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-1ON DENOTING FOR ARITHMETIC (PEANOS) AND IDENTIFYING A 5 th

NATURAL DIMENSION IN MATHEMATICS 1

I. M. R. Pinheiro 2

Abstract:

In this one more paper, we destroy one more idol in Logic: Arithmetic is actually complete,

suffices defining it correctly, as with what it refers to. As a side result, we also present a new set

of axioms for Arithmetic, and it is going to be an infinite set instead of a set with only 8

elements...We seem to bring actual progress to at least six major issues in Mathematics via the

present paper (Russells Paradox, Gdels most famous assertions, Peanos axioms, inclusion

relation basics, coordinate system of reference, parallels problem).1 According to [P. Schwarz 2008] , we hold three spatial dimensions, to which Einstein has added time, so that

they became four dimensions for reference of objects, what simply makes sense with our assertions in this paper.

With Superstring theory, however, they got to imagine 11 dimensions, and others, apparently. Here, we go up to

Einstein only, once Superstring theory is not yet fully understood by us. From superficial understanding, it does look

like they do not address our main issue, far simpler than what they deal with, primary issue: simply dealing with the

already existing Cartesian coordinates, but attaching always a fixed point in time, `t`, to them (`t is going to be a

constant, in each Cartesian time dealt with, and Mathematics will always deal with one at a time. We are creating

mathematical ways of refining language to the level already demanded by any mathematical problem in two

dimensions - issue neglected so far - in a way to generate confusional talk, as well as writing, for some. That goes

against the ethos of Science and, therefore, needs fixing and updating). We seem to be working a single step away

from what is not tangible, what we talk about supposed to be the own connection, if any exists, between what may

be seen by a rigid Science, Mathematics, and what may be experienced by a human being, from a physical/chemical

point of view. Thus, the 5th dimension might be, or might be not, the missing coordinate to precisely refer to the

physical/chemical world. Basically, people had skipped this coordinate and gone from Einsteins time to Superstring

coordinates. We then have managed to interpolate a coordinate to complete the theory, or make itmore complete.

2 Postal Address: Po Box 12396, ABeckett st, Melbourne, Victoria, 8006. E-mail: [email protected]. M. R. Pinheiro 1


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Key-words:

Peano, Arithmetic, completeness, consistency, complete, consistent, natural, proposition,

premise, parallels, inclusion, set, statement, denotation, Gdel.

Introduction:

Gdel has apparently written that Arithmetic could never be contained in any axiomatic system

which were not either incomplete, or inconsistent, but never both, according to a few sources.

That is a very strong statement

Basically, axiom is all which cannot be proven, all we assume to be true: the fundamental

stone for any theory .

However, lots of confusion seem to be living inside of everyone dealing with the words

regarding logical systems so far

First of all, the difference betweenproposition and statement does not seem to be as clear as it

should be.

Basically, a statement which does not hold logical value is simply a statement, whilst those with

logical value will be called propositions (they actually refer to them as premises these days).

For instance, uttering `uh: `uh is a statement, a declaration, but never a proposition, for nothing

may be inferred logically from that.

However, `the chair is blue is an assignment to chair of a match in the color spectrum and,

therefore, a proposition, or almost, once everything in language is context-dependent (to be a

scientific statement, that is, a proposition, we would actually need to specify this shade of blue

technically, lets say shade 55 in the color spectrum of factory X).

Another interesting point is: if we have `the chair is blue, `the table is red, `my sister is too oldI. M. R. Pinheiro 2


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and `I am then going to invite only Sue, the first two are statements, for they hold no logical

value for the conclusion, which seems to be deriving solely from the fact that the sister is,

indeed, too old. Notice, however, that if we increase the context size, that situation may change,

for logical situations are almost as dynamic as real life: `the chair is blue, `the table is red, `my

sister is too old, `therefore I am going to invite only Sue, `because red and blue are color for

kids. Now, the addition of another sentence made it all logically necessary for the conclusion, so

that they are now all propositions. Interesting enough that it was premature judging the three first

sentences in our case: It is all very interesting with language

Statement, involving Arithmetic, would be, for instance, `adding two numbers is merging the sets

corresponding to the number of units of counting of one number with the other, and then

providing the merging referent as response. This is astatement of definition , and therefore a

clear proposition (logical value,foundational ) for the operation of summing.

Another statement would be, for instance, that assigning a number to a certain set of units of

counting. Such statement would also be a proposition (foundational definition). Thus, only with

the natural numbers, there is an infinite number of them, and they are essential for Arithmetic to

be sound: the elements to be addedThis way, Peano was also a good joker when coming up

with only 8 axiomssomeone obviously forgot to state that, apart from the designations for

units of counting, we hold 8 axioms in Arithmetic(see [Alozano 2004], for instance).

In what follows, we will write first about Gdel and his statements on Arithmetic, criticizing and

explaining it all in detail, to then write our Conclusion, based on the development which takes

place before its writing.

