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8/6/2019 Infinis que sera tamen
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Infinis que seras tamen1
I. M. R. Pinheiro2
Abstract:
A new system of both geometric representation and reference for
mathematical objects is introduced. One of the many possible reasons
for swapping the Cartesian Plane system for the system presented in
this paper is lightly discussed in the own paper: The amount of
scientific mistake that has originated in the perpetuation of certain
erroneous paradigms, which ought to have appeared through
subliminal processes deriving from the use of the Cartesian Plane as a
mathematical tool.
Key-words:
Infinity, Cartesian, coordinates, space, time, world, number, limit,
reals, operations.
1 Copying the statement libertas que seras tamen, translated into freedom, even if
late, and adapting it to our notion: No borders, even if late… .
2 Po Box 12396, A’Beckett st, Melbourne, Victoria, 8006. E-mail:
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1. Introduction:
The term infinity apparently comes from the Latin word infinis, which
means no borders (see [S. Schwartzman 1994], for instance).
Neglecting the origins of the term, mathematicians have said that
there was an infinity of numbers between each randomly chosen
couple of rational numbers, for instance.
It is obvious, however, that the infinity the never-ending progress of
the horizontal Cartesian axis forms is of different nature -completely
different- from that infinity of numbers between any randomly chosen
couple of rational numbers: The in-between numbers are obviously
bounded by at least the couple of rational numbers that have just
been mentioned, for instance.
Basically, when we write about the infinity of numbers between two
rational numbers, we a sort of imagine an infinity that fits some very
well limited length, for which starting and finishing points are well
controlled by us.
Truth is that if both abstract objects ever deserved the same pointer,
such could only be in the English language, never in Mathematics.
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How could something that is reached inside of something else
compare to something nothing ever reaches?
Infinity is obviously a referent in the human language, but that does
not necessarily imply that infinity may be used as a referent in
Mathematics.
For each and every referent that we decide to insert in the world of
Mathematics, we need to prove to ourselves that the condition points
to a single world reference, with no mistake or confusion is satisfied,
once Mathematics has to be the scientific place of maximum amount
of control (over all inside of it).
The linguistic referent infinity, unfortunately, as we shall endeavour to
prove in this paper, is one of the referents that does not satisfy the
just mentioned condition.
That does not mean that the referent infinity is useless, just means
that it is not an acceptable mathematical referent.
Infinity is still a respectable referent, which has inspired several
people both inside and outside of Mathematics.
Someone in Mathematics, for instance, felt compelled to create Aleph
to represent the last number, which would not be infinity, but would
correspond to it (?)… .
Such a fact is found reported in [H. C. Parr 2003] together with a very
elucidative discussion on that. According to the source, Aleph-0 would
correspond to the order of the infinity to which the natural numbers
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would belong: A trial of measuring how much more infinite one infinity
is in comparison with another, matching our intuition, which tells us
that there are more rational numbers than natural numbers in the
world of Mathematics, for instance. The same source, however, brings
trivial argumentation as to why Aleph-0 is not a good measure for
how big the infinity of the natural numbers is: They multiply each
natural number by two and obtain the even numbers, claiming that to
be half of the previous size and still of Aleph-0 size... .
The discussion that we have just described is one more piece of
evidence about there being something absolutely wrong either with
the term infinity being used in Mathematics or with the world
reference chosen to describe infinity in Mathematics.
Aleph makes us realise that we need more than one term of the
human language to designate the current mathematical references
for infinity, once the current lingo leads people to at least sometimes
imagine that there is a cardinality attached to the concept of infinity,
cardinality that is of fixed size, like the current lingo leads people to
sometimes imagine that infinity is a physical length.
We would like to state that the right question to be asked is why?, not
how?, as for starters, in 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
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step size that we take over them (one unit, half unit, and etc.). This
way, walking in 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. Because mathematicians are lazy, instead of stating real
numbers have infinis, they say they go to infinity, meaning the
special place where no borders exist, or no place at all, once there is
never a stopping point.
