What is Infinity?

[Dave]

or Buzz Lightyear

 

You made me cry

[EL]

I can see this going Philosophical quite quickly

Will move it if necessary (edit: not meant as a threat' date=' just as a warning so you know where the thread is)[/quote']

Hello Dave,

Yes, I certainly agree with you that thread discussions under general mathematics should stay so, although there is nothing wrong with some philosophical aspects of mathematics, but it cannot be the core of discussion.

**

The word infinity comes from Latin : "In-finite" = not ended.

In mathematics, infinity is relevant to or the subject matter of articles such as limits, aleph number, class (set theory), Dedekind infinite, large cardinal, Russell's paradox, hyper-real numbers, projective geometry, extended real number and absolute infinite.

The Absolute Infinite is George Cantor's concept of an "infinity" that transcended the transfinite numbers. Cantor equated the Absolute Infinite with God. He held that the Absolute Infinite had various mathematical properties, including that every property of the Absolute Infinite is also held by some smaller object.

**

The Bolzano-Weierstrass theorem in real analysis states that every bounded sequence of real numbers contains a convergent subsequence. Also, Every bounded infinite set in R^n has an accumulation point.

**

In one of my older books, it is the Weierstrass-Bolzano Theorem that states that every infinite set has at least one limit point.

**

I have personally expanded in W.B.T. to produce a rigorous mathematical meaning of the Absolute Infinity rather than Cantor's philosophical approach.

For a spatial world with at least 3 degrees of freedom a referenced infinity should be isometric, such that the least number of limits must be one; If the limit was a spherical surface, then it can contain a negative infinity within as much as it can limit the positive infinity without. Thus |infinity| is a continuum from -oo to +oo through the spherical boundary as the least number of limits being that one.

The miraculous finding is that I accidentally defined the Zero Point boundary of the primordial chaos of the theory of chaos. Each ZP contains an infinite number of ZP, each of which contains an infinite number of ZP, ad infinitum. However, each level is identified by an order number of embeddedness as we depart from the original ZP of reference. This extension of W.B.T. allowed for the definition of the Universe as a bounded-infinite entity rather than the finite-unbounded entity, which Einstein have proposed.

As this post became too long, I leave you to imagine the uncountable consequences of such a find.

Regards.

[EL]

what i dont like about infinity is that infinity plus one is still infinity. so how can the new infinity be equal to the old infinity if one has been added?

Yes, that is a brilliant question, although not authentic. What I mean is that you are not the first to ask such a good question.

 

Let me explain to you something about "Significance Boundary".

Imagine being invited for a piece of cake, and while you place it in your dish, a teaspoonful of cream was left behind, but you relentlessly picked it up carefully and placed it in your dish.

Someone noticed and jestfully placed an insect magnifying glass on traces of cream and a particle of cake.

You smiled back showing that what you left out was really insignificant.

**

An arbitrated quantification system is created to be as practical as possible, and occasionally we have to deal with fractions of units. In a decimal number system, we care about the significant decimal places, right?

Now imagine a number so big, that a unit (1) is an incredibly insignificant small fraction of that big number so that adding it to that number as a fraction of it would practically dissapear.

 

This means that we can work our way up, from a zero point, as a referential limit, towards positive infinity as a boundary of significance, beyond which any number added is insignificant.

We may also work our way down, from a zero point, as a referential limit, towards negative infinity as a boundary of significance, beyond which any number subtracted is insignificant.

Not only those, but convergence of numbers on the zero point limit demonstrates the same number phenomenon on divisibility, such as the addition of all the fractions of the remaining fractions, (for example) the sum of 1/2 +1/4 +1/8 ......+ 1/2n where n tends to infinity must converge on the value 1.

 

I hope this explained a bit about such a huge subject.

 

Regards.

[Martyn]

thank you EL. someone who explains to me something, and not makes me slook stupid for being curious.

 

...but still, even tho the number added is insignificant, the new number is still bigger, so the numbers are not the same!

[matt grime]

EL helped? but mathematically his post was garbage, apart from what i am led to believe is simply a reposting of a wikipedia entry (without reference)

 

you are still talking about numbers that include infinity without thinking about which kind. if you'd like to know more about the proper mathematics of this then i am happy to talk to you about infinte cardinals and ordinals, as well as explaining what the maths behind extended number systems is. remember everything in mathematics is a human invention. if you want to know about the mathematics of 'infinity' then just ask about it, and if you abandon preconceptions about physics (which is unnecessary to the discussion) then you can learn more.