Arithmetic and Gdel

The proof of how impossible it is to `count the binary infinite sequences apparently generatesI. M. R. Pinheiro 3


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Gdels (1906-1978, see [E. Weissten 1996], for instance) proof for incompleteness. Cantor,

between 1873 and 1891, devised very particular proofs for this fact (see [George Cantor 1890]).

We have devised a far simpler way of proving that it is impossible to count binary sequences,

which are infinite (any sequence is supposed to be such, that is, infinite, according to a few

authors, everything else deserving status of set, rather than sequence), however. Suffices

pretending to be writing the natural numbers as binary sequences the following way: we assign a

sequence with the number one replacing the nth digit of the previously solely zero sequence

(every digit being zero) for each natural number n. Once this will cover every natural number,

and each one of the sequences attained this way, this far, may generate an infinite number of new

sequences, which may be generated by simply swapping a zero with a one, we reach the easy

conclusion that the number of binary sequences is incredibly higher than the number of natural

numbers: At least one infinity higher (picking any of the sequences representing any of the

natural numbers, and swapping one digit, at a time, to one, we reach an infinite number of new

sequences)!

So, it is all about comparing sizes of things

Gdel has also claimed that any well-built formula (built according to the well-posedness theory

for formulae in A.) in Arithmetic would have a Gdel number, that is, a string of numbers,

corresponding to it.

Gdel apparently assigned each mathematical symbol to a number and wrote each symbol of the

formulae that way to find a corresponding `image of the formulae via his assignment, so that

formula x, from Arithmetic, would haveG(x) as Gdels representation (see [Henry 2003]).

After mimicking the work performed by Cantor, Gdel ends up with a new formula, allowed by

the system for containing its allowed symbols, what proves that the number of formulaeI. M. R. Pinheiro 4


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available is higher than the ability of counting them via natural numbers. Gdel then (apparently,

according to a few) claims this proves that there will always be a well-formed formula, in

Arithmetic, which is not passive of deduction from the formulae already existing, so that it is

unprovable inside of any system with a finite number of axioms.

This assertion seems unreasonable.

The symbols used in a mathematical formula are not, necessarily, in direct correspondence with

the meaning of the formula: one may get several different formulae, which represent the same

information, for instance, or the same English statement.

So, this is one argument clearly against his claimed-to-be conclusion.

The second is that the deduction of a formula does not have to do with the symbols of the

previous formulae, but with the information content.

For instance:

`5+2=7 is the same as `7-2=5 and etc.

`For all tables of color shade 57, there are table cloths of color shade 45 is the same as `there is

no table of color shade 57 to which there is no table cloth of color shade 45.

So, we have, in both cases, different symbols for each variation of the same information content,

yet there is only one of them which should be listed in our enumeration, for that is already the

information intended: Basic mistakes when trying to fit the English language inside of

Mathematics in bijection.

One of the arguments used by Gdel has to do with a special sentence. It goes like this:

`P does not have a proof in axiomatic system T.

Call the previous sentence P (see [Kleene et al. 1986] , p. 6).

It is claimed that if a statement claims itself not to be provable then we have a sentence whichI. M. R. Pinheiro 5


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cannot be proved either false or true inside of the system under consideration. If we write that as:

`this statement does not have a proof in T, and we consider the possible truth values on it,

according to Classical Logic, we end up with `if the statement is true then it does not have a

proof in T, and `if the statement is false then it does have a proof in T, but its claim is that itself

does not have a prove in T and, therefore, there is contradiction. The confusion that Gdel

suffers from is not different from that suffered by people taking the Sorites seriously, as we have

explained before. Basically, the English words may be applied to more than one object, in the

same sentence, with no mistake, but the mathematical words cannot.

It is not because the writing `this statement does not have a proof in T is also labeled `statement

in English, that is, we may apply the word for both cases (that of `this statement does not have a

proof in T and the own original statement the problem refers to, which is not mentioned in detail

in the sentence), that the word points to the same reference in both cases: there is a `time issue

going on here, which is being disregarded. If one states the word `statement, that is like `x (for

Mathematics), when repeated in a single mathematical sentence: Rigid and inhuman, to make it

short. However, it is also beautiful and perfect, just like God, perfectly inaccessible to anything

which is human, unless via transcendence.

Basically, `this, in English, fits any possible thing seen by the speaker by time he/she utters the

sentence with that word. Of course, the same dynamics contained in Statistics, a human Science,

is present in real life: everything is updated and considers continuous modifications in the world

of reference. However, for Mathematics, one only develops reasoning if `freezing things at

precise time X, that is, if making use of Einsteins coordinate system with a fixed time `t

implied, but not stated clearly, each time the process occurs.

