Infinis que seras tamen
I. M. R. Pinheiro

Abstract: 
Infinity has always been a `problematic’ definition. Basically, some think that infinity is an excuse not to think any further. This way, when they are unable to state anything precisely, they make use of the word infinity. Others, however, like in Calculi, see infinity as a logically correct tag for everything which never ends. Dividing by infinity, multiplying by infinity: all seems to be a major problem, according to the mathematicians and logicians working in Science so far. In this paper, however, we continue our work from `On Denoting for Arithmetic’ and establish a new system of reference, different from that of the Cartesian Plane, yet supplementary. 
The major gifts from this paper are the completeness of analysis via logical reasoning: We close all gaps existing before regarding the concept infinity, and we do all based on what has been logically built before our writings, so that our findings do not create any new inconsistency in what already existed, but fix problems, what means actual progress.
Key-words: 
Infinity, Cartesian, coordinates, space, time, world, number, limit, reals, operations.

Introduction:

The word infinity apparently comes from the Latin word `infinis', which meant `no borders’ (see [S. Schwartzman 1994], for instance). Interesting enough that mathematicians also said that there were an infinite number of numbers in the middle of other two, what does not immediately relate to `no borders’, and far more interesting this fact became because it is completely consistent with what we will state, later on, in this very paper.
It is obvious, and trivial, that the infinity the never-ending-progress, of the horizontal Cartesian axis, forms, is of different nature -completely different- from that infinity of numbers in the middle of other two: Not only the numbers in the middle will eventually end in an integer, or rational, but if there were an axis there, special for the infinity visit, it would fold and come back, as we suggest here…Weird enough, if both objects ever deserved same pointer, such could only be in the English language, never in Mathematics, as we keep on feeling pushed to write, all the way through…
How could something, which is reached in the middle of something else, compares to something nothing ever reaches?
Now, infinity is obviously only a sigmatoid of language, not necessarily belonging to the universe of Mathematics, and, the way it stands this far, we dare stating, with certainty, it was never found inside of that universe either.
With this one more precursor work, we not only manage to transform the existing infinity into something more pertinent to Mathematics, but we also describe, at least partially, the universe in which infinity inhabits, universe previously inaccessible for any mathematician via own Mathematics.
The beauty of this work is that it fixes part of the old gaps regarding the concept of infinity, it fixes denomination, which was equivocated for Mathematics, even though it could easily be acceptable in language (another point we also keep on having to re-visit in our work), and it provides more utility to the so interesting concept of infinity.
Interesting, as well, is the fact that, in Mathematics history, someone felt compelled to create aleph to represent the last number, which would not be infinity, but would correspond to it (?)…
Such fact is found reported at  [H. C. Parr  2003], for instance, together with a very elucidating discussion on that. According to the source, Aleph-0 would correspond to the order of the infinity where the natural numbers would belong, what does make sense: a trial of measuring how much more infinite is one infinity than the other, matching our discourse…the same source, however, brings easy and trivial argumentation as to why Aleph-0 is not a good measure for how big the infinity of the natural numbers is…trivially, they multiply each natural number by 2 and obtain the even numbers, claiming that to be half of the previous size and still getting to be of Aleph-0 size…
The major thing is not the comparison between sizes of infinity, but between nature of them. Once more, we emphasize the fact that the right question to be asked is `why?’, not `how?’, as for starters of any theory.
The motivation for creating the denomination `infinity’ for the idea of never finding an end in the horizontal Cartesian axis, is obviously different from the motivation for creating a denomination for the idea of having an infinite number of real numbers between any other two…given the nature of the concern being of diverse nature, it could not be that, in Mathematics, those two denominations would be the same (for Mathematics must be precise to top of `precisification‘, language is another matter…).
The ideas are quite different as well, in all senses…what comes to mind, in the piece of ruler containing several infinities, is something like several line rolls, all mixed in a chaotic way, whilst the idea of the `never ending’ walk is that of the straight line…one fuzzy, another so organized it hurts…
We then discuss all these ideas in depth in the sections which follow.
We start with the Latin `infinis', progress to the uncountably many members of reals in between any couple of rationals, to then write about the Infinitum World, mentioned in [M. R. Pinheiro 2008]. As finishing, our conclusion.

