Godel's Theorems And Others'

ON THE CONSISTENCY AND THE COMPLETENESS OF ARITHMETIC I. M. R. Pinheiro1

Abstract: In this paper, we destroy one more idol in Logic: Arithmetic is actually both complete and consistent. As a side result, we propose a new set of axioms for Arithmetic, and it is going to be a set of infinite cardinality instead of a set with cardinality five. We explore at least six major issues in Mathematics: Russell’s Paradox, Gödel’s most famous assertion, Peano’s axioms, inclusion relation basics, coordinate system of reference, and the parallels problem. Key-words: Peano, Arithmetic, completeness, consistency, complete, consistent, natural, proposition, premise, parallels, inclusion, set, statement, denotation, Gödel.

1. Introduction:

It has always surprised us that something which seems clear, Arithmetic, has generated so much scientific debate by the time of Gödel, Church, Turing, and so many others. Because of this, we have decided to investigate the issue further. Before we proceed, we must make sure that the readers understand that whenever we make use of the sigmatoid 'Arithmetic', we actually refer to the word as it is used by the vast majority of the scientific authors. As a reference for what we mean, please visit [Causey 2006], p. 232 (basically, the natural numbers plus the operations which return values inside of the natural 1 Postal Address: Po Box 12396, A’Beckett st, Melbourne, Victoria, 8006. I. M. R. Pinheiro (32 pages)

E-mail: [email protected] 1

numbers set). In this paper, we actually wish for providing a definite proof that Arithmetic does not suffer from any possible problems: No inconsistencies, no incompleteness, nothing. We base ourselves in already published research, collecting bits and pieces here, as well as in our own argumentation. As a plus, we start writings on a `parallel world' for Mathematics, that is, an `alternative abstract space'. Our sequence of presentation is: 2. Gödel's assertions on Arithmetic; 3. Propositions, axioms, statements: How to tell the difference; 4. Some of Gödel's developments on his assertions on Arithmetic; 5. Some of Gödel's thoughts on axiomatic systems; 6. The human factor and the so striking, surprisingly unnoticed, differences between what is human and what is passive of mechanization; 7. Other philosophers whose reasoning seems to match Gödel's somehow; 8. Conclusions; 9. References.

2. Gödel's assertions on Arithmetic

Gödel has allegedly written, according to some (see, for instance, [Darling 2004], p. 136), that if an axiomatic system may contain Arithmetic then it must be either incomplete or inconsistent. Truth is that he never said that. Gödel has actually stated that the system, which is strong enough for the Elementary

Arithmetic of the natural numbers, would be syntactically incomplete 2 and that the consistency of Arithmetic could not be proved inside of itself (see [Jacquette 2002], for instance, p. 327).

In the lines which follow, we will then go through each one of the parts involved in Gödel’s assertions on Arithmetic.

3. Propositions, axioms, statements: How to tell the difference?

A declaration, in English, when seen by `scientific’ eyes, may be a proposition, a simple statement, or an axiom. Axioms are all which cannot be proven, all we assume to be true, and they form, in a mandatory way, the fundamental stones for any mathematical theory. 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, and also the sort of chair it refers to (like model, origin, and etc.)). Another interesting point is: If we have `the chair is blue’, `the table is red’, `my sister is too old’

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A system which is syntactically complete allows any proposition to be proved either true or false by considering the axioms of the system itself. I. M. R. Pinheiro (32 pages)

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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 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 colors for kids’. Now, the addition of another sentence made it all logically necessary for the conclusion, so that they are now all propositions. 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 for the operation of summing, possibly an axiom, it all depending on whether our axioms allow this statement to be proven, case it will be a proposition, or not. Another statement would be, for instance, that which assigns a number to a certain set of units of counting. Such a statement would be an axiom. 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 simply teasing us when he came up with only five axioms for Arithmetic over the natural numbers: Someone has obviously forgotten to state that, apart from the designations for units of counting, we hold five axioms in Arithmetic (see [Causey 2006] , p. 232, for instance). Besides, Peano has also forgotten to spell out the nature of the 'successor' function, and that is essential for us to single out our usual successor function, for there are infinitely many functions, which could be named `successor', and still satisfy his axioms, even adding two units, or getting the successor of the successor, making use of the term as in the English language.