Another primordial point to be made is that regarding the well-posedness theory for PhilosophyI. M. R. Pinheiro 6


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with mathematically, or intended to be dealt with, anyway)). There should be a way of making it

possible for Mathematics, instead of only Physics and Statistics, to include a human factor in its

analysisHowever, the human entity is too complex, as well as loosely determined, always with

some vagueness, to be passive of inclusion in the `world of Mathematics. For instance, some

text books are annoying and seem to demand that the student `draws an expected scene

(expected by the person writing the problem only), with which to work for the problem.

However, the ìmagination of the writer of the problem cannot, ever, be a mathematical being.

Recalling the most basic rule of well-posedness (solve all inside of the smallest context which

fits the whole problem and its expression), that of context, this sort of problem would be located,

as minimum placement, either in Statistics or in Physics, only for requiring ìmagination derived

from `personal interpretation of the words given by the `maker of the problemOne can see

that, even with severely limited constraints on all variables involved, not only mathematical

errors in formulation are found in unacceptable number, in already refereed work, but possible

unwanted interpretation of intended problem (not deviation from expected solution, but

understanding of it) is likely to happen, imagine with loosest scope of all (that involving not only

the imagination of the reader, but the imagination of the own problem maker). Insane are us, in

thinking such cannot ever be marked with a final `right or `wrong, not passive of discussion,

obviously! Look, the day mathematicians and logicians understand that whatever they do reaches

only 20% of human life, at most, and never the actual life, only an imagined life, where

everything is perfect and logical, they will definitely put far more work in order to refine

whatever they write and say to othersAs a simple example, a Mathematics teacher stated, as

final exam question, an exam which was supposed to tell who was àble to chase Mathematics

studies and who was not: A man is observing a painting from a distance of 1m. His eye reachesI. M. R. Pinheiro 8


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the top of the painting at an angle of 20 degrees. The line between his eye and the bottom of the

painting is a straight line. How long is the painting?

Sad enough, the student who was told to be àble had imagined a right triangle. However, the

student who was marked as ìmpossible to ever be able to learn Mathematics had chosen his

triangle to form more than ninety degrees with the walland actually wrote: I am sorry, but your

problem is not good enough for me to have a single answer. However, I will provide you with a

few alternatives: it is possible that the eye of the observer forms ninety degrees, if a line is drawn

from the bottom of it and the line is parallel to the èqualizer line of the surface of the floor. I

then have 20+90+70, what makes sense. In this case, I will get an easy answer, for there is one

meter of distance involved (one side), and all angles. However, if I make him sit, supposing he

were standing in the previous situation, the angle will be larger than 90 degrees, still possible, for

20+100+60 is also 180. In this case, I do not know how much larger I then cannot infer

anything else.

The teacher said ùnable to study Mathematics because I gave him several exercises, all same

way, I taught nothing else in class, so he is supposed to assume it was ninety degrees and make

the calculations, but he never made the calculations, only wrote remarks!

Oh, well, so Logic does not matter more than Mathematics? We all had this thought as premise,

however: that everything from Mathematics could actually be made by means of words only, that

is, without single symbol, but not vice-versa

Of course the teacher is wrong, and problems in Mathematics cannot be based in their own

heads, or teachings, they must be solely based on what is written, for that is what any person

assesses as valid: whatever is written, if anything written was given. Imagination, or habit, are

obviously not mathematical entities. If there are more interpretations allowed, there is no singleI. M. R. Pinheiro 9


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possible right answer, and if there is more than one, we have allowance for anything to happen,

including the student not doing anything. She asked the dimension of the painting, but one could

easily reply, using Logic, that such does not exist because there can only be one, however we get

more than one using different reasoning trends, what creates inconsistency, what blocks any

reasoning in Classical Logic, and stops the flow of the solution, or what entitles any response (in

conflict, everything implies, explosion law).

Unfortunately, Mathematics may only encompass the abstract world. If ever referring to humans,

the problem must contain a drawing of reference for that fact, so that the human part of the

problem is fully fit inside of the world of Mathematics. One cannot simply add a more complex

entity to a Mathematics problem and believe it will be passive of solution there. If the own entity

extrapolates the boundaries of Mathematics, it cannot be solved there, as we explained in [M. R.

Pinheiro 2007], with the well-posedness theory for Philosophy. It really does not matter how

many coordinates we create, in terms of reference, the complexity of a human being cannot, ever,

be reduced to Mathematics, as we have proved in [M. R. Pinheiro 2007]: not even the verbal

expression of a human being there fits

Basically, as we express in the second article of ours on the Sorites solution, it is important, in

Science, more than anywhere else, the `why we do things. If we overlook this step, everything

else implies, that is, logical classical rules rule and, therefore, there is no real progress, just

illusionary, or delusional, progress.

For instance, in another absurd trial, but more refined than the case with P, we find some people

referring to Russell as if Russell (see [A. D. Irvine 2003], for instance) thought a special set R,

supposed to contain normal sets, and those normal sets are defined as sets which do not contain

themselves (which, per se, is already absurd thought), to pose a problem for Mathematics, andI. M. R. Pinheiro 10


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not solely for language, and those who are experts in it, once more.