`Infinis'

There is no more adequate definition than  `that which has got no boundaries’ for what happens to the growth of the real numbers as time goes by,  in any step you take over them (one unit, half unit, etc). This way, walking the positive direction on the real numbers line would lead the walker to the conclusion that the only possible answer to the question `what is the highest real number?’ is `real numbers bear (infinis) no borders’, that is, they go nowhere, or they keep on going forever…basically. Because mathematicians are lazy, instead of stating `real numbers are with `infinis’’, they say they go to infinity…
Infinity would be the English word, derived from Latin, intended to mean the same `no finale’…However, there is substantial difference between a precise place called `infinity’ and a quality stated as `bearing no borders’. If there is a place where it dies, it is finite, …if it has no borders, it keeps on going forever, never dying…Therefore, it is hugest mistake of all stating that the real numbers `go to infinity’. No, they `keep on going forever’, never stopping, at most `with infinis`…However, if forcing nature and language into an impossible match, we must make it all clear in the Mathematics, for, in Mathematics, that is, part of Science, that is, daughter of Philosophy, no mistakes in definitions are allowed or double interpretations: If a term generates confusion, it is simply not good enough to there inhabit…
So, most sensical attitude of all would be erasing the concept, as well as symbol, for infinity everywhere in Mathematics and, as before its creation, the limit of something, which would not have a limit, would be told to be as it is: impossible to calculate because it does not exist. For example, the limit of the function f(x)=x, when x grows to maximum allowed in the reals (that means never stopping), has to be only delusional, for it does not exist! Basically, one would state, in good mathematical terms, that is, translated into the univocally interpreted language of Mathematics, in which every term points precisely to only one thing, that `the limit of f(x)=x, as x grows to maximum allowed in the real numbers, does not exist’ (writing things as they are is obviously most basic mathematical duty of all).
However, suppose we stick to the convention, some people being really fanatic about the lazy eight (as some decided to refer to it)…We then must create a way of stating clearly that such does not belong to the world of Mathematics; it does belong somewhere connected, of course, but of different nature than Mathematics, that is, to Metamathematics, for instance…
There, in Metamathematics, there is this being, not really a mathematical entity, but a funny symbol to point to nothing in particular, but to replace a language expression because we wish to adopt a symbol and write a single item instead of writing a sentence, or a set of symbols (`the limit … does not exist’ gets then replaced by `is equal to lazy eight’). Such disobeys the well-posedness principles for Mathematics, however. Therefore, we would be committing a crime if including it in it (similar to blasphemy when it comes to God).
We then, if ever willing to keep the `thing’, choose a `metamathematical’ world, meaning what is beyond Mathematics but still may be used by us there as long as we keep the picture and understanding of the object in use.
We propose here that we create a copy of the Cartesian system, call it `Infinitum System’, and, each time we refer to the expression `infinity’, we there visit.
We also state that if the numbers were ever human beings, Metamathematics would be to Mathematics the same thing metaphysics is for human beings: like a connection between what is human and what is beyond.
Anyway…were numbers humans, we could then say that when they `transcend’ their nature (human), they reach infinity (in human case, it would be God). 
Basically, the own number, poor thing, will never get to infinity, the same way humans will never get to God…there is always a lot in the middle, not mattering the names (soul, etc). It is as if God and Infinity do not actually exist, what exists is everything in the middle, but if we do not invent God and Infinity, we lose the `homocentric’ nature of things, and we are unable to understand all, once human beings apparently love being bossed, commanded, adoring things, and think they will be able to occupy the position of whatever to them is superior some day. If the figure of the boss does not exist, they apparently feel lost, in a World with no leader to follow, that is, nobody to be blamed for their mistakes who is not themselves…
Oh, well, let’s insist with the idea of liking the lazy eight…
We then have the `World of Infinitum’, where nothing will ever reach, mainly because it does not exist, but we imagine it does, and we work well with the imaginary Infinitum the same way we work well with the imaginary axis, for the square root of a negative number also does not exist.
The problem with infinity appears also in division and multiplication for nobody can do any mathematical operation with what is not mathematical. Try to, for instance, divide `your love’ by 3 and tell us how much you get, mathematically…
The day `love’ is mathematically measured, then we can, not before!
Infinity does not belong to the Mathematics world at all, so that even presenting it together with the standard mathematical symbols is unreasonable…
By simply forgetting it, we are left only with a few inconsistencies, all of them referring to zero (division by zero, multiplication by zero, etc).
The concepts which are not defined, to which we usually add the symbol for `infinity’ to get a result, if they get a simple `impossible’ answer, no inconsistency whatsoever… for there is then a very logical explanation, and it will only fit particular cases, in which the concepts cannot fit (square root, for instance, is only defined for non-negative numbers and there is no inconsistency there or incompleteness).
Let’s then work with both theories here: one will be for the `World of Infinitum’ and the other will be for the Mathematics world without the concept, and symbol, proved inadequate: `infinity`.

