Distance between two points

[BenSon]

I not sure which maths forum to stick this in so I'm putting it here I don't mind if it gets moved though. I was thinking about this earlyer and its been bugging me for a while now. Say you have two numbers 1 and 2 isn't there an inifnate amount of numbers between them? Like 1.999' So how can we ever get from one to the other? Does that make sense I'm going crazy here!

[martinluther]

fractions aren't real numbers

[Dr. Zimski]

I not sure which maths forum to stick this in so I'm putting it here I don't mind if it gets moved though. I was thinking about this earlyer and its been bugging me for a while now. Say you have two numbers 1 and 2 isn't there an inifnate amount of numbers between them? Like 1.999' So how can we ever get from one to the other? Does that make sense I'm going crazy here!

That's a big misconception. Yes, a number can be divided into infinite fractions, but that has nothing to do with an object's motion.

 

For example, let's say an object is moving at a speed of 10 cm per sec. In order for the object to follow your theory, it must slow down constantly and never stop moving. But in the real world, the object would keep a consistant speed (assuming it isn't acted upon by an outside force) and eventually pass whatever barrier you designate.

[BenSon]

That's a big misconception. Yes, a number can be divided into infinite fractions, but that has nothing to do with an object's motion.

 

For example, let's say an object is moving at a speed of 10 cm per sec. In order for the object to follow your theory, it must slow down constantly and never stop moving. But in the real world, the object would keep a consistant speed (assuming it isn't acted upon by an outside force) and eventually pass whatever barrier you designate.

I wasen't realy thinking 'real world' just mathematicaly. Its realy hard to explain what I mean but i'll try. If there is an infinate amount of numbers between any two numbers how can it not encompass all numbers. Example there in an infinate amount of numbers between 1 and 2 but 3 is not in there. How can you have an infinate that is in a defined range and not express all numbers is it realy infinate then?

[Connor]

infinite doesn't mean all.

 

there are an infinite number of letter combinations using abcdefghi (with repeats) but none of them make the word "slap"

 

One shouldn't think of passing through individual # positions as we travel through space, but of ranges. For example, we spend 1 second between point 0 and 10, 1/10 seconds between 0 and 1, 1/1000 seconds between 0 and 1/100, and as we approach a point, we essentially spend no time there.

 

...That was a horrible explanation, perhaps someone else will do a better job

[matt grime]

There are an infinite number of even integers, not every integer is even. So what's the problem?

 

Incidentally, we are not 'going from 1 to 2' in the real number system, or the rationals by 'going from one to the next', there is no such things as the 'next' number in the rationals. If you are thinking about in terms like that then you have a misconception.

 

Incidentally, it is spelled 'infinite' and something being infinite merely means it is not finite. If you are struggling to understand what something 'is' in mathematics then it is good to remember its definition.

[Mace Sin]

So this would disprove Zeno's Paradox?

 

 

On the assumption that matter' date=' space, and time are continuous and infinitely divisible (scale invariant), we can conceive of a point-like massless particle (say, a photon) traveling at constant speed through a sequence of mirrors whose sizes and separations decrease geometrically (e.g., by a factor of two) on each step. The envelope around these mirrors is clearly a wedge shape that converges to a point, and the total length of the zigzag path is obviously finite (because the geometric series 1 + 1/2 + 1/4 + ... converges), so the particle must reach "the end" in finite time. The essence of Zeno's position against continuity and infinite divisibility is that there is no logical way for the photon to emerge from the sequence of mirrors. The direction in which the photon would be traveling when it emerged would depend on the last mirror it hit, but there is no "last" mirror. Similarly we could construct "Zeno's maze" by having a beam of light directed around a spiral as shown below:[/size']

 

 