Basically, they claim that if R is normal, then it does not contain itself, but if it is normal, it

should also be inside of itself.

This is all ridiculous, and even primary students who are good in the English language are able to

see the fallacy involved: there is a temporal problem there, a very clear one. Mathematics is the

static picture of everything done in Statistics for a good reason: Time does matter!

Basically, whilst R is not built, there is no R to be analyzed. Therefore, R cannot, ever, possibly,

contain itself whilst is still being defined. On the other hand, once defined, it cannot be changed,

because it has become an axiom of the system involved, for it is a definition, for Gods sake!

What that means is that we cannot actually, formally, mention a name in terms of that set: it is an

`unnamable sort of set. If mentioned, only via elements and its definition. Why? Because giving

it a name will create same sort of inconsistency present in the case of the variable X, from our

earlier writing hereIt is true that the set of natural numbers contains itself? No! Never.

Basically, when it is being formed, we give it a name (or assign it to a pointer), which is `natural

numbers`. It cannot contain itselfa set has to be more than its elements, it is a pair (name;

elements). Getting rid of the confusion is essential workInteresting how Mathematics teachers

have been teaching wrong for ages: a set is equal to another if it is contained in the other and the

other is contained in itself (or some of them, anyway). The truth of all is found at [H. Langston

2008]: they are equal if and only if they are equalThis way, a set could never, possibly, contain

itself, for itself is an axiom of assignment (name; elements), and not only a letter, an empty place

holderA set containing itself is humanly impossible, and Mathematics was created by human

beingswhatever they cannot see, Mathematics will not either. Write a set by time `t. This is an

operation which is never completed if you are still writing it, you only know who is in the set byI. M. R. Pinheiro 11


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last element. Case you then add whole set as element, to be coherent and claim that the set then

contains itself, for instance, you need now to add everything as last element of it (`last not being

relevant here), what will create an infinite loop and, as Mathematics loves, it may only be true in

its limit of inclusion, that is, in this case, it will never, realistically, be true, but we get tired and

say it is possible where we cannot see, just like the parallels meeting at infinity

Basically, set B given, B contained in B is only achieved when n is infinity for the progressive

set of inclusions of the previous set development in current updated set. See:

Bo={a1,,aN}

B1={a1,,aN,{a1,,aN}}

B2={a1,,aN,{a1,,aN},{a1,,aN,{a1,,aN}}}

Bn=BN,

that is: lim Bn = BN, when n goes to infinity (this is also confusional statement, it is just better

than stating that a set may contain itselfwhat is absurd. This is confusional because it mixes

Physics with Mathematics, that is, human perception with rigid Science, what is not scientifically

sound for Mathematics: what we are able to cope with, our speed in writing, and the actual

truthjust like the parallels case, what we are able to see at a time, in a single picture, with no

further thinking, and the actual truth: basically, infinity fits a slice, of any size, of a ruler, as small

as we wish for (or as small as our eyes need)then take parallels there, they will reach infinity

in the reals, and yet they will never meet, proving the own thought of this to be as absurd as a set

containing itselfnot even the empty set could, possibly, contain itself for a name has been

given, that means a pair, same case as before).

A set is not what it refers to, Mathematics defined a set as being a pair of elements, instead. And,I. M. R. Pinheiro 12


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pair. Once the pair is axiomatized, then one may use only the name, as reference for the pair

formedthings made accurate, that is, scientific, things solved. Now, is R normal in the correct

case? No, R is abnormal, and no doubts about it!

(it is only possible in the 5th dimension, so that R is quite normal, rather than abnormal)

Another interesting thing is that writing R={(R; elements); R} is actually still incorrect, for the

symbols and calculations, inferences, etc, we still do not hold infinity fully dealt with in the axiomatic worldif we

try, we will go human, what ruins all: infinity is the figure we cannot reach ever(a person might not be able to

reach a figure in a ruler for physical impairment, so that is not good enough as universal definition). Perhaps we can

do better: infinity is what goes beyond any mathematical figure (spirit also does). One can see it is not a

`mathematical being. At the same time it is, for we even deal with it and understand its occurrenceso it must be a

metamathematical being, similar to metaphysics, which deals with whatever goes beyond Physics, by definitionit

is not èthical, then, to include infinity, or mix it, with pure mathematical elementsone would, perhaps, state: do

we have to change the Cartesian Plane? Infinity seems to be reached infinitely many times in a single unit step from

a single axis thereNobecause we are not `talking about it there, and we never will, for it is not something we

can talk aboutas soon as we state something like às x goes to infinity then the function goes to zero, we must

read that as às x goes somewhere else, where we cannot see, the function will be, in that world, zero. Where is

infinity? Not on the graph! If it were on the graph, we would have a precise number for it, or close, and, therefore, it

would be the limit of the function when x goes to that particular number (we see) from the graphnot infinity. We

believe it is trivial to understand that if you are able to locate a particular point in a graph, you are able to draw a

vertical line, forming right angle with the x axis, to determine precisely where the point is located on x, with very

little mistake, so that it will never be infinity, no matter how much we try (the universe of the rulers cannot reach

infinity, of course why? Because it is our own eyes and hands which built the ruler and our hands, eyes, and

instruments, cannot go beyond certain thickness, what will beset any trial of going places we are unable to locate

precisely, or almost, in the rulerinfinity is where the finger and the ruler cannot point and, therefore, no computer

program either, for the grid for the computer screen is also built based on what we can deal with, for we are those

building the machines, with our Logic). All that means is that infinity might be there, but will not make of theI. M. R. Pinheiro 14