Mathematics without Infinity: will it survive?

Notice, once more, the reassurance on everything we keep on writing about: mathematicians and logicians, so far, have never stopped to pay attention to language and Mathematics differences…
If one states, in the English language, `I will go to Infinity!’…is that right or wrong?
The answer is simple: there is no right or wrong at all there, for a person may utter whatever they like, sensical or not, …there are no rules for utterances. There are fixed rules for punctuation, for spelling, etc, but not for what may, or may be not, uttered…
They say `I will go to Infinity!’ and that is even fully understood…
Probably something similar to `I will go to nowhere where you can find me…’
In language, there is no inadequacy, ever.
Mathematics, however, aiming being Science, must hold severely tied discourse, which allows progress and abstraction over abstraction to develop. This way, we should always worry about the study of its terms, so that no mistakes occur, and specially inconsistencies with the own language, which was born before it.
In language, the idea of place, when applying the term `Infinity’, is unavoidable: Look at the infinity! (mean skies, usually), as most trivial example. We also say `myriad, or infinity, of coins’ (means uncountably many in our heads but, for Mathematics, when this is used in language, it actually means countably many, with no exception, for the number of coins produced on Earth is always finite), etc. The idea, in the case with the coins, is that a person gets exhausted of trying to even think of counting them, so that it compares to the size of the sky, a place…a place of fatigue, inability of determining things from a human perspective…
In language, it would never be natural to then extrapolate and apply this idea to a ruler or an axis. It was obviously a super-forced match, for nobody gets tired of starring at a mathematical axis…and nobody finds that hard to deal with…
Basically, if anything, it keeps on going forever because of what we try to represent with it: we do not really imagine a pile of numbers, like in the case of coins…or a spread of numbers all around us, like in the case of the skies…
It is really not common, in normal life, to associate this concept to what they have decided to associate it with in Mathematics.
However, it is necessary to start in real life, not the other way around, for us to being able to apply an existing sigmatoid of the English language to a mathematical element…
So, in the origin, it was already equivocated, simply a distorted view of a lazy person.
Infinity is then linguistically associated with a graphical image of overwhelming amount, something like a pile which covers us or things we cannot touch, which are everywhere around us…
Poor numbers, cannot even dream of being 3-D entities…
Then, one could easily state that we should simply drop the name and come up with another name, not common in language, created just for that specific end.
However, name of what?
Basically, what we wish for is a name which summarizes the whole statement `it never stops growing’. Unlimited? In this case, the limit is what does not exist…It cannot be somewhere, something, because it goes nowhere, to nothing, it simply keeps on passing everything and everywhere…
Perhaps `free’…reals are free, whilst natural numbers stop before negativity starts, being, therefore, half-free only.
Rationals are free.
Integers are free.
A={1,2,3} is a constrained set, or `arrested’ (?)…
In short, there are far better names, which will point to a single thing, making it all scientific talk, finally, instead of `infinity`…
So, what is the highest real number? 
- Real numbers are free, there is no absolute highest.
What about `infinity`, is it not the highest?
-Not only there is no number which is `infinity`, but there is also no set, what makes of it an empty referent, or sigmatoid, what makes of it a nonsensical term, which should just be erased from the history of Mathematics.
What we are going to do with everywhere we used to write Infinity, such as the limits, and etc?
-How can x go to infinity? Where is Infinity? 
X goes free over the reals, positive part…
Or x keeps on growing, for we are mathematicians, not mystery creators, fantasy creators, it is what it is…it simply keeps on growing forever…
-And then, for functions like f(x)=x, when x goes free over positive reals, where does the function go?
The function goes free, just like x, for it is the reals paradise (everyone free, everyone naked…)!
Is it not much more fun?
We are sure only boring people would still prefer `infinity` after our suggestion and explanation…
What about something divided by zero?
-That is such an interesting item…once more confusion…
What is division? Division is splitting something into parts…
If we divide a chocolate (and this you understand because it exists in real life) by 4, aiming to give equal parts to my best friends, how much of chocolate will we get, for each, if we have 8 perfect squares of chocolate together in our bar?
Easy! 2 for each.
Now, we have nobody to give our chocolate to. We wish to divide it even so…
(`why` is a good question, as we keep on saying…)
If we have nobody to give it to, and do not wish to keep the chocolate, logical and short answer is `dump it!’.
That means disappearing with the chocolate…that means dividing it by so many units of counting that it will disappear, it is the same idea…what are we left with? Zero.
But the chocolate is still there, whole…in the trash bin…now it is a decision between referential and division definition. Who matters? The chocolate bar, which is still there, either dumped untouched, or split into myriads of pieces, or our sight of things, and we do not see any chocolate anymore?
The answer is obvious, for we give away the 2 pieces in the first case, and we see nothing in our hands as well.
