System closed under all operations?

[Guest loarevalo]

Is there a system closed under addition, substraction, multiplication, and division?

 

Can we use this system to replace the current one? Is this system sufficiently complex to become the new foundation of mathematics?

 

As a mathematically oriented computer programmer, I always found it annoying having to create exceptions when defining operations and functions of numbers. One always needs to deal with the possibility that the user asks the program to divide by zero, or take the root of a negative number. The latter case is easy to fix: expand the system to include complex numbers. But I could not think how to fix the first case - other than to display the classic error message "Divison by Zero."

 

Peano arithmetic is not closed under division (division by zero is not defined in the system). Could we construct a better system, or ideally, a system closed under all operations?

[Dark Mind]

I don't know. But I think if we could it would've already been done :eek_big:.

[Guest loarevalo]

I wondered if Einstein also thought that:

 

"Hey. This Space equal Time thing seems pretty obvious. Surely, someone must have thought about it. If space could really be treated like time, I would've already read it or heard it in school."

 

We simply can't say that some system doesn't exist because we don't have it. It's the same as saying that every system that can exist or formulated has already been formulated. That thinking has been every thinker's enemy. Science, math, or any study, cannot progress without being a little rebelious and incomformist.

 

I understand your point. It seems that such as the number system is so fundamental that we should already know the answer to every important question about it - but maybe we don't! There has always been those professing to have "figured it out" and have "satisfactory" answers to all questions. Yet, if they were correct, then there has not been any progress, nor can there be any progress in the future.

 

Someday we will be able to say that our telescopes have reached the boundaries of the Universe. Maybe that already is the case in mathematics today - maybe Set Theory has already let us see the whole expanse of the number system (it should've, since such was the goal initially).

 

Yet, though we have seen It in its complete expanse, we definetly haven't seen It through all its depths.

[Turtle]

___The way to prevent division by zero causing a program shutdown by way of user input is the special conditions you dislike; Bullet-proofing.

___From Categories, to Groups, to Rings & Fields, it simply doesn't matter in regard to the result you seek. Godel proves that in ANY well defined closed system of mathematics, there exists irreconsilable propositions; QED.

___It is a mistake thinking all is known; always a mistake. Thinking so if nothing else prevents one from attempting to prove otherwise. Here's an unsolved challenge for you - use whatever system you care - All perfect numbers divide by two. :eek_big:

[Guest loarevalo]

Yes. Godel makes sense. But because he says mathematics can never be complete, not for that we drop our pens and move to the philosophy department (what is a pen? Does a pen exist? etc..)

 

I'm saying: Because of Godel's incompleteness, we don't stop looking for proofs of Riemann's. As I understand Godel, we will never FINISH mathematics, or construct a perfect system. But...we can try constructing better systems as long as we want. In other words, we will never reach the limit.

 

So, if we could better close the number system by adding complex numbers, in what other ways could we further close it (of course, while never able to close it entirely)?

[Guest loarevalo]

I got a question for Turtle (or anyone, of course):

Is 0 even? I think we could treat it as even - maybe not.

Is infinity even? I guess that if 0 is even, so is infinity - what do you think?

 

Sorry this is pointless...

[EWright]

I got a question for Turtle (or anyone, of course):

Is 0 even? I think we could treat it as even - maybe not.

Is infinity even? I guess that if 0 is even, so is infinity - what do you think?

 

Sorry this is pointless...

 

Infinity is even, and zero is the midpoint ;)

[Turtle]

___Getting right to the point, zero is even & infinity is not a number.

___On the question of 'other systems', yes there is room for more. My Katabatak system is congruent to modulo arithematic, & yet a different 'system'; a different method for arriving at similar results as another.

___The thread here on Buckminster Fuller's system of Synergetics is directed at exposing how poorly read this rich topic is. For example he comments on Euler's vertices, edges, faces geometry as incomplete; Fuller adds inside-outside among other qualities & shows the validity of his view.

___The investigation of math whether pure or applied, is never pointless. ;)

[C1ay]

Neither zero or infinity is defined as odd or even.

[alxian]

but 0 should be considered as neither odd nor even

 

it is between n and -n (where n in an odd number), its like saying all real numbers are even while n.5 is odd

 

what was the point of odd and even numbers anyway?

[CraigD]

Is there a system closed under addition, substraction, multiplication, and division?

 

Can we use this system to replace the current one? Is this system sufficiently complex to become the new foundation of mathematics?...