 

you need to ask yourself what you think numbers are, and what you think you wish to denote by infinity.

 

there is little reason to feel stupid: these things are intellectually difficult. but they do signify the difference between what mathematics 'is' and what lots of amateurs think it 'ought to be'.

 

for instanec, let us take ordinal numbers. these are ways of ordering sets. the simplest way to think of these is, perhaps, thus.

 

1 is a dot.

 

2 is two dots with a

 

3 three dots

 

and so on, label each by the number of dots in it.

 

now we can think of a line of dots with one dot for each whole number, call this line w. this is the first infinite ordinal. we can now create an infinite ordinal exactly one bigger than w, labelled w+1 by thinking of this infinite line of dots, then one more, say on a line above it. then we make w+2 by adding one more after that dot. we can now create nw+m by taking n lines of infintely many dots and a line above it with m dots.

 

another way to think about them is as follows.

 

take the numbers 1/2, 2/3, 3/4, 4/5... call the set up to n/(n+1) the ordinal n.

 

w is then the set of all of these as n ranges from 1 'to infinity' ie does not stop.

 

w+k is the 1+k/{k+1} so the numbers 1+1/2, 1+2/3, 1+3/4...

 

and aw+b can be gotten by adding a to all the fractions, a+1/2, a+2/3,...

 

each of these is an infinite ordinal (ordered number) and w and w+1 are different.

 

OR

 

we can take cardinals. we declare two sets to have the same cardinality if there is a bijection between them. this is going beyond what you know, i suspect. but a bijection between two sets means exactly that there is a way of associating to each element in one set exactly one element in the other. for instance, the set of positive whole numbers is in bijection with the set of even positive whole numbers because there is the mapping (association) n <--> 2n. does that make sense? sometimes the association is hard to spot but we can deduce it exists. we can show for instance that there is an association between the rational numbers and the positive whole numbers. just as we associated symbols to ordinals we can associate symbols to cardinals in a slightly more convoluted way. but the key thing here is that suppose we have the set of positive whole numbers, and the set of positive whole numbers plus zero. in this language they have the same cardinality because there is a way to associate the two sets, n <--> n+1, we just move the numbers one place to the right. so here the two infinite sets have the same cardinality though one has 'one more element than the other'

 

you see how it depends on the context?

[Martyn]

thank you matt grime. that did infact make very much sense, and it does bring some answers! i suppose you can have infinate numbers of anything, because some things dont have a limit. is it just that it is impossible to do simple sums with infinity? maybe because infinity is not simple!? and also i suppose, you cannot add one to inifinity, because infinity is an ever lastig number, so there is no end to add one to it?

[matt grime]

but again this all boils down to what you are attempting to describe by "infinity". mathematics is is simply about the deductions one can make about things that are defined to be mathematical, and there is no hard and fast rule about what is mathematical before you ask.

 

i can only tell you about the things that mathematics has declared to be associated to the word infinty (or infinite). (a quick check will tell you that 'primordial chaos' has nothing to do with mathematics)

 

perhaps you should step back and ask a metemathematical question: what do i mean when i use the word infinity?

 

in mathematics we say that natural numbers (1,2,3 etc) are finite. that any element of the integers (..,-2,-1,0,1,2,...) the rationals or the reals are finite. the term infinite means "is in some sense bigger than something in one of these sets of finite objects". we can use the naturals to describe the number of elements of some sets, so that {1,2,4,7} has 4 elements. if we consider the set of all natural numbers it is clear that no natural number is able to describe the set of all natural numbers in this sense, so this leads us to one way of discussing infinite things.

 

all of your questions are, to me, ill-defined. that is to say they introduce even more terms that have no meaning.

[jazzoff]

Does a/0 gives an infinite or undefined answer, if a is any number (real or complex)? And why does it give such an answer?

[jazzoff]

EL helped? but mathematically his post was garbage, apart from what i am led to believe is simply a reposting of a wikipedia entry (without reference)

 

Um...I know you are trying to support your point, but Ithink you are a little too harsh on EL...

[matt grime]

i don't like people posting authoritatively on things that they seemingly know little about. had he just stuck to the point rather than this "primordial chaos theory" i would'nt have said anything at all. there is far too much non-mathematical science (if it is even that and not a pseudo-science as i suspect) masquerading as mathematics

 

anyway, to your point.

division by zero is not a well defined operation within the real numbers so there the answer is not defined since division satisfies (a/b)*b=a, and thus (a/0)0=a, but we know that 0b=0 for any b, and we also know that x/x=1 when defined. thus there is no way to define it that is internally consistent.