The result is what happens to own chocolate.
Therefore, division by zero will return either pulverized chocolate, at a point of disappearing as a mathematical entity, or the whole lot.
Most logical choice is whole lot!
In this case, one chocolate divided by zero should be equal to one chocolate, and 1:0=1. However, 1x0 is not 1, what creates mathematical inconsistency. 
If you remember, we could never decide for sure, universally, between which way to disappear with the chocolate was best.
Mathematics cannot allow this to happen, what means that the division by zero cannot exist, it is simply not defined, just like square root for negative numbers…
They are defective operations, in that sense, with the universe being considered being that of numbers including zero.
So it is not inconsistency that it cannot be divided by zero, it is impossibility of definition there, for each operation is a function and, therefore, each number, which enters it, may only have one response, and that response must be universally accepted.
If zero returns more than one possibility, who is inadequate to belong there is zero…
The other issue is that of multiplying by zero.
Once more, because division and multiplication are `sister’ operations, if it does not make sense in division, it cannot make sense in multiplication.
Easy: multiplying is distributing something, as if you have as many copies of that something as necessary, always, over something else.
As example, multiplying breads, as Christ did…
If we multiply two chocolates by 10 people, we will get 20 because by making copies of the 2 each time needed for everyone to have the same as `we` did, we need 20 of them…
Now make copies of the number one to distribute to zero persons…
(?)
Whilst a copy for negative quantities is acceptable, for they may be seen as positive numbers ruler situated below surface, for instance, a location property, a copy to zero persons cannot make less sense. If there is no person, you need no copy, no operation.
We also need not dividing, for division implies having people to divide things with…
First of all, there is clear conflict, just like in the other case, with the own principle of the creation of the operation…so, why would it ever make sense?
However, if we swap things, making a copy of zero, instead, not mattering for how many, the operation will give us a sum of zeros, and therefore a zero!
This is possible here because multiplication holds the property of commutation, but division does not.
With these so simple considerations, there is absolutely no indeterminacy in Mathematics, what there is is disrespect for its principles…
Then the question comes…what happens if we have a function which `goes to one’ and lies in the numerator of a fraction, and another which `goes to zero’ and lies in the denominator? What is the limit?
If such a thing does not exist, one has two possibilities: considering something right next to zero, as many steps before as reasonable for the problem, or stating it is not possible to calculate.
In terms of limit, the final point is not relevant, so that we should go for closest, and we may divide 1 by any amount close to zero, as close as we wish for, yet not zero…
With `infinity`, of course we cannot define it, or think about it. If we got what is named infinity over infinity in the past, in terms of limits, unless we can apply one of the rules of simplification, we are realistically stuck. It is then called indeterminacy, but most of the time it is avoidable via mathematical tools.
The other case is zero divided by zero. That has to be one for one times zero is zero, and no inconsistency is there found. Besides, any number divided by itself means distributing it by itself, what means splitting it in as many parts as necessary, or persons. But no person was added to whoever was there before, it is zero…
The other case is zero times zero. Now we must make a copy of the units of counting for zero to distribute to nobody. Why making copies at all? We make zero copies…but then we will hold inconsistency with the inverse operation, if not careful enough, and remembering that zero is not invertible and, therefore, we will never get 0/0 from this…
It is, basically, a one-way operation.
It is consistent for otherwise same would apply to any number multiplied by zero, 5x0, for instance, would not be 0.
Therefore, eliminating the symbol for `infinity`, as well as the concept, and simply replacing that with the word `free’ and `impossible to calculate as it is presented now’, will bring nothing apart from gains to Mathematics.
Analysis results: not only it survives, but as in real life, it grows stronger without equivocated entities mixed with it…
Interesting enough, if the question asked regards how many numbers there are between any two others, in the universe of the reals, we may use the term Infinity, for the idea is that of you immersing yourself inside of the interval and counting, as if you could be reduced to that size and nature…it will be an overwhelming amount around you, so that you do hold, finally, Infinity! However, if ever kept in that situation, if we ever decide to use the symbol again, we must associate it with the idea of overwhelming and around, it is like you jump inside of the interval, anywhere you like, and you will never stop sinking…it is a physical idea, not mathematical, perhaps metaphysical…needs to be very well explained, and cannot be used in any problem, or solution, in Mathematics, for it may belong to Philosophy, but does not inhabit the world of Mathematics at all. It might be our informal answer to someone else, but not our answer in a class, which should simply be `uncountably many’, that is, correct to last letter, and accurate, in our piece of Science, which we are supposed to master.

