ON DENOTING FOR ARITHMETIC (PEANO’S) AND IDENTIFYING A 5th NATURAL DIMENSION IN MATHEMATICS
I. M. R. Pinheiro
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 (Russell’s Paradox, Gödel’s most famous assertions, Peano’s axioms, inclusion relation basics, coordinate system of reference, parallel’s problem).

Key-words: 
Peano, Arithmetic, completeness, consistency, complete, consistent, natural, proposition, premise, parallels, inclusion, set, statement, denotation, Gödel.
Introduction:
Gödel 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 between proposition 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, let‘s 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 old’ 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 a statement 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 added…This way, Peano was also a good joker when coming up with only 8 axioms…someone 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 Gödel 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 Gödel
The proof of how impossible it is to `count’ the binary infinite sequences apparently generates Gödel’s (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…
Gödel has also claimed that any well-built formula (built according to the well-posedness theory for formulae in A.) inn Arithmetic would have a Gödel number, that is, a string of numbers, corresponding to it.
Gödel 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 have G(x) as Gödel’s representation (see [Henry 2003]).
After mimicking the work performed by Cantor, Gödel ends up with a new formula, allowed by the system for containing its allowed symbols, what proves that the number of formulae available is higher than the ability of counting them via natural numbers. Gödel 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 Gödel 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 which 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 Gödel 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 Einstein’s 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 Philosophy and, therefore, Mathematics. Well-posedness is mandatory for a problem being passive of scientific analysis, and a problem cannot, ever, be well-posed, if not totally defined, and understood in full, by those reading it. The word `this’ besets any analysis in Mathematics because we do not know what it is referring to as the sentence appears on its own…One thing will always be true: if it is vague in language (may refer to more than one reference as presented), it is definitely unsuitable for Mathematics the way it is. First, one must master the language, be able to write expressions, and sentences, which will make everyone who reads them see what they see, to then be able to translate, univocally, as the well-posedness principle for Philosophy demands, that into good and sound Mathematics. 
Ambiguity, or vagueness, besets any possible mathematical analysis, that is, classical logic one. Therefore, a sentence as described above is not `ready’ for mathematical analysis yet and could not, possibly, be included as logical proposition of the system under consideration.
Now, suppose we specify `this’ to mean what is being written by time we write: interesting that the only way to do it is writing 
`X cannot be proven in system T’.
Calling this sentence X makes the statement mathematically incorrect, for one cannot use the same place holder/variable to refer to different objects of reference in the same mathematical statement. If the sentence is true, then X cannot, indeed, be proven in T. If it is false, `X can be proven in system T’ and, therefore, the statement bears no logical confusion whatsoever. Notice, as well, that Einstein’s world (according to our sources here listed) of reference is also implied here (whatever is taken to be x will also hold a time coordinate as mathematical locator, that is, the `thing’ x points to definitely encompasses at least four coordinates, if not more (eye of beholder? - depends, of course, on what is being analyzed/dealt 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 analysis…However, 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 `imagination’ 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 `imagination’ derived from `personal interpretation’ of the words given by the `maker’ of the problem…One 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 others…As a simple example, a Mathematics teacher stated, as final exam question, an exam which was supposed to tell who was `able’ to chase Mathematics studies and who was not: A man is observing a painting from a distance of 1m. His eye reaches 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 `able’ had imagined a right triangle. However, the student who was marked as `impossible to ever be able to learn Mathematics’ had chosen his triangle to form more than ninety degrees with the wall…and 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 `equalizer’ 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 `unable 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 single 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, and 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 God’s 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 here…It 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 itself…a set has to be more than its elements, it is a pair (name; elements). Getting rid of the confusion is essential work…Interesting 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 equal…This 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 holder…A set containing itself is humanly impossible, and Mathematics was created by human beings…whatever 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 by 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 itself…what 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 truth…just 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 itself…not 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, as seen, if (R; elements) is our result, and we say that R={(R; elements);  R}, then it is wrong mathematically, because that R would have to point to two different entities in Mathematics, not the same…one is a set and its name, the other is just the name, and if R is element of itself, there is no R which does for Mathematics to be sound,… this fact has to place R, as minimum obligation, in the 5th dimension (the one of the infinity case). The axiom of formation brings the pair. Once the pair is axiomatized, then one may use only the name, as reference for the pair formed…things 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 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 such…R 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 and that would create eternal, or infinite number, of steps, with new names all the way through, and a new set always containing previous axiom and previous set (as shown earlier on here)…so, if mathematicians and logicians can spell and speak, write and read, then they cannot get updated or see the time issue (so far)…conclusion: we are all problematic, but mathematicians and logicians, so far, are realistically lunatic, unable to perceive time change!
Basically, a set which is still being defined cannot be a member of itself because even the own set under analysis does not know who it is, can you all not see? 