Infinity, according to one of our most popular dictionaries ([Merriam-
Webster-1 2009]), would mean:
Main Entry:
in·fin·i·ty
Pronunciation:
\in-'fi-nə-tē\Function:
noun
Inflected Form(s):
plural in·fin·i·ties
Date:
14th century
1 a: the quality of being infinite b: unlimited extent of time, space, or quantity :
boundlessness2: an indefinitely great number or amount <an Infinity of stars>3 a: the
limit of the value of a function or variable when it tends to become numerically larger
than any preassigned finite number b: a part of a geometric magnitude that lies beyondany part whose distance from a given reference position is finite <do parallel lines ever
meet if they extend to Infinity> c: a transfinite number (as Aleph-null)4: a distance so
great that the rays of light from a point source at that distance may be regarded as
parallel
However, there is substantial difference between a precise place
called infinity (meaning number 3a from the dictionary extract) and a
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quality meaning bearing no borders (meaning 1b from the dictionary
extract).
It has to be the hugest mistake of all stating that the real numbers
keep on going forever, go to infinity . No, they keep on going forever ,
never stopping, at most with infinis.
Notice that the set of the real numbers does not have any of the
borders, but the set of the natural numbers, for instance, does not
have the right border only, so that we should have two different
referents, one for each one of the mentioned situations, if willing to
apply human language to Mathematics, for, in Mathematics, no
inaccuracies are allowed: If a term generates confusion (confusion
might simply mean multiple interpretations), it is simply not good
enough to be part of the mathematical lingo.
Given so much inaccuracy in the Mathematics that is generated by
the use of the term infinity, the most sensible attitude of all is erasing
the concept, as well as the symbol, for infinity everywhere in
Mathematics and starting over.
In Mathematics, the symbol ∞ and the word infinity are applied to at
least the following situations:
a) Mathematical description (and calculation) of limits;
b) Mathematical description of intervals; and
c) Primary school mathematical operations.
The limit of the function f(x) = x , when x never stops growing (or
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when x goes to infinity , according to the language used so far in the
world of Mathematics) in the set of real numbers, to use one of the
situations that we have just mentioned, has to be delusional, for there
is no accumulation point in the real numbers line that does for the
end of the set as a whole.
Therefore, the right decision here would be excluding, from the realm
of possible questions in Mathematics, what is the limit of the real
function f(x) = x when the domain member grows to infinity?.
Basically, we need to state, in this case, that the limit of f(x) = x, as x
never stops growing in the set of the real numbers, is not a valid
mathematical request because there is no accumulation point at the
end of the real numbers line (writing things as they are is obviously
the most basic mathematical duty of all).
Notice that we had to appeal to the use of common language,
therefore to the use of terms that are not part of the mathematical
lingo, to explain what was going on in the just mentioned situation.
The explanation is simple: We have proved that the just mentioned
request is not sound in Mathematics, therefore does not belong to the
world of Mathematics. If something lies partially, or totally, outside of
the world of Mathematics, it has to be dealt with at least some terms
that are not part of the mathematical lingo.
With the intervals description and the operations, the problem has to
be that it is unacceptable to equate ∞ to 5. We cannot, obviously, use
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∞ and 5 in the same mathematical situations, for they are entities of
different nature. ∞ is not a location in the real numbers axis, for
instance, but 5 is.
We obviously have to find alternative ways, which will probably be
more complex than the ways we use this far, of describing our ideas
in the Mathematics sceneries.
To mention one mathematically sound alternative, instead of writing
[5,∞), we should probably be writing {x є R/ 5 ≤ x}.
Once the intervals, in Mathematics, seem to have been inspired by
the Cartesian Plane representation, we should need to fix the
Cartesian Plane in order to finish with the problem involving infinity
and the intervals in Mathematics.
Notice that when we get |0 |1 in the Cartesian Plane, we shorten it
to [0,1], for instance.
When we get |0 >, we shorten it to [0,∞) instead.
The representation [1,∞) should be a translation of the idea {x є R/ 1
≤ x}, but it is, as we have already proven in this paper, another
statement instead: It actually means
|1 |∞.