Again the total path is finite, but has no end, i.e., no final direction, and a ray propagating along this path can neither continue nor escape. Of course, modern readers may feel entitled to disregard this line of reasoning, knowing that matter consists of atoms which are not infinitely divisible, so we could never construct an infinite sequence of geometrically decreasing mirrors. Also, every photon has some finite scattering wavelength and thus cannot be treated as a "point particle". Furthermore, even a massless particle such as a photon necessarily has momentum according to the quantum and relativistic relation p = h/l, and the number of rebounds per unit time – and hence the outward pressure on the structure holding the mirrors in place - increases to infinity as the photon approaches the convergent point. However, these arguments merely confirm Zeno's position that the physical world is not scale-invariant or infinitely divisible (noting that Planck’s constant h represents an absolute scale). Thus, we haven't debunked Zeno, we've merely conceded his point. Of course, this point is not, in itself, paradoxical. It simply indicates that at some level the physical world must be regarded as consisting of finite indivisible entities. We arrive at Zeno's paradox only when these arguments against infinite divisibility are combined with the complementary set of arguments (The Arrow and The Stadium) which show that a world consisting of finite indivisible entities is also logically impossible, thereby presenting us with the conclusion that physical reality can be neither continuous nor discontinuous.

 

Or am I missing point? The article states, I believe, that the physical world and speaking of philosophical means are two different things. So, would Zeno's Paradox be on track or off track with this particular subject?

[dkv]

Good question.

Infinite set of Numbers and Infinite set decimals are two basic assumptions of Maths.

All such computational truths are limited by the computational strengths.

 

Consider my definition of a meaningless number:

A meaningless number is number which is known but can not be used for any computational purpose thus making the knowledge useless.

For example 0/0 , Infinity or some arbitarly complicated Chess Game in which the rules change as the game proceeds.

Unlike imaginary number "i" which has found its physical use.

 

In principle there is a limit to finding a meaningful number from an infinite set.

 

Today which is largest number which can be represented and used?

Ofcourse we will find some answer. It is not unanswerable.

 

Tomorrow this limit may get extended but it still remains finite.

 

Thus the possibilities are infinite but realization is finite.

 

When an ant covers a distance between two ends it doesnt take infinite steps.

In reality things happen Quantized way.

[matt grime]

Good question.

Infinite set of Numbers and Infinite set decimals are two basic assumptions of Maths.

All such computational truths are limited by the computational strengths.

 

Consider my definition of a meaningless number:

A meaningless number is number which is known but can not be used for any computational purpose thus making the knowledge useless.

 

we have a distinction already between computable and non-computable numbers.

 

For example 0/0 ' date=' Infinity or some arbitarly complicated Chess Game in which the rules change as the game proceeds.[/quote']

 

none of those is a real number (Ie in the set of Real numbers, which is the cauchy completion of the rationals), it is spurious to say something is a meaningless number when it isn't even a number in the first place.

 

0/0 is meaningless, fullstop (ie we do not assign it meaning, numerically or otherwise), infinity is ambiguous as a 'number', though handy. no idea what chess has to do with anything.

 

Unlike imaginary number "i" which has found its physical use.

 

strictly speaking i is not in R, but in an extension of R, so you need to explain what you consider a number to be exactly before you start dismissing things as meaningless examples of them.

 

 

Incidentally, all known discrete models of space are 'inaccurate': they make physical predictions that are observably false, so it is a little premature to say everything happens in a quantized way.

[dkv]

we have a distinction already between computable and non-computable numbers.

REP:Yes this distinction exists for the convinience of Complete Computability.

Maths doesnt deny the existence.All it says is such and such no. is non-computable to the extent of defintion of our Reference Point.

==============================================

none of those is a real number (Ie in the set of Real numbers, which is the cauchy completion of the rationals), it is spurious to say something is a meaningless number when it isn't even a number in the first place.

0/0 is meaningless, fullstop (ie we do not assign it meaning, numerically or otherwise), infinity is ambiguous as a 'number', though handy. no idea what chess has to do with anything.