Infinitum World

If we consider smallest piece we may physically get, in a ruler, to mean Infinity,  we   then may say that the smallest piece is a unit of measurement for the ruler as well. Consider, then, that each unit forms `one infinitum‘, then the set of all `infinita’ will form the whole ruler, or axis…With this, instead of referring to Infinity as a boundary, we will refer to supremum of the infinita or the infimum of the minus infinita…both concepts with no reference in the ruler, or in the actual world we live -empty referents- but, at least, right idea in Mathematics, with no mistake. With this, we may now write accurately, in mathematical terms, what before was wrongly stated, and used in calculations of formal Mathematics even so, obviously generating mistake (for every time such happens a huge mistake is added, precisely like the cases we have enumerated this far, specially on the paper `On Denoting’).
Just like the limit, sometimes it will fail giving us results because it does not have a result for that universe, so that not being able to tell the supremum of the infinita does not create inconsistencies…
It is simply a mathematical referent, that is, a mathematical being to point at that idea, that is, forcing insertion of our language statements there, in a language that the beings from Mathematics will, finally, understand the same way we do (see the same thing).
This also solves the problem pointed by  Parr, that of Aleph thoughts…size of infinity…
This way one immediately understands how large the axis is when compared to its pieces, what is simply logical, and makes it all coherent…
Small pieces are Infinitum, larger pieces are Infinita, and the whole lot is the interval between the infimum of the negative infinita and the supremum of the positive ones.
This way, we also create clear understanding of what really goes on in the Cartesian system, just as if the Infinitum World were the `soul’ of it, or the metaphysical correspondent for a far less thing than a human entity, that is: a metamathematical object.
We then understand that 0.5 in the axis is actually 0.5 infinitum, what immediately makes us associate that with uncountably many units contained in it, what is definitely different from considering half a unit only (like half a chocolate?).
We actually consider all the little bits of it, just like considering all cells in a human body…the human body is just one but, for a person to claim to be a biologist, they must see it as set of cells, uncountably many (so far) if their ability of counting is the reference.
Therefore, it could not be mathematicians drawing and writing before…We always thought it to be weird only mentioning the integers, or ratios, in an axis…
This way we can represent all those figures we used not to represent, which are still there, without having to write them, as wished for…
We simply state it is a set of uncountably many, some of which are such and such…
With this, we have created a whole universe in Mathematics, in terms of units, which did not exist before, with us considering the physical units before, what does not match the independency of Mathematics as another piece of Science…
It is not uncommon, given the confusion thus created, that people do not write a.u., v.u., etc, when calculating the area of a triangle, or the volume of a cylinder, for instance, in Mathematics.
From now onwards, we may write x infinitum meaning simply real numbers, yet correct also to deal with the physical universe, managing to still keep in the mind, of those using the concepts, the idea of what belongs to Mathematics and what belongs to Physics, what is trivially necessary.
It will be square infinita, for instance, instead of square meters. This dissociates Mathematics from reality of things, making it all more sensical, for the world of Mathematics will never fit reality with 100% accuracy, only the world of a ruler, of a mathematical object.
Necessary dissociation, precisely to reduce the amount of vain and useless discussions, unsolvable problems and paradoxes.
They will stop, by time of generation, if things are correctly annotated in Mathematics…
As best suggestion, both Infinitum and Infinita would be place holders, so that they would `adapt’ to each situation the graph will treat. 
The beauty involved in doing things our way is actually infinite (!).