We do not write |1 |∞ because we do not believe that it is possible to fit the length
in our piece of paper (we write |1 > instead).
In order to translate the idea {x є R/ 1 ≤ x} into an interval, we have,
therefore, to create another symbol, a symbol that does not create
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confusion, as explained before in this very paper.
Once more, writing things in a different way: Because using the
Cartesian Plane as a mathematical tool would lead us to, amongst
other things, believe that [1,∞) is a viable option, the Cartesian Plane
must be replaced with another space.
This special space, beyond Mathematics, or simply outside of it, would
have to satisfy the identified psychological need of mathematicians of
seeing things from a geometric perspective and should not imply any
mistake in the Mathematics through subliminal psycholinguistic
processes.
As a first requirement, this special space, to allow us to state that we
are presenting something more scientifically bearable than the
Cartesian Plane, must provide us with a way of stating clearly
(therefore also geometrically) that the referent infinity does not
belong to the world of Mathematics; for infinity belongs to somewhere
connected, of course, but somewhere of different nature to that of the
world of Mathematics, that is, to what we could call
EXTRAMATHEMATICS (see [Bunyan 1997], for instance).
As an illustrative allurement, Extramathematics could be told to be to
Mathematics the same thing that Metaphysics is for human kind: A
sort of connection between what is of its nature and what is beyond.
Basically, the own numbers, poor things, will never get to infinity if
keeping their nature, the same way humans will never get to the
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spiritual world if doing so: Trascendence is a necessity in both cases,
that is, acquiring temporarily a nature that is different from the usual
nature, that goes beyond.
It is as if God and infinity did not exist in reality (reality = usual
nature's world?), what existed were everything in the middle, but, if
we did not invent God and infinity, we would not have an
anthropocentric world and we then could think that progressing in the
direction of the absolute knowledge or the absolute control over all
that there is is an impossible task, what could lead us to stop
investing in Science, for instance.
Also, to the side of why, human beings apparently love either being
bossed, commanded, adoring things and people, or thinking that they
will be able to occupy the position of whatever, or whoever, to them is
superior some day. If the figure of the boss did not exist, the vast
majority of the human beings apparently would feel lost, in a world
with no leaders to follow, that is, in a world with nobody to be blamed
for their mistakes who is not themselves, what could also lead to
absence of motivation to invest in their human existence, for
instance.
To insert all that we have placed in the EXTRAMATHEMATICS world in
the Mathematics world once more, we will make use of our World of
Infinita, where nothing ever reaches, mainly because it does not exist
but we imagine that it does, and we will work with the infinita the
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same way we work with the imaginary numbers, for the square root of
a negative number exists as much as the physical place that no
number ever reaches does.
3. World of Infinita
Apparently, both mathematicians and logicians, so far, have never
really acknowledged the enormous differences between Language
and Mathematics, perhaps not even as school disciplines.
For instance, if one utters, in the English language, I will go to
infinity!; is that right or wrong?
The answer is simple: There is no right or wrong here, for a person
may utter whatever they like, sensible or not; there are no rules for
utterances. There are fixed rules for punctuation, for spelling, and
etc., but not for what may, or may be not, uttered.
They say I will go to infinity! and the message that they wish to
convey is sometimes even fully understood by the intended recipient.
Probably something similar to I will go to nowhere where you can find
me.
Mathematics, however, whilst part of Science, must hold severely tied
discourse, which allow for abstraction over abstraction to develop in
the direction of the true progress. This way, we should always worry
about the study of its terms, making sure that no mistakes occur in its
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lingo, specially inconsistencies.
In human language, the idea of place, when the term infinity is used,
is unavoidable. To mention one example, Look at the infinity! brings
the term infinity associated with the idea of skies, therefore place.
We also say myriad, or infinity, of coins (means uncountably many in
our heads but, for Mathematics, the infinity of coins in this sentence
actually means countably many, for the number of coins produced on
Earth is always finite) and the idea that is intended by this sentence is
that of a subject getting exhausted of trying to even think about how
to count the coins, so that the sensation of the person uttering this
sentence is that of the amount of coins comparing to the size of the
sky, a place: A place of fatigue, fatigue that relates to the repetitive
unsuccessful trials of fully controlling things (from a human
perspective).