REP: Infinity is ambiguous?? No it is part of Real Set of Numbers without which many Limits wont work and offcourse Induction Principle requires it to make the results Universally True(Making it true to the last attainable and any possible unattainable Number). As far chess is concerned let us replace it with an Computer being asked to perform Binary Search for based on 9x9 chessboard or 10X10 Chess Board and so on...notice that at some point of Time Computers shall take Very Large Time to Reach a conclusion.Thus making the Mathematical Problem redundant for practical use in immediate tournament. Humans can play in a much more complicated computing space which is a combination of Complex Congition skills and logic. Here we can say that a big Maths Problem (to the extent of Number of Branches it generates) gets replaced by Human Mind. Making the Problem appears Mathematical without taking away the pleasure of chess which still remains computable in Human Mind.Thus all numbers are not useful as they are limited by computational strengths to produce useful results.

Consider another example : 2^32847928374987978987923 is my secret code which I intend to use in communication.Now I can compute it completely to know its property for encryption but I can also use it as series of digits and alpha numeric codes. e.g let 8 stand for *.. the meaning changes it no more remains a number.It becomes a comprehensible code. Thats why chose the Chess as an example to define a Large Number with precisely defined purpose.

 

0/0 is meaningless to the extent of Convinience.... All it means is that there

no property of such a state which can used to improve understanding.

Square root of -1 is non-sensical but works because it has well defined properties. Loosely this applies to Infinities as well . But it remains a problem as well in the form of singularity in Physics.

 

Anyways the question was on Infinity. We say is Infinity can not be reached as it is at the Horizon of Mathematical Thought.

Can we say the same for 0 ?

Let us look very closely at 0 and its meaning.

0 doesnt represent Nothing. It is always represented as a finite State in Computational Domain. It can be -ive potential or some relatively lower finite potential.0 Apples doesnt mean Nothing.It means there are no Apples.

What if I choose to call 0 as 1? I mean isnt it possible?

Let Absence of an Apple be called 1 then we have 2 , 3 apples defined accordingly.

Cant we perform all the Mathematical operations without falling into the trap of Infinity and Indeterminate State?All we will have to change the corresponding Understanding as well.. and thats all.

 

This shows that Problem is purely Mathematical and it does not mean or should not mean incomprehensibility of Mathematical Situation.But here we have shifted the Problem some where else. Thus to attain complete comprehensibilty using Maths Infinite Number of Transformations will be required.Which by defintion of Infinity is not Possibile. Therefore Universe can not be understood entirely using Maths.

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strictly speaking i is not in R, but in an extension of R, so you need to explain what you consider a number to be exactly before you start dismissing things as meaningless examples of them.

REP: The reason I gave are straight forward.

==================================================

Incidentally, all known discrete models of space are 'inaccurate': they make physical predictions that are observably false, so it is a little premature to say everything happens in a quantized way.

REP: Inaccurate in a fundamental way. It says there is a limit to accuracy which can achieved....Physics shows it using Uncertainity Principle.

The reasons for this inaccuracy I think is embedded in our approach towards the understanding of Universe which is entirely Boolean.

===========================================

[matt grime]

Infinity is not part of the real number system in the sense I gave. Point out one reference that states it is.

 

Since you are simply using it for limits and induction, then it is clear that you aren't actually using infinity as a 'number' (which you didn't appear to define; your writing is hard to read if you don't use breaks), merely as the quality of not being finite. Which is good, and all it should be used for really, but that doesn't make it 'part of the real number system' in the sense I used of 'being a real number'.

 

If you are going to use infinte cardinals then there is a proper class of them. That is why it is ambiguous. You talk of infinity as a 'number' without saying how you're using it.

 

 

i as the square root of one is not nonsensical at all. It is merely difficult for people to accept because of the nonsense they were taught in schools about maths. Look at the historical analogy of the (probably apocryphal) murder of the person who proved that the square root of 2 was irrational. Surely you accept sqrt(2) is fine, so why is i nonsensical? Because if you square a real number it becomes positive? Sure, that just states that you can't take square roots of negative reals and get a real, but that doesn't mean that there is no such way to extend square roots to accept negative arguments. Just as we shouldn't restrict to square rooting only numbers that give rational answers.