We now make of Mathematics graphs finally a logical place for the numbers to be at…
Not only our origin may change, according to our needs, but also our units, making of scaling something coherent with the rest of Mathematics, once nobody, so far, has connected function graphs to scaling, in any sort of text book we had contact with, what means they simply disregard the personal factor involved with building a function from a personal table, what is actually absurd. Mathematics should deal with top logical and refined statement of things, yet as simple and clear as it is possible for the statement of things to be…
Basically, so far, it all looks disconnected, and useless, inside of the own Mathematics. Why learning how to produce scaling if they are not actually incorporated in the future Mathematics disciplines (once you learn, everything should be updated to also encompass that learning, not to go backwards in time to when you did not know how to produce a scale)?
With the concepts of the World of Infinitum, we finally close one more gap and put right things together…
From now onwards, each mathematical `Cartesian’ representation should come accompanied of brackets containing the so famous physical referential (who is seeing, from where=origin, that is, center of the system…) as well as the adequate mathematical referential (what is the best scale for us to see a useful picture of the material we need to present? What about most valuable member of the set of data to be plotted, that is, the one showing more times, or to which we would like to attach/compare all other results, or members of the domain? - of course, all this information may be worked out, with no doubt, by Statistics, as long as the reasoning for that choice is always clearly stated on the own graph as well).
Notice that the objective of drawing a graph is making the wealth of a table of values immediately meaningful to those who do not belong to Mathematics, or Statistics, or any piece of Science involved…
The most logical way of doing things has to be the target of Mathematics, and scale is already included there, as well as axis-labeling, as well as origin determination, so that we are not adding elements to Mathematics which do not belong there, someone else has already done it…
The idea of graph regards mental images -personal observation- and goes through all human problematic, what means they should not, probably, be a mathematical tool, at most a mathematical display tool, mostly for outsiders…
Obviously a graph, per se, cannot be a mathematical tool of decision, only the written analysis of it, made rigorously according to the mathematical principles…
At most, in Mathematics, graphs may be used for aid in memorization of concepts…
Things being so, we are just improving one of the communication tools of Mathematics, communication also in terms of evaluating the understanding of the concepts by those inside of it, also used for quick verification of results by experts before they read the theory involved.
Basically, we then suggest that another element, besides axe and title, becomes an essential element of any mathematical graph, the quadruplet (size of infinitum, size of infinita, origin, time of creation), always with fixed place for each, so that anyone reading the graph will know what the graph is about, as most complete set of graphical information one could possibly get from anything of that sort, or most complete tool for analyzing the accuracy of mathematical calculi, for analyzing perfection of description of the actual world via Mathematics, for speaking to outsiders of Mathematics about any mathematical concept, etc. Time should be the time when the person drawing the graph believes to hold all needed data for the plotting.
The Infinitum World gets to be presented then as a supplementary world to the Cartesian, with all overlapping, that is, we will get all the elements of the Cartesian Plane but, from this moment onwards, the only acceptable unit for our graphs will be Infinitum and Infinita…
Notice that, so far, the Cartesian Plane actually did not have rules for placing explicit elements from domain or image in it…
What we are doing is axiomatizing all of it (whatever is sensical in it), once every mathematical object, if it is mathematical, must be `born’ in some sort of axiom of definition, as minimum requirement for it to `belong’ to Mathematics…