 

now, that doesn't stop us positing the existence of a diferent structure extending the real numbers, and indeed in the extended complex plane (which is the more widely used, so i'll talk complexes for now) a/0 is taken to be the point at infinity (as long as a is not zero or infinity itself, for even then 0/0 and infinity/0 are not defined).

[jazzoff]

Thanks, Matt.

[Graeystone]

Hrm. . .could finite beings such as ourselves be able to truly understand infinity? Something for both science and philosophy to consider.(I'm not saying trying to understand infinity is a waste of time either.)

[matt grime]

well, whatever your question is it is not one about mathematics since we are prefectlyu happy with it here. i suggest you post your question in a physics or philosophy forum so you don't start another one in the mathematics forum where your question isn't really suited.

[Graeystone]

Time doesn't really exist. It isn't built into the basic physical laws of the universe. Whether time can end or not is irrelevant, because it does not exist in the first place.

 

If time doesn't exist then what about quantum mechanics? Isn't there aspect of time in it?

 

Don't know how old you are, but someday you'll be feeling it.

[brad89]

I thought that infinity was kind of like having all the numbers packed into one variable. X can equal anything, 250, 5, 9.2435 * 10^7, but to be infinity, it means that it is moe than one number at a time.

[jdurg]

I've always likened infinity to a fast moving object. It exists, but in order to do something with it you have to stop it. When it stops, however, it's no longer infinity.

[matt grime]

Err, well, not in mathematics. In mathematics there are objects with the label infinite, and there are points at infinity and we can work with them all quite easily if we stop trying to read real life into them.

[EL]

Infinity and Zero are true physical quantities that the human mind handles through a filter called significance.

Divergence spoils quantification but convergence creates the illusion of exactness when significance dictates the bounds and limits.

[DQW]

What ? I didn't understand a word there. Would you please explain what that was all about ?

[EL]

What ? I didn't understand a word there. Would you please explain what that was all about ?

 

I am trying to be concise here.

Ok. I will try to give an example to make it simple.

You are on the railway (no trains are coming), and the lines are perfectly parallel from your POV standing in their middle. You stare at the horizon where they vanish from your perspective. In fact, the rails continue beyond the point at which you significantly see, and into the zone of insignificance where you cannot see.

The lines converge into a single point at infinity from your perspective, but that infinity was dictated by your senses differentiating between the significant and the insignificant.

[Phi for All]

In fact' date=' the rails continue beyond the point at which you significantly see, and into the zone of insignificance where you cannot see.

The lines converge into a single point at infinity from your perspective, but that infinity was dictated by your senses differentiating between the significant and the insignificant.[/quote']I don't recognize this "zone of insignificance" as a mathematical theory. This is beginning to sound like speculation. Also, railroad tracks are certainly not infinite, even to our visual perceptions.

 

Let's stay on topic here.

[brad89]

Thing is, I have always thought that infinity is forever going with no beginning or end. So is a Ray infinite? It has a beginning. But a line doesn't. I thought that the line was infinite, and the ray wasn't.

 

Also, what is negative infinity?

[EL]

I don't recognize this "zone of insignificance" as a mathematical theory. This is beginning to sound like speculation. Also' date=' railroad tracks are certainly not infinite, even to our visual perceptions.

 

Let's stay on topic here.[/quote']

 

I am deliberately using words that you cannot "recognize" as anything you can find by applying a search engine.

However, you can search your brain for a meaning and you should get the point.

Remember that my elaboration was meant to be for a layman and not a professional language.

Now take my bounds and limits and add convergence onto infinity or zero limits and you can find enough mathematical words to search the internet to find this subject in professional details.

I did not speculate anything, and you did not apologize yet.

[matt grime]

Your being wildly unmathematical and misleading and quite sepculative, EL, professionally speaking. None of this is helpng to explain the mathematics.

[EL]

Do you think that if I explained the closed complex plane and a point at infinity denoting the class of all sequences of complex numbers {a_n} with lim a_n = oo, where (n---> oo) and talking about the Riemann sphere, that the subject will be perfectly clear?

You are the expert, so why don't you tell us why do we have to truncate the infinite decimal figures of pi?

Infinity is evident, but it contradicts with practical methods of demonstration and there is where significance becomes the criterion with which we decide when to be satisfied with our infinitely converging infinitesimals.

If you see a mistake in my logic or expressions, feel free to correct me rather than giving out unfounded verdicts.

I am willing to learn to improve my expressions as long as I live, so please contribute actively, and not stand passively and only complain.

Regards.