Some people seem to have extended the same allowances of the
human language to the mathematical lingo but whilst, in language,
almost anything can be accepted, perhaps under special conditions at
least sometimes; in Mathematics, very little can be accepted, the
most limiting condition of the mathematical lingo being that every
term in it must be univocal.
Those people have wrongly called infinity both the myriad of numbers
around a specific number in the real axis and the place that is not
determined, is never reached in reality by both eyes and imagination.
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The two ideas are substantially different in nature and, therefore,
could only increase mistake, and probability of mistake, in Science if
taken to be the same, that is, if being assigned the same referent in
the mathematical lingo.
Each term of the mathematical lingo should always, if possible, satisfy
a condition that we could call perfect bijection, that is, should
univocally point to a certain object (of either concrete or abstract
nature) and have the pointed object univocally pointing to it in return.
The idea that comes easily to our minds, after assessing the just
exposed issues, is that we should drop the referent infinity and come
up with other names, names that are not common in language,
created for the specific end of pointing to the world references that
were previously designated by the term infinity. That is obviously the
only way to go, once the referent infinity is part of the usual language
and is not univocal, what makes the term useless for mathematical
purposes, as explained before.
Proceeding this way, we start with taking the smallest piece we may
physically, and easily, get in a ruler to mean X . We then say that X is
a unit of measurement for the ruler. Next, we call each unit 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 the
supremum of the infinita, or the infimum of the minus infinita; both
concepts with no reference either in the ruler or in the world we live
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-empty referents (or sigmatoids), but at least expressing the right
idea in the Mathematics.
Our theory also solves the problems pointed by Parr, the issues
involving Aleph: size of infinity, types of infinity, and etc.
Now, 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.
We then understand that 0.5 in the axis actually means 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?).
From now onwards, we write x infinita meaning simply x , the real
number, and such a writing will help us understand and deal with the
physical universe, once it helps us keep in the mind the difference
between what belongs to Mathematics and what belongs to Physics,
what is trivially necessary.
What follows is that it will be square infinita, for instance, while things
are in the context of Mathematics, instead of square meters. This
dissociates Mathematics from reality of things, making it all more
sensible, for the world of Mathematics will never fit reality with 100%
accuracy; it will fit this way only the world of an abstract ruler (for the
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concrete ruler usually bears mistakes of at least physical order) or the
world of a mathematical object in general. This is a necessary
dissociation, precisely to reduce the amount of vain discussions,
unsolvable problems, and paradoxes in Science (those will be stopped
by the time of generation now, as a consequence, at least those
relating to the issues we deal with here).
We then suggest that other elements, besides axe and title, become
essential part of every mathematical graph of the Cartesian type.
These elements form a triplet: size of the infinitum, origin, and time3.
These elements should always respect the given order, so that
anyone who read the graph know what the graph is about.
Notice that, so far, the Cartesian Plane did not have rules for placing
explicit elements from the domain, or from the image, of a function
inside of it. Yet, the Cartesian Plane is one of the most important
mathematical entities ever created and everything else in
3 The time that we refer to here is the time of creation of the graph or of collection
of the data, the latter being preferred, if available, to the former. Notice what a
difference it all makes, in terms of mathematical elements, when it comes to
analyzing graphs and understanding them: Now Mathematics will acknowledge the
relevance of time, for just one second more could mean a centimeter more for
element of domain 5, for instance, x=5. Now, graphs are finally also what
Mathematics is about: Static pictures of a reality that has already passed and is fully
under control (for further details on how important this little modification in the
specifications of graphs in Mathematics is, see our work on Russell's Paradox ([M. R.
Pinheiro 2008])).
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Mathematics is found completely tied to rules.
We are, therefore, starting to axiomatize this piece of Mathematics,
once every mathematical object, if truly mathematical, must be born
of some sort of axiom of definition.
It is worth mentioning that our choice of infinitum size should be
logical.