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name would have to be another, or we are using Maple, where such is possible: we cannot, in

Mathematics, have this occurring at the same time, once after the new set, containing axiom and

previous R, is being formed, we cannot give it the same name of an already existing set, it has to

be another. It cannot, ever, happen at the same time. Only Maple may account for that with the

`:= symbol. That means: forget previous definition of R, it is now suchR stops existing as

before, and starts existing as now defined, otherwise inconsistency(which is not allowed in

Mathematics, ever). Another way of seeing that is that we would have to give it another name

Cartesian graph something inconsistent. However, claiming it is there will make our mathematical discourse

inconsistent, so it is better, for our own sake, to always state that the world of infinity does not fit the world of the

Cartesian graphs, and is not accessible by us physically (yet?), but it is accessible by our abstract entities, for the

numbers must definitely know who infinity is, same way the souls will always know the way to GodSo, we would

have a coordinate system with five elements (3 from the 3D Cartesian, 1 from time, and 1 from infinity), where the

infinity coordinate would accuse 0 if it does not appear or 1 if present in the system of reference. However, it may

appear in any of the original reference system coordinates, so that the coordinate for infinity should at least bear 3

place holders, instead of one, so that we know to which axis it refers. Even though infinity is reached several times

between a single real number and another, we are unable, at this point in time, to come up with a single practical

example in which mathematical reasoning would lead us to refer to any of those infinities there, in between. For this

reason, the system last suggested, as reference, seems to be complete to account for the Mathematics world so far.

Interesting enough, it is one more dimension if not activated, but it becomes 3 as soon as activated, that is, as soon

as the progression of the figures `calls for it, establishing the needed connection, not existing that farthat is why

we state infinity would be the closest the mathematical world could possibly get to the human world:

transcendenceit is definitely not inside of Mathematics (perhaps yet), yet there is a primary trial of `boxing it

therenow, there is a difference between this fact and the assertion that the world of Mathematics is then

incomplete, or inconsistent, because of suchit is, perhaps, missing quite a few axioms for the element infinity,

what may mean simply going from the reference system suggested here onwards, what we shall soon endeavor to do

ourselves.I. M. R. Pinheiro 15


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insaneif it ever did, it would be there as element, what means already defined by time of

`pointing; what cannot, ever, be the casea name is a complex entity and, as soon as we

associate it with a complex entity, we understand that time of baptism is extremely relevant (like

a nuclear bomb at our door!). A person is born little, they then grow, they get to keep their name

only because it is in the English language. Mathematically, and scientifically, however, they are a

different being each, and every, even thousandth part of second(the complexity of a being

cannot, ever, be described scientifically, never in real timeit will never be possibleby time

the machine produces reading it is already something elseobviously and triviallyby time of

the human eye observing the fetus, the image arriving to mind, the actual being, is already

another being5the time a human sees is also different from the machine) the same chain of

mistake error we find between baby in tummy, machine, observer/reader of scan, and finally

doctor, we find with the translation, as we pointed out before, just in smaller scale because the

doctor is also a dynamic entity, not written somewhere, not abstract, like the text. If we

understand all this, and accept it all as what gives relevance to life, what makes it interesting, we

will also understand how trivial Mathematics is in this so complex universe, or how trivial IT

SHOULD BE, anywayit is all about static pictures of things that will change all the time, so

that it is always wrong for real life and will never be good enough for those who are really nasty

about correctness. However, it is perfect for the abstract world if laws of definition, which

sustain its perfection, are finally respected. Otherwise, even there, it will fail and produce

5 Interesting enough that this is like an example for parallel worlds: a world is that which is mathematically

happening, which is never going to be accessed by the being reading it, and another is what they read from their own

limited perception of the actual worldof course, there is also the possible `Matrix effect (reference to last movie

of the series): an actual high chance that the world is yet another thing, which is not the actual world, time wise, or

the world perceived by vast majority of peopleI. M. R. Pinheiro 18


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inconsistencies.

