How complete is algebra?

What is the basis for reason? And mathematics?

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Philosophy Explorer
Posts: 5636
Joined: Sun Aug 31, 2014 7:39 am

How complete is algebra?

Post by Philosophy Explorer » Sat Mar 24, 2018 9:04 pm

When one talks about math, we're normally concerned with its consistency and completeness. Consistency means (hopefully) no self-contradictions and completeness means the subject area of math covers all
cases.

When it comes to algebra, there are a dozen axioms. The axioms are said to be consistent and complete for the real numbers. So for example, we have the commutative law of addition which is closed. When we finally reached the square root of negative numbers, we had to invent a new number (the complex numbers and imaginary numbers) which is a type of advanced math.

So I believe the axioms for the real numbers are consistent and complete. How about you?

PhilX 🇺🇸

wtf
Posts: 622
Joined: Tue Sep 08, 2015 11:36 pm

Re: How complete is algebra?

Post by wtf » Sat Mar 24, 2018 10:12 pm

Philosophy Explorer wrote:
Sat Mar 24, 2018 9:04 pm
When one talks about math, we're normally concerned with its consistency and completeness. Consistency means (hopefully) no self-contradictions and completeness means the subject area of math covers all
cases.

When it comes to algebra, there are a dozen axioms. The axioms are said to be consistent and complete for the real numbers. So for example, we have the commutative law of addition which is closed. When we finally reached the square root of negative numbers, we had to invent a new number (the complex numbers and imaginary numbers) which is a type of advanced math.

So I believe the axioms for the real numbers are consistent and complete. How about you?

PhilX 🇺🇸
Do you understand that you're equivocating two different meanings of completeness? There's logical completeness, meaning that every proposition can be either proved or disproved from the axioms; and algebraic completeness, meaning that every polynomial has a zero.

And there's also topological completeness, meaning that every Cauchy sequence converges. In this sense the rational numbers are not complete but the real numbers are.

Philosophy Explorer
Posts: 5636
Joined: Sun Aug 31, 2014 7:39 am

Re: How complete is algebra?

Post by Philosophy Explorer » Sat Mar 24, 2018 11:24 pm

wtf wrote:
Sat Mar 24, 2018 10:12 pm
Philosophy Explorer wrote:
Sat Mar 24, 2018 9:04 pm
When one talks about math, we're normally concerned with its consistency and completeness. Consistency means (hopefully) no self-contradictions and completeness means the subject area of math covers all
cases.

When it comes to algebra, there are a dozen axioms. The axioms are said to be consistent and complete for the real numbers. So for example, we have the commutative law of addition which is closed. When we finally reached the square root of negative numbers, we had to invent a new number (the complex numbers and imaginary numbers) which is a type of advanced math.

So I believe the axioms for the real numbers are consistent and complete. How about you?

PhilX 🇺🇸
Do you understand that you're equivocating two different meanings of completeness? There's logical completeness, meaning that every proposition can be either proved or disproved from the axioms; and algebraic completeness, meaning that every polynomial has a zero.

And there's also topological completeness, meaning that every Cauchy sequence converges. In this sense the rational numbers are not complete but the real numbers are.
I would say pick what you're most interested in or comfortable with. This is one reason why I put up this type of thread so that we can learn.

PhilX 🇺🇸

wtf
Posts: 622
Joined: Tue Sep 08, 2015 11:36 pm

Re: How complete is algebra?

Post by wtf » Sun Mar 25, 2018 4:11 am

Philosophy Explorer wrote:
Sat Mar 24, 2018 11:24 pm
I would say pick what you're most interested in or comfortable with. This is one reason why I put up this type of thread so that we can learn.
Well then I'll ramble about completeness a bit.

Actually you raised a good issue, which is that technical terms can have different meanings in different areas of math. There are at least three uses of completeness I can think of, and for all I know there are others I don't know about.

* Logical completeness. This is when we have a formal system, which is a collection rules for forming legal sentences in some formal language; along with some axioms; and some inference rules that allow us to draw deductions (theorems) from the axioms. The entire process is syntactic, utterly devoid of meaning. Any meaning is in our heads, as a way of gaining meta-understanding of what the formal system is talking about.

If you have an axiomiatic system, a model of that system is a set of objects that satisfy the axioms. A given set of axioms may have many different models. For example the axioms of geometry have both Euclidean and non-Euclidean models.

Gödel's incompleteness theorem says that if you have some axiom system that's strong enough to describe the natural numbers; then there is some proposition P such that neither P or not-P are theorems of the system. In any given model of the axioms, the proposition P will either be true or false. So in that model, P or not-P is a truth that can not be proven.

That's one kind of incompleteness.

* Topological completeness in a metric space. Consider the rational numbers. They have "holes" in them. For example the sequence of rational numbers 1, 1.4, 1.41, 1.414, etc. gets closer and closer to the square root of 2. But the sequence doesn't converge in the rationals, because the rationals have a hole where the square root of 2 should be.

The real numbers are the "completion" of the rationals. If you take the rationals and plug all the holes, you get the reals. The real numbers are topologically complete.

* Algebraic completeness. This refers to whether a given set of numbers contains roots of all the polynomials whose coefficients are in that set. For example in the real numbers, the polynomial x^2 + 1 has no root (or zero). So the real numbers are not algebraically complete. If you toss in the imaginary unit i and all the numbers of the form a + bi where a and b are real, the resulting system, called the complex numbers, is algebraically complete.

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