 

 

There is no consistent way to extend the fractional notation of a/b to accept 0/0, that is why it is genuinely meaningless in mathematics.

 

If you have to resort to arguments about apples then you're either insulting my intelligence or your own. Stick to talking mathematically when discussing maths.

[x__heavenly__x]

there are an infinite number of letter combinations using abcdefghi (with repeats) but none of them make the word "slap" - Conor

 

Wow!! That was a really good explaination for "From infinite to definite."

Anyway, I used to have these imagination "shocks"...Just imagine a person as small as a molicule....for him 1 km wud be infinite!!!! But a normal man, just 1 km. So basically its a frame of reference, from where we look to what. Depends on size of the observer!

 

Hope it helped somewhere.

[matt grime]

It in now way depends upon the size of the observer. 1km is 1km. There is a difference between 'infinite' and 'being too big to contemplate'. One is mathematics, the other is your subjective opinion.

[dkv]

Without insulting anyone let me state it this way

R={-1,0,1,2,3 etc ....}

No where I wrote Infinity.

But Now the question is Infinity part of the real set of numbers or not...?

Whether it is defined or not is different question.

The answer is it has to be part of the Real Set otherwise the Numbers System can not be extended in Principle. We say that the Set can extended Infinitely...

How many members are there in a Real Set of Number? (The question is self referral)

As the theory dictate the answer has to be Infinite..

No one run away from the theory.

But practically there is always a limit to what can be achieved..

Well practical is not part of Maths in many countries... therefore we will have to statisfy ourselves with Limit tending to Infinity.

Now the question is Limit a Mathematical Tool or is it real?

Consider another argument:

In logical Deductions we are accustomed to tweaking of process to make it ok for physical applications.... When in reality this should not be the case.

Assuming Binary reasoning applies to Real World.

Then application of Science of Truths should not lead to oscillation between Maths and its application (Physics or Economics or Biology).

At every step in the derivation we should be dealing with a Segregated Truths.

By Boolean logic it becomes mandatory to maintain the integrity of Derivation.

 

How is Infinity Defined...?

Well this question is as good as asking what is maximum computing strength of Universe?

How many points are available to represent a Number at any instant?

In other words what is the Maximum Value Surface Area of Universe can represent?

It is unkown but should be finite.Thus removing the need for Infinity.

==============================================

I am personally very liberal with numbers including i.

But what does it represent in Reality.

Note that Ideally Truth should not osciallate between Maths and its application.Maths is an Abstract Science... consistent within itself.

A good question .. I hope

====================================================

There is no consistent way to extend the fractional notation of a/b to accept 0/0, that is why it is genuinely meaningless in mathematics.

REP: What I meant here was that you can translate the Number System to any origin and compute.... Thus a computational problem may not be able the actual problem...

If I choose 0 as -1 then -1/-1 becomes indeterminate and not 0/0.. the respective understanding will also change. However it will help solve Computational Problem of 0/0 .Translation helps solve the problem of Computation but not actual problem which retains its relation to actual reality.

=================================================

[matt grime]

Without insulting anyone let me state it this way

R={-1' date='0,1,2,3 etc ....}[/quote']

 

writing it like that doesn't suggest the real numbers at all,

 

 

No where I wrote Infinity.

But Now the question is Infinity part of the real set of numbers or not...?

 

if you mean 'an element' by 'part', then of course not. if you meant is there some part of the properties of the real numbers that is infinite then of course there is. what you wrote doesn't let one decide if you're incorrect or speaking about something else.

 

Whether it is defined or not is different question.

The answer is it has to be part of the Real Set otherwise the Numbers System can not be extended in Principle. We say that the Set can extended Infinitely...

 

nope, that doesn't make mathematical sense.

 

How many members are there in a Real Set of Number? (The question is self referral)

 

 

no it is not self referring

 

As the theory dictate the answer has to be Infinite..

 

that doesn't mean infinity is an element of the set of real numbers.

 

REP: What I meant here was that you can translate the Number System to any origin and compute.... Thus a computational problem may not be able the actual problem...