On same realm of things (see [A. D. Irvine 2003], for instance), we find another gem`From P

we may infer logically P Q, but from P and P Q, we will infer QThey mention this as a big deal. Sincerely, at the same point in time, either you have P or its negation. How is it

possible, IN MATHEMATICS (!), having both, please? What are the mathematical entities which

would be there and not be, at the same time, as Ontological Paraconsistency would like to state

as truth: Whatever is not, is obviously something else than whatever isfor Gods sake! If they

do not occur at the same point in time, how can that generate any problem in any deduction

whatsoever? Logic is, once more, just like well-written, well-posed, mathematical problems, or

even logical: attached to a context, a context which is human, and that is all own humans may

deal withIt does include, obviously, minimum human environmental conditions (time,

location, participants of relevance, probably others we are currently unable to mention). A few

mathematical entities may exist per se, of course, and so will, most of the time, such as triangles

we create from our own imagination, or circles, or functions: Whatever is abstraction over

abstraction, already axiomatized in full, will be passive of creation by us with no context

whatsoever. In this case, however, we must keep in our minds the `trigger rule: if human matters

are implied by time of assertions, then it is not abstraction over abstraction.

What is P? P must be an assignment of some sort, must mean a previous axiom, even if

temporary, that is, something for that specific problemshow us then a P, which is a

mathematical entity, that `may be, and `may be not, at same point in time, as well as same

conditions, and we will believe this is a problem for scientists to worry aboutby the time it is

not true anymore, it cannot be the same P: that is trivial! P must bear the four point reference,

always, at least, even if such is not spelled out in the statements!I. M. R. Pinheiro 19


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If multiplication is defined as it is, based on the sum, then it is proven that Arithmetic is complete

for first order as well.

They then claim that the Second Order Arithmetic (that involving quantifiers) is not complete,

providing a reason for Gdel to be correct.

Second Order Arithmetic is obviously not complete because to create a `for all statement, one

does not need to check each element of the first order: the statement may be born on second

order and be not deductible from whatever existed in first order because it is impossible to

enumerate all natural numbers, for instance, in the clearest case. If we tie the application of

quantifiers to what we may count, however, then quantifiers may be included in Arithmetic and

we do then have completenessFor instance, take A={a, b, c, d}. If we claim that `a belongs to

the set of natural numbers, and so dob, c, and d , we then have a valid logical inference: `for all

x, x inside of A, it is true that x is also inside of the set of natural numbers. And there is no doubt

about what is included in A and what is not, so that any assertion about A is easily told to be true,

or false, in those regards.

This way, there is at least one sentence which is not provable from the first order propositions, or

even from the second order ones: `For all may only be inferred from another `for all or from

`there is not a single element which does not.

If one defines Arithmetic to only be that of first order, then it is complete, and that may only

make sense, for Arithmetic is about summing, not about generalizations. Apparently, the

problem held by many researchers in the area, in what regards completeness, is the acceptance of

the induction process, as formal sets of foundations, to support conclusions. The other issue some

had was Russells issue, on sets of sets.

Oh, well, bad on them, who did not read the Bible of Mathematics with the Bible of the EnglishI. M. R. Pinheiro 21


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language, at the same time, so that it is all compatible.

Induction is a better reason than Russells paradox (again, is there a real paradox? We start

thinking they are all like Parallax mistake, only an equivocated observation, superficial, of

things), in terms of Science. However, it is pretty clear that induction does generate righteous

conclusions and one may easily go from conclusion to theorem if told there was induction and

vice-versa, so that there is both soundness and completeness going on there. One cannot simply

state Arithmetic is incomplete, or inconsistent, or any system containing it: a proof is necessary.

However, we had proof of the opposite and all examples, in the sense they would like to have

seen, were proven wrong here.

Right names given, Arithmetic is, finally, complete and consistent. If the deduction of a sentence,

with a quantifier, cannot be attained via axioms from Arithmetic, it is because the sentence does

not belong there: as easy as that. The other point is that if a true sentence, that is, a proposition,

sound in Mathematics (well-posedness) is ever found to exist, but it is not passive of deduction

via the system rules, then one of these things has to be true: the person uttering that proposition

has incurred in a fallacy or the system is incomplete, in the sense that it should have included

that proposition as axiom of foundation, what simply means `include that proposition in the set

of axioms of the system and re-build it.

Logical systems do hold their symbols, which are mandatory to exist in higher number than the

symbols from the system formed by them for they may generate more than one system: different

ones.

In the literature, they mention the self-referential statements. However,P (1), for instance,as

mentioned before here (2), in this very paper (3) (and notice how many referents (3 in total),

in the English language, we feel obliged to use to point precisely to the P we wish for, by now,I. M. R. Pinheiro 22


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possible, once they are unprovable in the system, undesired presences).

x = y -> Sx = Sy is another axiom of Arithmetic (see [Storrs McCall 2008], for instance).

Easy to see how the last axiom is also a direct consequence of the statement of definition the way

we wrote it.