4. Some of Gödel's developments on his assertions on Arithmetic

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]). Gödel has also claimed that any well-built formula (built according to the well-posedness theory for formulae in Arithmetic) in Arithmetic would have a Gödel number, that is, a string of numbers, corresponding to it. Gödel has apparently assigned a number to each mathematical symbol, consequently succeeding in translating any mathematical formula into strings of those numbers, 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 already existent formulae, so that this particular formula is not provable 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 I. M. R. Pinheiro

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information, for instance, or the same English statement. Thus, this is the first argument clearly against his claimed-to-be conclusion. The second argument 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’. We then have 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.

5. Some of Gödel’s thoughts on axiomatic systems

One of the arguments used by Gödel to prove the incompleteness of the axiomatic system, which could possibly contain Arithmetic, 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 of it, according to Classical Logic, we end up with either `if the statement is true then it does not have a proof in T’ or `if the statement is false then it does have a proof in T’, but its claim is that it does not have a proof in T and, therefore, there is contradiction emerging from the possibilities, or from both possible interpretations attained under Classical Logic. The confusion that Gödel suffers from is not different from that suffered by people taking the Sorites seriously, as we have explained before (see [Pinheiro 2006-2009b]). Basically, the English words may be applied to more than one object, in the same sentence, with no mistake, but the scientific 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, which is being disregarded, going on there. If one uses the word `statement’, that is like `x’ (for Mathematics), when repeated in a mathematical sentence: Rigid and inhuman, to make it short. Basically, `this’, in English, fits any possible thing seen by the speaker by the time they utter 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. Notwithstanding, for Mathematics, one only develops reasoning if `freezing’ things at precise time t, that is, if making use of Einstein’s coordinate system with a fixed time t implied, but possibly not stated clearly, each time the process occurs. Another primordial point to be made is that regarding the well-posedness theory for Science: `Well-posed’ is a mandatory quality of a problem which is passive of scientific analysis. Words such as `this’ need to appear, in a scientific text, accompanied by what they refer to, in order to I. M. R. Pinheiro

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allow of scientific analysis. One thing will always be true: If it is vague in language (see [Sorensen 2006], for instance, to learn more about vagueness), it is definitely unsuitable for Science the way it is. First of all, 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 being able to translate univocally, as the well-posedness principle demands, that into scientific lingo. Ambiguity, or vagueness, besets any possible scientific analysis, especially in Mathematics, a Classical Logic environment. Therefore, a sentence, as described above, is not `ready’ for logical analysis yet and could not, possibly, be included as a logical proposition in the system under consideration. Now, suppose we specify `this’ to mean what is being written by the 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 logically 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 a mathematical locator, that is, the `thing’ X points to may only be precisely `located' if at least four coordinates are attached to it as descriptors).

6. The human factor and the so striking, surprisingly unnoticed, differences between what is human and what is passive of mechanization.

There should be a way of making it possible for Mathematics, instead of only Physics and Statistics, to include human factors in its analysis. For instance, some text books are annoying and seem to demand that the student draw an expected scene (expected by the person writing the problem only), which will be necessary for the addressing of the problem. However, the `imagination’ of the writer of the problem cannot, ever, be a mathematical entity. 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 the intended problem (not deviation from the 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!). We do think that such things cannot ever be marked with a `right’ or a `wrong’, which is not passive of discussion. 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 another simple example, a Mathematics teacher has proposed, as a final exam question, an exam which was supposed to tell who was `able’ to chase further Mathematics I. M. R. Pinheiro

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studies and who was not: A man is observing a painting from a distance of 1m. His eye reaches the top of the frame at an angle of 20 degrees. The line between his eye and the bottom of the frame is a straight line. What is the length of the frame? 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 being 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. Nonetheless, I will provide you with a few alternatives: It is possible that the eye of the observer forms ninety degrees, case in which a line is drawn from the bottom of it and the line is parallel to the 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 I know all angles. Nonetheless, 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, for instance). In this case, I do not know how much larger... .I then cannot infer anything else. The teacher has written `unable to learn Mathematics because I have given him several exercises, all repetition, I taught nothing else in class, so he is supposed to assume it was ninety degrees and make the calculations, but he has 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 a 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 has been given. Imagination, or habit, is obviously

not a mathematical entity. If there are more allowed interpretations, 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 for the dimension of the frame, 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 in Classical Logic). 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 have explained in [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 [Pinheiro 2007], for 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, there is no real progress, just illusionary, or delusional, progress.