Only after a set is defined, and the axiom of definition is created, what means we have already generated a clear association, which, if supposed to last, has to be a pair containing the name and the elements, we know what the set is…how can it be included in itself before we know what it is? Interesting that if that were ever true, a person would also be contained in themselves, just like the set of Russian dolls (see [Dale Group  2001], for instance)…Is it not obvious that such is not possible at all? A person is, at most, equal to themselves, trivially! If adding a finite, but large, number of Russian interpolating dolls, the difference between last insertion and that before the last may make our (faulty, always) perception `see’ as if the Russian doll may, indeed, contain itself (remember that the properties of the being do include its size, on top of everything else…). Why? Human perception fails, always, what means that not even there the assertion will ever be true…mixing things, however (human universe with Mathematics), as it is usual for statements involving infinity (not in the case of Calculus, for limits; once, there, it is obvious that the limit will be reached when the infinity step on the real numbers is finished), may lead us to confused, or hybrid, talk and writing. However, logicians, so far, would be realistically lunatic if accepting that as being `any set is contained in itself’… Now, once we know who they are, what is included in R is the elements, once more, not the pair…why? Because the English name was given to the elements, and R is replacing that, only, that is the only purpose of R…in that point of time, on what was seen there…perhaps, to make it all precise, we should then complete the English words which gave birth to our R (set of sets which do not contain themselves at this point in time, 3pm of the 30th), for Mathematics may look like `Casa da me Joana’, but it is certainly a very well defined place, and we doubt any mathematician of respect would like their own houses loose like that…then, the right name for the updated R is `set of sets which do not contain themselves at this point in time, 3 pm and one second, of the 30th, soon after we have just written what the set R is`. With Maple (inside of the machine world), we may then simply re-define R, update and keep the name, but if with (inside of the world of) Mathematics, we are now obliged to come up with a new name, so that the set from 3pm will be called R, for instance, but the set of 3pm and one second has to be at least R’ and so on, so forth. It is either Mathematics, and everything in scientifically defined, or it is language and we discuss it there, as explained in [M. R. Pinheiro 2007]. We must understand that Philosophy is the mother of Mathematics and, apparently, we have fathered the well-posedness theory for it in 2007, so that we have recently fixed the gap, what made this all possible theoretically. It was obviously not gap in language, for language never had problems, or issues, of that sort, but absence of understanding of the fundamentals of language by those doing Logic and Mathematics, what just corroborates our statements in `Translation avec Samba’: make people learn more subjects well and they will not create problems which will only waste humanity’s time…
OK, so just to make it shorter: even wondering about the possibility of a set containing itself is insane…if it ever did, it would be there as element, what means already defined by time of `pointing’; what cannot, ever, be the case…a 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 time…it will never be possible…by time the machine produces reading it is already something else…obviously and trivially…by time of the human eye observing the fetus, the image arriving to mind, the actual being, is already another being…the 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, anyway…it 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 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 Q’…They 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 is…for God’s 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 with…It 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 problem…show 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 about…by 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!
Apparently, Russell has created a complex theory to explain all this, and called it `theory of types’. Once more, scientists show incompetence in understanding life and language in depth: Things are simple, Science aims simplicity, most basic principle of it, as we keep on stating…
Now, once we know, we are back to what we have stated before: the mathematical prohibition of naming two things the same way. The English words might be the same, but we must either write them all or use a different name in Mathematics, for they are not even close to be the same in Mathematics…the confusion is always generated by superficial understanding of the English language. In English, we can do it, not with variables (and the name of a set, as usually seen in mathematical technical lingo, happens to be one variable)!
It is interesting to see that, nowadays, some mathematical journals oversee this sort of incorrectness, or absence of perfection, in mathematical proofs, and even accept the writing of the computer program Maple, which allows the same variable to become itself plus one, for instance. 
That is OK for computation purposes, but one must remember the origins of the variables and the fact that they may only hold one assignment at a time, not two in the same logical proposition. And, even in Maple, by the time x becomes x+1, x disappears and will never be recovered from the system…Thus, Maple is not against mathematical principles (thanks all), only those making use of it mistakenly, or of its symbols…
With this, the argumentation used by Gödel to prove incompleteness is knocked down.
To be able to utter that X cannot be proven in T, we obviously would have to exhibit the value of X, which we are not able to do so far.
In fact, First Order Arithmetic has already been proved to be complete (see [D Jabcquette 1991], for instance).
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 Gödel 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 completeness…For instance, take A={a, b, c, d}.  If we claim that `a belongs to the set of natural numbers, and so do b, 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 Russell’s issue, on sets of sets.
Oh, well, bad on them, who did not read the Bible of Mathematics with the Bible of the English language, at the same time, so that it is all compatible.
Induction is a better reason than Russell’s 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,… it cannot be easy to have something scientific being passed onwards via written language…), is not a person and, therefore, cannot refer to itself. Who is making it look as if this is the case there is the person writing it, who happens to be a nonsensical writer. It is impossible to write such a thing in Logic, for the rules of assignment are context dependent, and include only one actual World reference to a name in each complete problem, or statement.
It is simply ridiculous even bothering…
Once more, however, it is there, so we must bother!
Another interesting point is that if we ever find out that certain statement does not have proof in a logical system T, we are not going to use this information in any deduction, rather the opposite: we will never mention that statement for it is not a valid proposition in the own system considered and, therefore, it cannot, possibly, be called proposition (instead, we call it statement, English statement). 
All this confusion is always the same we write about since the Sorites paradox solution: logicians and mathematicians seem not to study the basic items forming their own communication (see [M. R. Pinheiro 2007]). Of course there is a difference between the English language and what is logical in it. Only what is logical may be used in deductions (of scientific nature, only ones we are able to refer to in Science,  if ever aiming conversation of scientific level).
They call axiom from Arithmetic, for instance, the fact that any number summed to zero is the own number, that is: x+0 = x (see [Storrs McCall  2008] , as possible source).
Interesting that this is part of the statement defining the operation of summing: It is trivially included there.
So, if we take the statement of definition as axiom, we have that automatically! (This fact makes the axiom invalid, for it is redundant, and well-posedness demands smallest amount of axioms as 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 paradigm…Good 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 Gödel 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 be 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 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 it…Those 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 person’s 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 winning…What 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 ethical…make it 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: let’s 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 designers…we 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 everywhere…Basically, 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?
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 at 
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] Gödel 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 Gödel 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 at www.geocities.com/mrpprofessional, as seen on the 30th 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. Gödel, Kurt. Eric Weissten’s 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.