To mention an example, we may get a graph where the domain
elements differ by 0.5, the lowest-in-value domain element is 1, and
the image elements differ by 0.2. The person thinks that they are
ready to draw the graph by 2 pm of the 15th of June of 2011. In this
case, we may elect 0.5 our infinitum size in the horizontal axis
(smallest necessary slice of the domain) and 0.2 our infinitum size in
the vertical axis. Because the maximum detail of the data is one
decimal place, we'd better choose this format for every spelled
member of the axis. On the other hand, it may be better placing the
origin at (1;0). We would need all the details about the data to make
sure that our choice is the logical choice, but we do not have the
image of the points for our example. Assuming that this is all we
know, our graph, to follow our rules, should bring the triplet ((0.5;0.2),
(1;0), 15 /06/2011 at 2 pm) soon after the title.
The difference between the World of Infinita and the Cartesian Plane
might be subtle, but it helps absolutely everyone involved at both
subconscious and psycholinguistic levels. Now, everyone is kept
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aware of the presence of an infinite number of members in each non-
degenerated interval of the real line. On the other hand, with our
triplets, we also become aware of the difference between the
infinitum of size 0.5 and the infinitum of size 1.3, for instance, that is,
we understand that there are less elements in one than in the other.
Some changes will not be a big deal. For instance, all the operations
that used to return infinity in Mathematics so far will return one of the
options below instead:
a) Sup Infta;
b) Inf -Infta.
Besides, all the operations that used to contain infinity4 will now have,
instead, one of the elements above (a or b).
Worth remembering that the way infinity is currently found in Mathematics makes us
think that infinity is an actual number , passive of becoming a member of divisions,
multiplications, and etc. However, this is incompatible with the current knowledge of
the meaning of the entity created by John Wallis in 1655.
With our notation, we now have something that is, in Mathematics, passive of proper
4 Originally, the symbol used for infinity was found replacing a determined numeral
in the sequence of the Roman numerals (see, for instance, [Weisstein 1999]).
Notwithstanding, in Mathematics, we cannot accept symbols, or words, for its lingo that have already
been used somewhere else to mean something else in the own Mathematics, for everything in
Mathematics should be univocal. Therefore, we are actually fixing more historical mistakes than the ones
we have initially waved with by producing this new notation, or these new referents, for the mathematical
lingo.
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inclusion in a division, in a multiplication, or in any other operation, as a member.
Even the immediate understanding of what is going on in the operation of division
between 1 and x, for instance, using the example from [Weisstein 1999], when x goes to
what should be the largest real number available, is now incredibly improved, for all is
consistent and coherent with the pre-existent mathematical concepts that form a
requisite, in terms of learning, for us to be able to understand what goes on in the
operation calculating the limit of .
While, before we change things, a teacher would have to teach infinity soon after
introducing the concept of limit, now they may simply add the concept of limit instead,
following all other mathematical concepts, so that things do not look as if they are being
manufactured , forced , or forged .
4. Practice: One example.
We now present one example of a graphical situation in which, previously, we would be
including the Cartesian Plane as environment of display to show how easy it is to
replace the Cartesian Plane with the World of Infinita.
From [Dendane 2008]:
Problem 5: The cost of producing x tools by a company is given by
C(x) = 1200 x + 5500 (in $)
a) What is the cost of 100 tools?b) What is the cost of 101 tools?c) Find the difference between the cost of 101 and 100 tools.d) Find the slope of the graph of C?e) Interpret the slope.
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To draw the graph describing the problem, the best thing to do is dividing the given
function by 1000. In this case, each unit of the previous function will be replaced with
one thousandth of it in the graph.
Once the main intents of the problem are seeing the slope of the graph of the function,
we should choose at least two domain numbers to work with.
Suppose that we choose x = 0 and x = 1, attaining C 10=5.5 and C 1
1=6.7 .
Going beyond the need (calculating the slope), we also choose x = 2 and form one more
ordered pair: (2; 7.9).