On the other hand, nothing prevents the person from keeping the integers, for instance, as alternative reference (also marked on the graph designed this way…), it is just not desirable. The appearance of the integers in the graph does demand strong justification, once our Infinitum, as well as our Infinita, should be all logically based and, therefore, were the integers ever relevant, our Infinita choice, at least, would reflect that.
As example, we may get a graph where every domain element varies by 0.5 and they start at 1. The person thinks they are ready to draw the graph by 2 pm of the 9th of April of 1982. In this case, we may choose our Infinitum to be 0.5, and our Infinita to be 1 unit (step over our Infinitum). On the other hand, better placing the origin on one, perhaps, even though we need all details about the data to make the best logical decision, and that includes image of the points…This way, our graph would read (0.5, 1, 1, 09/04/1982 at 2 pm) soon after the title, for instance, hoping to get our initial suggestions as universal conventions.
If we think that knowing zero was left behind is relevant, we may plot the zero on the horizontal axis as well, as long as we write something like `the reference for all data was a machine placed one Infinita before the origin’, for instance.
The difference between the Infinitum World and the Cartesian might be subtle, but helps absolutely everyone involved, in both subconscious and psycholinguistic level, to always be attentive to the presence of an infinite number of members in each slice, not degenerated, of the real line. On the other hand, with our triplets, we also become aware of the difference between the infinitum of 0.5 and the infinitum of 1.3, for instance, that is, we understand there is more elements in one than in the other but, at the same time, we tie all that to uncountably many elements, what only makes sense of all in Mathematics…
By transferring the concept to a unit of measurement, similar to those from Physics, we understand that that notion does not entirely belong to Mathematics. As interesting as it may sound, Mathematics is able to deal with it in full, but does not access it entirely, just like the enlightened human being in relation to God.
Basically, we are in the Metamathematics World, and if such is made clear to everyone, we will quickly understand what is fully determined and limited, therefore in Classical Logic and Mathematics, and what is not yet fully determined, and cannot be limited, therefore in almost real life Logic and beyond Mathematics.
This is far more sensical than including all that in a graph and presenting it as Pure Mathematics.
Now the graph is presented as it is: a tool for any piece of Science to play with the way they need, with no need to treat a graph in Statistics a way and in Mathematics another way.
We also make it clear that what is accessible by the Mathematics world is the infinitum, not each and every number from reals…and that is the worst loss contained in making use of the Cartesian Plane: the megalomaniac feeling induced by that activity.
We can only deal with pieces of the real set of numbers, not with each and every individual number from it.
At least not at the same time…
Some changes will not be a big deal. For instance, all operations which used to return Infinity in Mathematics so far: they will return one of the options below, instead…
a) Sup Infta
b) Inf -Infta.
As we know, they are still empty sigmatoids (meaning no reference). However, it is normal for a word in Mathematics, a term living fully inside of it, not to have reference sometimes, yet being passive of representation (not operation, only representation). What cannot be normal is a term from the English language without reference, such as `infinity`, once everything is created as pointer for the universe elements, not the other way around…the need to express to another comes before the term…communication needs…
Notice that sometimes we will be able to tell what Sup Infta, or Inf -Infta, are, but sometimes we will have to state there is none.
However, one can feel the psychological striking difference between the understanding, or mental ability to cope with, of the individual making use of the Infinitum World and the individual working with the Cartesian one…What is gained is understanding, and that does help Science progress.