I believe that the system you’re describing is the union of the set of the rational Rational numbers (Q) and the set consisting of the transfinite number aleph-one (A1), that it is a group under the 4 fundamental arithmetic operations, and that it is at least as complex as Q. Not all members have inverses under all operations.

 

My understanding of Cantor’s formalism is too poor to be certain of this. I’ve read some reference that seem to directly contradict my claim.

 

A practically programmatic implementation of Q+{A1} strikes me as fairly easy, but for one question. Division by zero results in positive or negative A1. Any operation involving A1, including multiplication by 0, results in A1, with the same sign rules as members of Q. Although I believe A1-A1=A1, I have misgivings that –(A1-A1)= -A1+A1= -A1.

 

In my personal, string-based rational arithmetic module, division by 0 returns the zero-length string.

Is 0 even?

I can think of 2 relevant definitions of “even” that give different answers to this question.

1. The usual definition under Q is: N is even iff N * 1/2 = M/1. Under this definition, the answer is yes.
   
2. a less common definition, usually applied only to the Natural numbers (N), though easily extendable to Q-{0}, is: N is even iff the prime factorization of N contains a factor of 2 with non-zero exponent. Under this definition, the answer is undefined. Remarkably, under this definition, 1/2 is even.

Is infinity even?

Under [1] above, the answer depends on how you constructed the number set. For Q+{A1}, the answer is undefined – A1 is not a Q, so has no denominator. For (N+{A1})/(N+{A1}), A1/1 is even.

 

Under [2], the answer is, I believe, yes.

 

Re: Godel incompleteness: So long as you’re not attempting to prove all of, or one of the unprovable, theorems in a formal system, Godel’s theorem shouldn’t matter.

 

:P Out of curiosity, what sort of mathematically oriented computer programming are you interested in? What is an example of a mathematically oriented computer program that interests you?

[Guest loarevalo]

;) Out of curiosity, what sort of mathematically oriented computer programming are you interested in? What is an example of a mathematically oriented computer program that interests you?

I was just referring to an application - a program that evaluates mathematical expressions, like a program that graphs a user-defined function, in essence, a calculator.

 

What does "Mathematica" do about 1/0? or 0/0 within no context?

 

The system you propose of Q + {A1} seems interesting, though somewhat artificial and coming short of including ALL the numbers. Also, simply defining or attaching a symbol to 1/0 has been attempted innumerable times (See Projectively Extended Real Numbers), and has never worked, because it implies that this symbol be the a "last number" as zero is the smallest number - what about A2, and innacesible cardinals, aren't their reciprocals 0 or less than 0 (but not negative)?

 

Furthermore, such a "last number" is not defined, for it would be The Absolute Infinity. So, 1/0 doesn't exist, in the sense that The Absolute Infinity doesn't exist.

 

I believe Set Theory came to be partially to resolve the paradoxes involved in defining Infinity as one single number, rather than a family of numbers. In the past, this Infinity would be one single number, and would be the "last number." Set Theory's approach define infinite numbers without defining any "last number" - thereby avoiding the paradoxes.

 

CraigD and EWright might be interested in another thread I started titled "If there is no biggest number, is there a smallest?", dealing with what I wrote in this post.

[Guest loarevalo]

___Getting right to the point, zero is even & infinity is not a number.

True, INF isn't a number, that is, it isn't a definite single number, but a vague concept, we should rather use instead of the noun "infinity" the adjective "infinite" - it means the same.

 

Instead of arguing our philosphical points, let's argue something definite:

 

y = cos(n*pi)

gives 1 for even n, and -1 for odd n. Clear enough.

 

cos(0*pi) = 1, so zero is even (also observe (-1)^even = 1 and (-1)^0 = 1)

 

For n = infinity, we can see the limit is undetermined, there is no limit.

For

y=cos(pi/n) and n = 0, we encounter the same problem. At n=0, y=?

 

Perhaps, eveness applies to transfinite ordinals, but not to transfinite cardinals - for the latter, eveness is indeterminate. Who knows? ;)

[Qfwfq]

Of course zero is even!!!!!!!! But not because the cosine of zero is 1, that only shows that the cosine function must have an even part but that's a "slightly" different definition.

 

The whole point is that the definition of a field includes the property that the set without the additive neutral must be a group for the multiplication operation. Because of the distributive property, this element can't have an inverse (because 0*n = 0 for every n).