If I choose 0 as -1 then -1/-1 becomes indeterminate and not 0/0.. the respective understanding will also change. However it will help solve Computational Problem of 0/0 .Translation helps solve the problem of Computation but not actual problem which retains its relation to actual reality.

=================================================

 

*sigh* 0 represents the additive identity, whatever label you choose for the additive identity (and they are just labels) 0/0 does not make sense. you cannot change the meaning of a symbol and then claim that it solves the original problem.

[dkv]

Traditionally Real Numbers are numbers which are rational or irrational.

Or simply can be represented using non-terminating decimals.

As I understand you believe Imaginary Numbers to be part of it...Am I correct?

 

I had used only a subset of Numbers.

 

And yes Infinity is an Element of Real Number...

Set of a Real Number is a Universal Set.All the measurable qunatities can be expressed as using a Subset of it.

In essence any Real question refers to Real Number System for Answers.

Now my question was How many Numbers are there in the Universal Set...(If the question is real and understandable..which I think is)

Obviously the answer must be found within the Set as it is a Universal Set of Real Numbers unless you give me a solid reason to Believe it otherwise.

Thus the question is Self referral and Infinity must be an Element as well as part of it.

 

I am not playing with symbols... I am translating a Computational Problem to a new origin. Thus seperating the Real Problem from a Mathematical Problem.

 

 

Now the next question is : Is it not true that often the Application of Maths results in oscialltion of truth between Maths and the Applied Field.

[matt grime]

Traditionally Real Numbers are numbers which are rational or irrational.

Or simply can be represented using non-terminating decimals.

As I understand you believe Imaginary Numbers to be part of it...Am I correct?

 

no' date=' as you can tell where i said that i is an element of an extension of R in my first reply to you, and therefore asked you to explain what criteria you were using to say things are and aren't numbers.

 

And yes Infinity is an Element of Real Number...

 

No it isn't. This again is why i wanted you to state what you meant by 'number' if you were going to make any claims about them.

 

 

 

Set of a Real Number is a Universal Set.All the measurable qunatities can be expressed as using a Subset of it.

 

is that your definition of 'universal set'? all the measurable quantities of what? the cardinality of the real numbers is 'a measure of quantity' and is not a real number.

 

In essence any Real question refers to Real Number System for Answers.

Now my question was How many Numbers are there in the Universal Set...(If the question is real and understandable..which I think is)

 

it's (the set of real numbers) cardinality is c, the cardinality of the continuum. according to the continuum hypothesis (which is known to be independent of ZF) this either is or is not the same as aleph-1, the first uncountable cardinal.

 

Obviously the answer must be found within the Set as it is a Universal Set of Real Numbers unless you give me a solid reason to Believe it otherwise.

Thus the question is Self referral and Infinity must be an Element as well as part of it.

 

Only if you're assuming that the reals are 'a universal set'.

 

I've no idea why you'd assume that or even what your definition of a universal set is. Generally maths does not operate in a set theory that allows for a 'universal' set, ie the set of all sets (this is not the same as presuming that our sets exist in some universe though). but i doubt you're using any of these terms with the accepted standard meanings, given that you believe there is an infinite cardinal in the set of real numbers.

 

 

I can give you the definitions of what the real numbers are. that should be a solid enough reason. this isn't about 'opinion', dkv, but about the deductions from the definitions. R, the set of reals, is the completion of the rationals in the euclidean norm, it is the set of dedekind cuts of Q, it is the totally ordered field. it does not contain 'infinity' as an element (there is no cauchy sequence of rationals 'converging to infinity'. it is not an element of R. if you think it is then you are simply mistaken.

 

A model for the set of reals is the set of decimal expansions, so that the reals are numbers of the form

 

[iNTEGER PART].{fractional part}

 

(modulo the relation that we identify certain strings such as 0.9.... and 1)

 

and obviously given any real number there is an integer [iNTEGER PART]+1 that is bigger than it. Even if you didn't see it before surely now you can see that there is no such real number as 'infinity' if given any real number we can find an integer larger than it.