This way, our infinitum could be worth 1 for our horizontal axis and 1.2 for our vertical
axis. The time of collection of the data may be assumed to be today and now, for it is
not relevant in the problem proposal or we cannot investigate the origin of the data. The
origin of the system may be the usual one: (0;0). This way, we would end up with the
following graphical description for our situation:
World of Infinitum or W. I.
((1 tool; 1.2 dollars (original cost ÷ 1000)), (0;0), 15th June 2011 at 3:00 pm)
C. D. Infta
D. Infta
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In the World of Infinita, we call our Cartesian look-alike axes Domain of Infinita and
Counter-Domain of Infinita5
, instead of Domain and Counter-Domain, which are
usually referred to as x and y, and represent their names by means of acronyms.
Were our W. I. displaying the graph that describes the Problem 5 in full, our horizontal
axis would show the numbers 0, 1, and 2 and our vertical axis would show the numbers
5.5, 6.7, and 7.9.
The complete graph for the Problem 5 should read, in the W.I.:
Cost per tool, company unknown, slope problem
((1 tool; 1.2 dollars (original cost ÷ 1000)), (0;0), 15th June 2011 at 3:00 pm)
7.9 C. D. Infta
6.7
5.5
D. Infta
0 1 2
Notice that the person dealing with the representation of a situation in the W. I. will
have no difficulties in understanding that because the numbers hold clear separation, or
5 We do not call the horizontal axis tools and the vertical axis cost because the axe cannot be called that
way. Basically, if we do such, we will end up with a slope that is 1.2 dollars/tool , rather than 1.2, what
is unacceptable in the Mathematics (remember that the slope is the tangent of the angle in which the
line lies over the horizontal axis).
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distinction, amongst themselves, and the physical objects, described by them in the
graphs, are continuous in nature, or because the physical entities are not passive of
fractioning beyond a certain level and the mathematical entities are, there is a huge
distinction between the physical and the mathematical universe, which can only be dealt
with through an artificial, highly manufactured, and exotic, world, such as the World of
Infinita that we here propose.
5. Conclusion:
In this paper, we have proposed the World of Infinita as a replacement
for the Cartesian Plane.
The World of Infinita (W.I.) is, finally, a proper mathematical tool, once
its use does not generate mistakes in the Mathematics.
We have also proposed that we abandon both the symbol and the
referent infinity in the Mathematics, once we prove that they both
constitute mistaken conceptions when the foundations of both
Mathematics and Language are considered.
To replace the referent infinity , we have created the referents
infinitum and infinita and we have made them be represented by
abbreviations plus existent mathematical symbols.
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6. References:
[Bunyan 1997] A. Bunyan, T. Benwell, M. Errey, et al. Prefixes. (1997).
Englishclub.com. http://www.englishclub.com/vocabulary/prefixes.htm. Accessed on
the 15th of July of 2009.
[Dendane 2008] A. Dendane. Linear function problems with solutions.
(2008). Analyzemath.
http://www.analyzemath.com/math_problems/linear_func_problems.ht
ml. Accessed on the 17th of July of 2009.
[H. C. Parr 2003] H. C. Parr. Infinity. (2003). Available online at
http://www.cparr.freeserve.co.uk/hcp/Infinity.htm. Accessed on the 3rd
of May of 2008.
[Merriam-Webster-1 2009] Merriam-Webster's authors. Infinity. (2009). In Merriam-
Webster Online Dictionary. http://www.merriam-webster.com/dictionary/Infinity.
Accessed on the 15th of July of 2009.
[M. R. Pinheiro 2008] M. R. Pinheiro. Completeness and consistency of
Arithmetics. (2008). Preprint located at scribd, illmrpinheiro and
illmrpinheiro2. Accessed on the 1st of August of 2009.
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[S. Schwartzman 1994] S. Schwartzman. The Words of Mathematics:
An Etymological Dictionary of Mathematical Terms. MAA.
ISBN:0883855119.
[Weisstein 1999] E. W. Weisstein. Infinity. (1999). MathWorld.
http://mathworld.wolfram.com/Infinity.html . Accessed on the 16th of July of 2009.
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