There is a basic mistake, solidified with time, which appears in the Arithmetic theory: the own

definitions have to be axioms of the system, but they usually do not include it there. Included,

however, as we propose, perhaps we get no inconsistency, or incompleteness, or thoughts going

on about them whatsoever

Inconsistency means we may infer two conclusions, fully contradictory, from same set of

premises/propositions (a premise is like an English statement for us, as before explained. It may,

or may not be, a proposition, which is something context-dependent, as also explained before in

this very paper. One may wonder why we state a premise is like an English statement. The reason

behind that is that the word premise is used by several people, outside of Science, to mean

whatever they hold as paradigmGood scientific terms must hold maximum uniqueness as

pointers, so that `premise` could not be a good one for that end).

That would be a very weird assertion to be made about Arithmetic. As far as we know, such

contradictory set of conclusions was never mentioned in the literature

And, in fact, Gentzen, 1936, has proven the consistency of Arithmetic (see [E F Robertson 2005],

for instance).

With this, what remains to Gdel to wish for is that Arithmetic were incomplete.

Incompleteness of a logical system means that not the whole set of possible logical statements

may be proved inside of own system, that is, there will always be a possible statement, from the

allowed set of propositions (well-formed, according to rules of the system), which may not beI. M. R. Pinheiro 24


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proven to be true, or false, using the postulates of the system.

With Arithmetic, such a statement does not exist.

Interesting that it suffices renaming things to what they actually are, and there is no statement not

passive of being proven true, or false, in Arithmetic.

For instance, the initial assignment of names to numbers, to the actual sets containing the units of

counting, is arbitrary and, therefore, could not be seen as anything different from `naming

things: It is baptizing objects from our imagination with something we can refer to in written, or

spoken, language. Baptism may only be considered axiom, once it is an arbitrary assignment:

there is no possible logical universal agreement on baptism, but it has to be imposed upon things

so that we can talk, and write, about those things the name refers to. As it is necessary for the

theory to be referred to, must be included on the set of axioms for that system. Therefore, for

Arithmetic, it will be the whole set of the natural numbers: an infinite number of axioms added to

our previously found 8!

If the definition of summing, which is also a name, is an axiom, why would not the assignment

of names to quantities?

So, they are all axioms, all contained in Arithmetic, and Peano did a lazy job presenting only a

reduced number of axioms for Arithmetic when there is actually an infinite number of them

(including all number assignments, it is an infinite number of them).

Conclusion

Arithmetic is both complete and consistent: any claim opposing this must be accompanied by

other counter-examples, different from the ones so far, all proven to be equivocated, mostly for I. M. R. Pinheiro 25


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same reason (shortage of understanding of human kind by those doing Mathematics or Logic).

The work performed by people like Frege and Russell is the most important factor for any

Science: the right language for communications. Unless a scientist is eternally on Earth, how can

Science ever progress without most objective lingo of all? The important is communicating

things to the level Science demands, that is, so that a person simply reading the paper of

someone else will understand everything to top detail, no matter who they are, as long as they

have been introduced to the rigors of Science. A paper can only be good if the majority of people

in Science, or Science literate, may assess itThose defending the opposite, that the good thing

to do is `hiding, is `making it mysterious, are obviously committing crimes against human kind

as repulsive as torture, brain-washing, slavery: they cannot, ever, be considered scientists at all.

Prizes to scientists must consider first rule: simplicity and assessibility (as well as accessibility).

Why? Because if even with a whole editorial board we get papers like those we mention (see H-

H inequality and S-convexity, or others of ours), imagine if we intentionally limit the amount of

people who are able to criticize/read them? Vast majority of scientists does not have time to spare

criticizing research of others, that is, contributing to another persons research (actually, also

deserving remark: for free?), imagine making it all difficult? It is obviously a huge favor if a

paper of someone gets to be well criticized before acceptance for imagine finding out, on your

500th published paper, that the logical scheme you took for granted as right is actually wrong, and

you have been using it in almost your every mathematical proof? Better falling from the horse at

the beginning of the race than almost at the final line, and winningWhat we want is that those,

who are top students, find Science a comfortable place to be at, not the exam cheaters instead:

Do it right, and it will be a welcoming place for those who matter for whoever got degrees

cheating will never love learning, or teaching and, therefore, will never be ethicalmake itI. M. R. Pinheiro 26


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bearable for the good students and everything is coherent with the democratically chosen

principles: Science is definitely not a place for everyone, or should not be, starting with those

who wish to make it inaccessible intentionally.

In short: lets do the right thing, no matter how historically relevant (how do we actually measure

that is another issue: quantity of papers produced? Number of non-thinkers copying?) the author

is: If something is blatantly incoherent with all foundational theory that far, like the set-

containing-itself-thing, we immediately yell it is, not seeing the king naked and, taking politeness

as excuse, telling others he was dressed with most modern fabric ever, from top designerswe

go one, or several, wrong steps back but we re-do it right so that when we progress it is for real,

for it would not be Science otherwise, only schizoid delusion!!