7. Other philosophers whose reasoning seem to match Gödel’s somehow

In another absurd trial, but more refined than the case with P, we find some people referring to I. M. R. Pinheiro

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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. 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 it 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. What that means is that we cannot actually, formally, mention a name in terms of that set (thinking the way we have been induced to think this far, as from what has been considered up to now): It is an unnamable sort of set. If such a set gets mentioned, it can only be via its elements or its definition. Why? Because giving it a name will create the 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`. A set has to be more than its elements, it is a pair: (name; elements). Getting rid of the confusion is essential work! Mathematics teachers seem to 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, which, in principle, is an empty place holder. A set containing itself is humanly impossible and Mathematics has been created by human beings: Whatever they cannot see, Mathematics will not see either! Write a set by time t: This is an operation which is never completed if you are still writing it. You only know what the set is by its last element. In case you then add the whole set as an element, to be coherent and claim that the set then contains itself you need to now add everything as a 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 in the case with the parallels (meeting at infinity). Basically, set B given, B containing B is only achieved when n is infinity for the progressive set of inclusions of the previous set development in the current updated set. See: Bo={a1,…,an} B1={a1,…,an,{a1,…,an}} B2={a1,…,an,{a1,…,an},{a1,…,an,{a1,…,an}}} … Bn Є Bn , that is: lim Bn Є Bn , when n goes to infinity (this is also a confusional statement, it is just better than stating that a set may contain itself, what is absurd. The statement lim Bn Є Bn , when n goes to infinity, is confusional because it mixes Physics with Mathematics, that is, human perception with rigid Science, what is not scientifically sound for I. M. R. Pinheiro

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Mathematics. It is all about 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 in a ruler, as small as we wish for (or as small as our eyes need). Then take two parallels there, they will reach infinity in the reals, yet they will never meet, proving the own thought to be as absurd as the thought of a set containing itself. A set is not what it refers to; Mathematics has defined a set as a pair of elements instead. And, as seen, if (R; elements) is our result, and we say that R= {(R; elements)} then it is wrong mathematically because that R would have to point to two different entities 3 in Mathematics at the same point in time: One is a set and its name, the other is a reassignment of reference to the same sigmatoid (R), what makes the statement inconsistent in Mathematics, therefore unacceptable. This fact has to place R in the 5th dimension (the one of the infinity case4). The axiom of 3

Notice that (R; elements) is part of an axiom, even if temporary, and (R; (R; elements)) is part of another one.

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From [P. Schwarz 2008], we infer: To the three existent spatial dimensions, Einstein has added time, so that we ended up with four dimensions for each object, in terms of references to locate its points. From [Zwiebach 2004], we infer: String Theory has come to make it possible we hold even more than eleven dimensions for each minuscule part of each object (point). In Mathematics, we are able to build abstraction over abstraction, so that it is possible go creating new spaces and new dimensions without even justification, what is apparently the main criticism to the Superstring Theory. However, the well-posedness principle for Mathematics would oblige us to stick only to those dimensions which are actually necessary for the most objective description of an object as possible, everything else being passive of becoming part of perhaps exercises of reasoning in Mathematics, but not of proposed theories. This way, we actually hold four needed dimensions, we know are needed for sure, in any mathematical description of objects (see, for instance, [Rucker 1985], p. 145). As for our work here, we seem to be working a single step away from what is not tangible, what we write about being supposed to be the own connection, if any exists, between what may be seen by a rigid piece of Science, Mathematics, and what may be experienced by a human being from a physical/chemical point of view. The 5th dimension may then be the coordinate to refer to this physical/chemical world. We are then placing infinity in the 'extraphysical' world, where nothing and nobody can really get. If we