 

The cardinality of any infinite set is not a finite number and not in the set of real numbers.

[dkv]

Thanks for the cool reply.

I do not doubt the precision with which Infinity has been left out...

Let me take it from a Group Prespective,without getting into definition debate.

I must admit that your defintion is very precise.

But what all I am saying is even after the exclusion from the Real Domain why does the Infinity remains ambiguous and occures every now and then? Or why cant Physics get rid of it? Or why cant Maths be more Universal in its approach?

 

Lets take Real Numbers as something belonging to a Group.

Groups having various properties(additive , associative etc)

But as all the laws are mathematical we will not dicsuss the real world..

Only those invented Mathematical elements which adhere to the properties of Real Number Group are included... Fine.

Now this gives us an obvious reason to keep out the infinity from the Group.

 

Interestingly the Group also claims to contain all the possible answers i.e. if you apply xyz operation defined by the Group on the Group members then you will get something within the Group.But the catch is you can end up with Infinity which we had outcasted ...!!

So as a Mathematician what do we do? we define another Law in Abstract Space that such an such operation is not allowed or is not defined or requires special treatment.Which is not obvious and generic to the Nature of Groups..

Groups do not require it for their existence in Maths.. We ask for it. Therefore such a rule is an exception to the Universal Nature of Mathmatical Space.So the choice is clearly between a more Self consistent and holistic Theory and a Theory surrounded by restrictions within the defintion of Group.

 

After all the processing done .. and all the relations found one has to come up with a set of possible Numeric Answers(Answer may or may not be True depending upon real Scenario.. the answer can be a simple constant which can be subjected to verification.. if its only exact final solution is something like "i" then I wonder why should I call it a solution in the first place.I dont know of any possible means to verify it. At best it remains consistent within Mathematical Space....

Now if "i" is acceptable then what holds us back in Accepting Infinity.

Probably becuase the monster goes against the rules of its creator and forces us to outcast it.

Once again I wish to state the same fact again that Infinity(whether good or bad) comes as a consequence of its existence

in a Generic Real Number Domain.

If and if there is no "real" challenge or equivalent to it then we must admit that Truth indeed oscialltes between Actual Reality and its Abstract tool(Maths is an Invention btw).

==========================================================

the cardinality of the real numbers is 'a measure of quantity' and is not a real number.

REP: Can you give me an example?

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[matt grime]

The real numbers are only a group under addition. No finite number of additions of real numbers yields infinity as an answer. It cannot or it would not be a group. You write the operation multiuplicatively. The reals are not a group under multiplication. The nonzero reals are. again no finite product of nonzero real numbers is infinity.

 

'infinity' as you are talking about is purely an analytic thing and merely is short hand to describe behaviour *that is not finite*. It is unwise to use infinity as a noun in this way for precisely the misapprehensions it has created in your understanding.

 

What does the phrase

 

[math] \lim_{x \to 0} 1/x = \infty[/math]

 

actually mean? I can write it our without ever using the symbol [math]\infty[/math] or the word infinity. It is equivalent to the statement, for all e>0 and all L in R, there is an x such that 0<|x|<e and |x|>L.

 

See, I didn't mention the word infinity once.

 

Further, I can replace this infinite sum

 

[math]\sum_{n=1}^{\infty}x_n[/math]

 

with one that does not use the symbol or word infinty as well

 

[math]\sum_{n\in \mathbb{N}} x_n[/math]

 

 

Nothing holds us back from 'accepting' infinity, or more properly 'using symbols that are in some sense larger than any real number' for algebriac operations, they are just not elements of the set of Real numbers. Just as we can extend the reals to allow i, and get the complex numbers. Incidentally the term 'real' in real numbers in no way is supposed to imply that these are 'real' in the ordinary language use of the word. Look up non-standard analysis or hyperreal numbers. Also try the extend real line.

