally.

 

That will save us both a great deal of time.

 

Thank you Dapthar

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Why don't you just send me all of your field axioms. Yours personally.
I don't really have a personal version per say, I'm just quoting from memory. If you'd like, I can dig up my old Abstract Algebra book and post those field axioms here. Alternatively, I can just tell you the page number if you already have access to his book; Algebra by Michael Artin.

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I don't really have a personal version per say, I'm just quoting from memory. If you'd like, I can dig up my old Abstract Algebra book and post those field axioms here. Alternatively, I can just tell you the page number if you already have access to his book; Algebra[/url'] by Michael Artin.

 

No i don't have access to that book unfortunately.

 

Hmm.

 

Well I have a list, why don't you criticise mine.

 

Let me see...

 

Let a,b,c denote arbitrary Real numbers.

 

Closure under addition and multiplication:

 

a+b is a real number

a*b is a real number

 

Commutativity of addition and multiplication:

 

a+b=b+a

a*b=b*a

 

Associativity of addition and multiplication:

a+(b+c)=(a+b)+c

a*(b*c)=(a*b)*c

 

Multiplicative and additive identity elements:

 

There is at least one real number called zero, denoted by 0, such that:

 

0+a=a

 

There is at least one real number called one, denoted by 1, such that:

 

1*a=a

Multiplicative and additive inverses:

 

Given any real number a, there is at least one real number -a, called the addive inverse of a, also called "negative a" such that:

 

a+(-a)=0

 

Given any real number a, if not(a=0) then there is at least one real number 1/a, called the multiplicative inverse of a, also called the reciprocal of a, such that:

 

a*(1/a)=1

 

Distributive axiom:

 

a*(b+c)=a*b+a*c

 

Axiom of addition:

 

If a=b then a+c=b+c

 

Axiom of multiplication:

 

If a=b then a*c=b*c

 

Non-triviality axiom:

 

not(0=1)

 

Order axioms

For any real number x: not (x<x)

For any real numbers, x,y,z: if x<y and y<z then x<z

For any real number x, there is at least one real number y, such that x<y

For any real number x, there is at least one real number y, such that y<x

 

And I use some definitions.

 

Is the list above complete, is it minimal? Is there a preferrable set of field axioms? I'm not clueless as to the answers to my own questions, but I am interested in seeing what others call "the field axioms".

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I went and got my textbook "Elementary Classical Analysis"

 

by Jerrold E. Marsden and Micheal J. Hoffman

 

I'm not sure if that's the same Hoffman who wrote my linear algebra text.

 

At any rate he lists the field axioms as:

 

commutativity of addition/multiplication

associativity of addition/multiplication

additive/multiplicative identities

additive/multiplicative inverses

distributive axiom

non-triviality axiom

 

He also lists six order axioms

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