Conclusion:

In this paper, we have just created the World of Infinitum, which we believe to be highly useful, in several senses, for Mathematics and people working with its concepts. We have also discussed things such as division by zero, multiplication, etc.
We seem to have eliminated quite a few indeterminacies which existed so far in Mathematics, on the top of everything else. We have also proposed, according to well-posedness theory for Philosophy, a new plane in which to plot any mathematical graph, making the needed connection between Mathematics and those depending on its tools: statisticians, physicists, etc.
We have also destroyed the possibility that the Russian doll thesis (presented at [M. R. Pinheiro  2008]) could ever be true mathematically (graphs cannot form foundation for mathematical reasoning, only foundation for insight of mathematical nature). With the last contribution to the scholarship, we have actually stolen all remaining hope that Russell’s Paradox could ever be a paradox, in any possible sense, for Mathematics, or even a problem for it. Not to be taken away, however, is the value of the issues raised by the would-like-to-be paradox, all helping us writing better Mathematics, or building better concepts, closer to what would be expected from any scientific field.
In this paper, we have also abolished both symbol and notion of `infinity`, existing so far in Mathematics, all here proven to be mistaken conceptions, when the foundations of Mathematics and Language are considered. To replace that concept, in what is sensical, we have created the terms `infinitum` and `infinita`, and the most basic operations with them. 
On top of that, we propose a close-to-perfect graphical expression in Mathematics, taking into account Psycholinguistics, its influence, even in terms of blocking research insight.
We have established that 0/0 is one, but division by zero of any other number is not acceptable because of the definition of division.
We also have determined that any number multiplied by 0 is 0, with no incoherence between this fact and the effective division by zero. 
Via eliminating infinity from the mathematical lingo, no issues of indeterminacy with it may be found. 
If something is not defined, it is not defined, we then go for possible approximations, or tricks of transformation, whenever possible.
We have also associated The World of Infinitum to Metamathematics. In the World of Infinitum, we should hold a four-point tag addition to the basic Cartesian axe representation. In it, we should also perform some scaling and Statistics work on the data before being able to commence plotting, and add the result of the analysis in those regards to the graph. We have modified the Cartesian Plane quite a lot in order to attain our World of Infinitum, as we prefer to call it. Once the number of modifications made to the original Cartesian Plane are in significant number, we chose calling the new mathematical graphic environment World of Infinitum instead of something like Modified Cartesian Plane.

References:

[H. C. Parr  2003] H. C. Parr. Infinity. Available online at http://www.c-parr.freeserve.co.uk/hcp/infinity.htm, as accessed on the 3rd  of May of 2008.

[M. R. Pinheiro  2008] M. R. Pinheiro. On Denoting For Arithmetic (Peano’s) and identifying a 5th Natural Dimension in Mathematics. Preprint located at scribd, illmrpinheiro, or www.geocities.com/msorfiap2, submitted.

[S. Schwartzman   1994] S. Schwartzman. The Words of Mathematics: An Etymological Dictionary of Mathematical Terms. MAA.  ISBN:0883855119