 

 

 

By defintion 'the size of a set' is called its cardinality. Look up transfinite numbers or infinite cardinals. I already gave you two examples, c, and aleph-1, which may or may not be the same. Aleph-0 is the cardinality of the set of natural numbers. Approximately a cardinal is the class of isomorphic sets.

 

 

"Infinity(whether good or bad) comes as a consequence of its existence" is in my opinion nothing to do with maths. I can't even decide what that means, if it means anything.

[the tree]

But what all I am saying is even after the exclusion from the Real Domain why does the Infinity remains ambiguous and occures every now and then?

Infinity is not ambiguos and it occurs because we create situations that involve it.

Or why cant Physics get rid of it?

Because physics is reliant on what is, not what seems convenient.

Or why cant Maths be more Universal in its approach?

You know that doesn't make sense, we know that doesn't make sense, why bother?

Lets take Real Numbers as something belonging to a Group.

Yes, that's what we've been doing hundreds of years. [math]\mathbb{R}[/math] is hardly a new concept.

Now this gives us an obvious reason to keep out the infinity from the Group.

Inifinity is not "kept out", it just never was part of the group.

Interestingly the Group also claims to contain all the possible answers i.e. if you apply xyz operation defined by the Group on the Group members then you will get something within the Group.

Wrong.

Example: [math]2^{\frac{1}{2}}\notin\mathbb{Q}[/math] where [math]\mathbb{Q}[/math] is the rationals.

But the catch is you can end up with Infinity which we had outcasted ...!!

This is no more extraordinary than performing some function with items of a group of even numbers, and finding an odd number.

So as a Mathematician what do we do?

Leave it, as it is an expected result.

[Tom Mattson]

fractions aren't real numbers

 

Of course they are. Fractions are elements of the rationals, which are a subset of the reals.

[matt grime]

For the tree, the rationals in latex are \mathbb{Q}

 

note that the reason for this 'problem' is that both of you are using operations that are not the group operation to obtain something, so it is no surprise closure fails since no one is saying that the reals are a group with respect to multiplication, and square roots aren't even binary operations.

[dkv]

The real numbers are only a group under addition.

No finite number of additions of real numbers yields infinity as an answer. It cannot or it would not be a group. You write the operation multiuplicatively. The reals are not a group under multiplication. The nonzero reals are. again no finite product of nonzero real numbers is

infinity.

REP: Thats exactly what I am trying to say. Ideally I believe that all numbers should hold same democratic right in the Abstract World.

The exclusion should not have been so ruthless.

I also believe that the current construct of a Group is incomplete.

There must exist larger framework for handling it.

========================================================

Nothing holds us back from 'accepting' infinity, or more properly 'using symbols that are in some sense larger than any real number' for algebriac operations, they are just not elements of the set of Real numbers.

REP: I am disappointed. Infinity no doubt can be understood as a Verb.

But When I ask an Objective and Valid Question on the number of Elements in the Real Set ,what do I get in return...A verb(or the mathematical Sum). Consider the case of Parallel Lines. They are supposed to meet at Infinity in normal geometry.A set of parallel Line can thus very well represent infinity.Now if someone is asked to understand Infinity... then he will carry on the journey forever.

He will never return from his logical bus to tell you whether he understood it or not.But we come back with intuitive reasoning that we understood it.And today we know that no such parallel line can be drawn in principle.

The intuitive reasoning therefore has its own limitation.

The situation of Infinity is itself hypothetical as there is nothing in the real world to relate to.(as of now)

==========================================================Just as we can extend the reals to allow i, and get the complex numbers. Incidentally the term 'real' in real numbers in no way is supposed to imply that these are 'real' in the ordinary language use of the word. Look up non-standard analysis or hyperreal numbers. Also try the extend real line.

REP: I do not intend to relate real to the mathematical defintion of Real(Reality is much more complicated).

"i" plays a role more fundamental in Physics.Therefore I wanted to know the real world significance.

I looked up the Non-standard Analysis and found it very interesting.It only helps me to believe more strongly in the Number Democracy.