Another point is that about the symbol `subset of in Mathematics. There is something which

needs to be fixed there, for long creating confusion everywhereBasically, we have been using

the same delimiters for parts of sets and sets, that is, artificial boundaries, which do not exist,

which we insert to write about pieces of sets, and actual boundaries, which exist due to our

definition of a new set, such as the natural numbers.

We should have a symbol for when the boundaries actually do not exist but we are creating a

subset from an existing set (so that it is not a new definition, or axiom, only new presentation of

the elements of a set) and another for when the set were created via axiom, so that there is no

possible confusion, or mistake, in the notation (remember that the English words should match

the Mathematics symbols, and different meanings should deserve different mathematical

symbols for Mathematics has to be precise, and univocally interpreted, what is not a condition

for language).

The imaginary line could then be something like `!!, why not?I. M. R. Pinheiro 27


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This way, via axiom we create the set B, for instance B={a, b, c, d}, but to express the idea of a

subset, which is not there physically as a set inside of another, we write !!a, b!! C B, for instance.

The reason for this further fixing is obvious: if things remain as they are now, we will have to

agree that a set is contained in itself (no distinction between the physically existing line around a

set and the imaginary line around a subset, so that the elements of the set contained in the set are

equated to the whole axiom attached to the set, what is absurd and creates inconsistencies in

Mathematics, what is unacceptable).

References:

[Storrs McCall 2008] Storrs McCall . The Consistency of Arithmetic, found online at

http://www.mcgill.ca/files/philosophy/The_Consistency_of_Arithmetic.doc, as seen on the 27th

of April of 2008.

[E F Robertson 2005] J J O'Connor , E F Robertson . The real numbers: Attempts to understand.

Found online atI. M. R. Pinheiro 28
http://www.mcgill.ca/files/philosophy/The_Consistency_of_Arithmetic.dochttp://www.mcgill.ca/files/philosophy/The_Consistency_of_Arithmetic.doc


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http://www-history.mcs.st-andrews.ac.uk/HistTopics/Real_numbers_3.html, visited on the 30th of

April of 2008.

[D Jabcquette 1991] Mojzesz Presburger, Dale Jabcquette. On the completeness of a certain

system of Arithmetic of whole numbers in which addition occurs as the only operation.History

and Philosophy of Logic, Volume 12, Issue 2 1991 , pages 225 - 233.

[George Cantor 1890] George Cantor. Uber ein elementare Frage der Mannigfaltigkeitslehre.

Journal of the German Mathematical Union (Deutsche Mathematiker-Vereinigung) (Bd. I, S. 75-

78 (1890-1)).

[Henry 2003] Gdel numbering. Planetmath, accessible via

http://planetmath.org/encyclopedia/GodelNumbering.html, seen on the 27th April 2008.

[Kleene et al. 1986] Eds: S. Feferman, J. D. W. Junior, S. C. Kleene, G. H. Moore, R. M.

Soloway, J. V. Heijenoort. Kurt Gdel Collected Works Volume I Publications 1929-1936.

Oxford University Press, 1986.ISBN-13: 978-0195039641.

[M. R. Pinheiro 2006] M. R. Pinheiro;A Solution to the Sorites , Semiotica , 160 (1/4), 2006.

[M. R. Pinheiro 2007] M. R. Pinheiro. The inferential step in the Sorites paradox: logical or

human? Submitted, preprint located atwww.geocities.com/mrpprofessional, as seen on the 30th

I. M. R. Pinheiro 29
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of April of 2008.

[Alozano 2004] Alozano. Peano Arithmetic. Planethmath as seen at

http://planetmath.org/encyclopedia/PeanoArithmetic.html, on the 27th of April 2008.

[E. Weissten 1996] E. Weissten. Gdel, Kurt. Eric Weisstens world of biography, accessible via

http://scienceworld.wolfram.com/biography/Goedel.html, as seen on the 27th April 2008. 1996.

[A. D. Irvine 2003] Irvine, A. D., "Russell's Paradox",The Stanford Encyclopedia of Philosophy

(Summer 2004 Edition) , Edward N. Zalta (ed.), URL =

http://plato.stanford.edu/archives/sum2004/entries/russell-paradox/.

[P. Schwarz 2008] P. Schwarz. Looking for Extra Dimensions,

http://www.superstringtheory.com/experm/exper5.html, as seen on the 01st of May of 2008.

[Dale Group 2001] Dale Group. Matryoshka Doll (How Products Are Made). Date: January 1, 2001.

Accessible online at http://www.encyclopedia.com/beta/doc/1G2-2897000064.html.

[H. Langston 2008] H. Langston. Discrete Mathematics, Lecture 2, Logic of Quantified

Statements, Methods of Proof, Set Theory, Number Theory, Introduction and General Good

Times, p. 35, found online at http://www.cs.nyu.edu/courses/summer06/G22.2340-001/lect/lecture_02.pdf , seen on the 01st of May of 2008.

I. M. R. Pinheiro 30
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