Field axioms with or without closure

[honestrosewater]

I'm not studying algebra yet, I just happened to notice this and am curious. Mathworld's entry for the field axioms doesn't include closure axioms, but I have seen other authors include closure axioms in the field axioms. Does anyone know why this is or what difference it makes? Can closure be deduced from the other axioms?
Without the closure axioms, could you prove, for instance, that the sum and product of a nonzero rational number and an irrational number are irrational? The only way I know how to prove this is with the closure axioms.

 

[Hurkyl]

Which axioms do you mean by closure axioms? Are you referring to the requirement that if a and b are field elements, then so are a+b and a*b?

Usually, those are rolled up into how + and * are specified: functions FxF --> F.

 

[honestrosewater]

Which axioms do you mean by closure axioms? Are you referring to the requirement that if a and b are field elements, then so are a+b and a*b?

Yes.

Usually, those are rolled up into how + and * are specified: functions FxF --> F.

Oh, okay. Thanks.

 

[fourier jr]

+ & * are binary operations, which always leave an algebraic system closed

 

[stevefaulkner]

Hi,

The Field Axioms prescribe the theory of fields which is a first-order theory. First-order theories don't need an axiom for closure although one is often shown. An axiom for closure for groups is not needed either, although one is almost always shown. The reason for one not being needed is that all first-order theories are modeled by mathematical structures. The structures modelling the Field Axioms are the fields under operations of addition and multiplication. A structure is closed in any case.

Steve Faulkner
Foundations of the Quantum Logic
http://steviefaulkner.wordpress.com

 

[sponsoredwalk]

Hi,

The Field Axioms prescribe the theory of fields which is a first-order theory. First-order theories don't need an axiom for closure although one is often shown. An axiom for closure for groups is not needed either, although one is almost always shown. The reason for one not being needed is that all first-order theories are modeled by mathematical structures. The structures modelling the Field Axioms are the fields under operations of addition and multiplication. A structure is closed in any case.

Steve Faulkner
Foundations of the Quantum Logic
http://steviefaulkner.wordpress.com

I keep coming across this distinction of what a first order theory is but honestly do not
understand it. Would there be an easier way to learn this distinction you mention as
regards fields & closure without having to read an entire book on logic? Would you be
able to recommend a good book on logic that would ensure I could specifically
recognise the distinction you've made once I've read it? Do you think these books:

https://www.amazon.com/dp/0486638294/?tag=pfamazon01-20

https://www.amazon.com/dp/0387302948/?tag=pfamazon01-20

would allow me to specifically recognise the distinction as regards fields, closure etc...?
It's just annoying reading these things & pretending you know what's going on

 

[stevefaulkner]

I'm so fed up with this site. I just wrote a long reply to you and as I submitted it, I was told I had been automatically logged out!

Hi, thanks for your question.

I'm afraid I can't help you anything like as much as i'd like to. There is a gulf between physicists and logicians. Genrally logicians are interested in gaining further understanding of their subject rather than making it accessible to physicists like me. That said, they are already aware of answers to problems in physics but physicists are not listening.

I learned much of the logic I know from an Open University course: http://www3.open.ac.uk/study/undergraduate/course/m381.htm I didn't attend the course I just bought the course units second hand. You'd probably find a copy on the web somewhere. The units do teach rather than list what you need to know.

I understand Joe Mileti is writing a book on logic for mathematicians. His website: http://www.math.grin.edu/~miletijo/ suggests he will send a copy of his draft to people who write to him. If he sends you a copy I'd be very pleased to see it.

My paper: http://www.vixra.org/pdf/1101.0045v1.pdf covers basic concepts and discusses differences of approach between first-order theory and mathematical physics. Have a look there at the way I have written the Field Axioms. First-orde theories consist of propositons rather than equations.

The wikipedia is a good first reference so long as you know a few terms to look for. Read about bound variables, free variables, quantifiers, propositions, sentences.

Do persevere with this. I think it will become important in physics. If you do, keep in touch, there aren't many of us. I don't use this site now, so use my email: StevieFaulkner@googlemail.com

Best wishes, Steve.

 

[sponsoredwalk]

I'm so fed up with this site. I just wrote a long reply to you and as I submitted it, I was told I had been automatically logged out!

Been there alright

Very helpful response thanks, I'll just go at logic straight on over the next few months &
if I still can't answer this question I'll send u an e-mail, I must say that

This article is one of a series explaining the nature of mathematical undecidability
discovered present within quantum mechanics. The centrepiece of the project
is an axiomatised version of quantum theory, which derives indeterminacy
and furnishes a mechanism for measurement.

from your article is a very interesting idea & something I'll definitely be checking out
over the next few years.

 

[Office_Shredder]

I keep coming across this distinction of what a first order theory is but honestly do not
understand it.

The basic idea of first order logic is that you can make mathematical statements, but you can only quantify over elements. For example, you could say every natural number has a prime factorization, because "every natural number" is talking about individual elements. However, "every subset of the natural numbers has a least element" is not a first order statement because "every subset" is not an element of the natural numbers, it's a subset.

Then you have axioms of how logic works, and can prove theorems about first order statements etc. But that's not particularly useful for the discussion at hand


All the closure axiom says is that if a and b are in your field F, then a+b is also. But let's consider how + is defined. + is usually defined as a function:
$$+:F\times F\to F$$
In this context, the range/image/codomain (pick your favorite word) of + is entirely in F because that's how the function is defined. So you're being told that a+b is contained in F, it's just not spelled out as a separate axiom.

 

[mathwonk]

steve, that happens to me a lot. I hate it too. The solution is either to go right away to "Go advanced" or, more sure, to type your response into a word processor outside the site and then paste it in.