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"Infinity(whether good or bad) comes as a consequence of its existence" is in my opinion nothing to do with maths. I can't even decide what that means, if it means anything.

REP: Its existence is denied by the defintion of Mathematical Constructs to avoid the grand collapse of its own defintion.

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Infinity is not ambiguos and it occurs because we create situations that involve it.

REP: A simple equation found in Nature E=k/t explodes to infinity at exactly t=0. t=0 is not denied mathematically but infinity is therefore t can not be exactly equal to 0.Thus t=0 also gets denied in the process.If we deny Infinity as measure then obviously in the above case

we are also denying 0 as a measure of t.

Here I have taken 0 but it can be any Number.(E-E1)=k/(t-t1)

Similarly you have other equation's blowing up to the Verb at Speed of Light.

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Because physics is reliant on what is, not what seems convenient.

REP: Convenience at the cost of Universal Formalization.

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[dkv]

I only asked questions and didnt gave you any answer...

But let me try ..

What is meant by an Abstract Science?

Is Maths really an Observer Independent Science?

I am going to prove it otherwise.

Why do we come out of Infinity and trust our understanding of it is right?

Process of understanding infinity becomes easy if we construct a Logically Infinite Statement.

for example :

If there is any last representable Number N then additive property allows

N + 1 . No where I am using time here to question its computability... Abstract world doesnt recognize Time as an Essential Element of Mathematical Thought.

Therefore it is useless to argue using Time to the Mathematicians.

Now I ask a different question:

How many minimum number of Mathematicians are required to construct a real number set? Universe doesnt share this secret on the leaves of the tree and every child needs to learn these basics again.

One has to deduce it from observation.Therefore observation is essential part of Mathematics...

As the Maths is Boolean in Nature..and true and false reasoning is allowed to take place in Nature we need minimum of two Observers to explain the Mathematical World.

In one observer world the deductions can turn out to be wrong.

For example I may see only one and only one. Everything can appear to be a single Entity.Just like a Fish in the uniform pond with occasional occurence of waves..how this can happen in Actual Universe is a different question but is answerable using Physics.

So I begin with relaizing myself as one but as I come across another observer.. he disagrees and claims to have a seperate identity.

After much discussion we agree upon two.

If two can happen then why not 3 or 4 or 5 and so on.(Multiplication is short hand for addition) depending upon the logically separate identity found in Universe.

The disagreement continues and we end up with a very large number.

But at some point during logical discussion we should understand something as vanishing from what was already understood as seperate entity.

For example : A collection of Birds flying in Unison. or a group of Cells in Human Body or a close knit society.

 

Therefore I suggest that after reaching a Sensible Number Limit we should reach to the One again in a logical Group.Instead of finding a flat spread of Numbers we should find a Circular one.

Loosing all the information about numbers... except one.

Let us call this operation N&1 = 1

Now the question is how can we loose information.

We are so well connected. We have computers to assist.

For an individual it means Death or a similar state.

Which is inevitable.

This is an observer centric answer.

[matt grime]

You've note defined 'number' still, yet keep talking about them as they have some well known definition that you're using.

 

You don't mean 'verb' you mean 'noun', and you got back a definition. The cardinality of R is c. That is the symbol we use to describe its cardinality. Just as I use the symbol 3 to describe the cardinality of {a,b,c} (assuming they are all distinct). All sets that are bijective with R also have cardinality labelled c. With the axiom of choice it is possible to well order cardinals so that given two sets either |S|<|T|, |S|>|T| or |S|=|T|. As i said, roughly speaking we can think of cardinals as being equivalence classes of sets. After all, what is 3? You demonstrate 3 to a child be counting off 1, then two then 3. It's actually quite deep really. We can unambiguously state when some set has 3 elements by declaring it to be so if it is in bijection with some canonical set with 3 elements. See eg the peano axioms.

 

 

A group is a group is a group. There are simpler algebraic objects, and more complicated ones. It is not 'incomplete' as a definition. It might not do what you want but that is a problem with you misusing something. Get a better tool.