Math Is Fun Forum

anonimnystefy

Real Member

From: Harlan's World

Registered: 2011-05-23

Posts: 16,049

Axioms of Q?

Hi

Can somebody tell me what the axioms of the rationals are exactly?

Last edited by anonimnystefy (2012-06-26 01:26:26)

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“Here lies the reader who will never open this book. He is forever dead.
“Taking a new step, uttering a new word, is what people fear most.” ― Fyodor Dostoyevsky, Crime and Punishment
The knowledge of some things as a function of age is a delta function.

Bob

Administrator

Registered: 2010-06-20

Posts: 10,582

Re: Axioms of Q?

hi Stefy,

Don't you have access to google ?

http://www.physicsinsights.org/numbers-rational-1.html

Looks a good place to start although he does say it's not 'standard' whatever that means.

Bob

---

Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob

anonimnystefy

Real Member

From: Harlan's World

Registered: 2011-05-23

Posts: 16,049

Re: Axioms of Q?

I don't see how those are different from the axioms of integers.

---

“Here lies the reader who will never open this book. He is forever dead.
“Taking a new step, uttering a new word, is what people fear most.” ― Fyodor Dostoyevsky, Crime and Punishment
The knowledge of some things as a function of age is a delta function.

Bob

Administrator

Registered: 2010-06-20

Posts: 10,582

Re: Axioms of Q?

I'm not surprised.  I expect you could create a suitable 1:1 mapping to show they are the same algebraically.  Note:  I'm guessing so this might be rubbish.

I'll leave it as another challenge for you.

Bob

---

Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob

anonimnystefy

Real Member

From: Harlan's World

Registered: 2011-05-23

Posts: 16,049

Re: Axioms of Q?

Hi Bob

There already exists a proof about the equivalence of the sets N and Q.

But there should be some axioms that are special for Q, since it is a different set from N.

---

“Here lies the reader who will never open this book. He is forever dead.
“Taking a new step, uttering a new word, is what people fear most.” ― Fyodor Dostoyevsky, Crime and Punishment
The knowledge of some things as a function of age is a delta function.

Bob

Administrator

Registered: 2010-06-20

Posts: 10,582

Re: Axioms of Q?

hi Stefy,

I didn't mean Cantor's proof that they're the same size.

What I had in mind was this:

Find a bijection F between {integers} and {rationals} such that

if a, b and c are members of {integers} and @ is any binary operation defined on the field {+ - x ÷} then

a @ b = c if and only if  F(a) @ F(b) = F(c)

If I've understood the term correctly, this would make the two sets isomorphic, from which it would follow they have (need) the same axioms.

All I'm asking is, find me an F.

Bob

---

Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob

anonimnystefy

Real Member

From: Harlan's World

Registered: 2011-05-23

Posts: 16,049

Re: Axioms of Q?

What if there isn't one?

---

“Here lies the reader who will never open this book. He is forever dead.
“Taking a new step, uttering a new word, is what people fear most.” ― Fyodor Dostoyevsky, Crime and Punishment
The knowledge of some things as a function of age is a delta function.

Bob

Administrator

Registered: 2010-06-20

Posts: 10,582

Re: Axioms of Q?

Well now you are at the crux of the matter.  Mathematics is like that.

I have no idea if F exists.  Maybe it does; maybe it doesn't.

If you find an F, that settles it.  If you don't .... well maybe an F can be found but you haven't found it yet.

Or you prove it is impossible to have such an F.

Welcome to the world of mathematical research. 

Bob

---

Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob

anonimnystefy

Real Member

From: Harlan's World

Registered: 2011-05-23

Posts: 16,049

Re: Axioms of Q?

This is the kind of stuff that can give you a nasty headache.

But how are we supposed to know if the sets N and Q are isomorphic if we don't know what it is (we haven't got any of its axioms).

---

“Here lies the reader who will never open this book. He is forever dead.
“Taking a new step, uttering a new word, is what people fear most.” ― Fyodor Dostoyevsky, Crime and Punishment
The knowledge of some things as a function of age is a delta function.

Bob

Administrator

Registered: 2010-06-20

Posts: 10,582

Re: Axioms of Q?

I'll give it some thought. 

Bob

---

Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob

anonimnystefy

Real Member

From: Harlan's World

Registered: 2011-05-23

Posts: 16,049

Re: Axioms of Q?

Ok. xD

---

“Here lies the reader who will never open this book. He is forever dead.
“Taking a new step, uttering a new word, is what people fear most.” ― Fyodor Dostoyevsky, Crime and Punishment
The knowledge of some things as a function of age is a delta function.

Bob

Administrator

Registered: 2010-06-20

Posts: 10,582

Re: Axioms of Q?

OK.  Some thinking later.

The axioms cannot be the same and no F  exists.

The rationals are closed for + - x ÷

The integers are not closed for ÷

Have a look at what the integer axioms say about the inverse of multiplication.

Bob

---

Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob

anonimnystefy

Real Member

From: Harlan's World

Registered: 2011-05-23

Posts: 16,049

Re: Axioms of Q?

Only 1 has it.

But I still don't know what the all of the Q axioms are.

---

“Here lies the reader who will never open this book. He is forever dead.
“Taking a new step, uttering a new word, is what people fear most.” ― Fyodor Dostoyevsky, Crime and Punishment
The knowledge of some things as a function of age is a delta function.