formation brings the pair. Once the pair is axiomatized 5, one may use only the name as a reference for the pair formed. Now, is R normal in the correct case? No, R is abnormal, and no doubts about it! Therefore, writing R= {(R; elements)} is actually incorrect, for the name would have to be another name, 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 an axiom and the previous R, starts being formed, we cannot give it the same name of an already existing set, it has to be another name. It cannot, ever, happen at the same time. Only Maple may account for that with the `: =’ symbol. Such a symbol means: Forget the previous definition of R, it is now such. Then R stops divide by zero, we get infinity; if we divide something by infinity, we get zero. Therefore, zero is actually found `married to' infinity somehow, as exotic as it may seem (the `all' with the `nothing'), at least in the 5th dimension. This is a case for the Möbius band reasoning then, that is, what Priest has claimed to like in 2000: The Ontological Paraconsistency (where things may 'be' and 'not be' at the same time). Basically, in Mathematics, we say that something divided by infinity has become zero but, in the actual world, it can only be the case that the thing has become the own infinity, for now we hold the same physical object split into infinitely many units of whatever it has become (if all things are continuous, they all may be split into infinitely many pieces for the cardinality of the ruler is infinity in every 'infinitesimal' piece of it, in terms of the population of numbers, or fractions (in the English language sense) of the material. Once there is a bijection between any piece of the ruler and infinity, then the division has to be possible and the result is obviously not zero. Notwithstanding, the piece is so minuscule that our eyes see it as `zero' would be seen. In the Mathematics, we write it has become zero but, in the physical world, it can only have become the own infinity, for now we hold the same physical object split into infinitely many units of whatever it has become. That world, of infinity, is not accessible yet not even via Physics, as far as we know). We mathematically force the result to be `zero', then, an entity from Mathematics, even because Mathematics cannot change the nature of the material, only Physics may account for that. The world of reference for Mathematics is pre-fixed, from start of analysis, and, in it, one thing may not become another, for it does not have enough tools in its systems to account for that. Once what we see, from the world of Mathematics, is nothing left, only because we are also stupid enough to attach things to our observation when it is Mathematics, it then is told to be zero. But zero what? Zero units of whatever that was there before! Thus, the nature of the matter of the object under consideration has been fixed in time and will be kept until the end of the I. M. R. Pinheiro (32 pages) 15

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 the previous axiom (as shown earlier on here). It appears to be the case that 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).

analysis in any mathematical setup. Oh, well, infinity can fit Mathematics, as in the ruler, even several times. However, the nature of infinity is different from the nature of the entities which belong to the Cartesian Plane: It has to be located in an extra dimension of some sort; It is located in the world which is not represented precisely in language yet. If it is only graphical, it is incomplete, once Mathematics must be able to define each one of its elements with precision, primarily in language, and, if possible, in pictures, not the opposite. Mathematics is obviously about symbols and calculations, inferences, and etc., and we still do not hold infinity fully dealt with in the axiomatic world. If 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 a 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 entity’. At the same time it is, for we even deal with it and understand its occurrence! So, it must be an extramathematical being, similar to those from Metaphysics, a piece of Science which deals with whatever goes beyond Physics, by definition. It is not `ethical’, then, to include infinity, or mix it, with pure mathematical elements. One would, perhaps, ask: Do we have to change the Cartesian Plane? Infinity seems to be reached `infinitely’ many times in a single unit step on an axis there… !The answer is no, because we are not writing about it there and we never will, for it is not something we can write about: As soon as we state something like `as x goes to infinity then the function goes to zero’, we must read that as `as 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 graph, not 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 a right angle with the x axis, to determine precisely where the point is located on the x axis,

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! 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, similar situation to that of the set of Russian dolls (see [Dale Group 2001],

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 have built the ruler and our hands, eyes, and instruments, cannot go beyond a certain thickness, what will beset any trial of going places we are unable to locate precisely, or almost precisely, in the ruler. Infinity is where both the finger and the ruler cannot point and, therefore, no computer program either, for the grid for the computer screen has also been built based on what we can deal with, for we are those building the machines, with our own logic. All that means is that infinity might be there, and that fact will not make of the 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, the same way the souls will always know the way to God. Thus, we could perhaps 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. Notwithstanding, it may appear in any of the original reference system coordinates, so that the coordinate for infinity should bear at least 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 three as soon as activated, that is, as soon as the progression of the figures `calls’ for it, establishing the needed connection, not existing that far (that is why we state infinity would be the closest the mathematical world could possibly get to the human I. M. R. Pinheiro (32 pages) 17

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 the 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 (human universe with Mathematics6), as it is usual for the statements involving infinity (not in the case of the limits for Calculus: There, it is obvious that the limit will be reached when the infinity step over the real numbers is taken), may lead us to confused, or hybrid, talk and writing. With Maple (inside of the machine world), we may 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 3 pm will be called R, for instance, but the set of 3 pm and one second has to be at least R' and so on and so forth. It is either Mathematics, and everything is scientifically defined, or it is language and we discuss it there, as explained in [Pinheiro 2007]. OK, so just to make it shorter: Even wondering about the possibility of a set containing itself is insane… : If it ever world: Transcendence... . It is definitely not inside of Mathematics (perhaps yet), yet there is a primary trial of `boxing’ it there). Now, there is a difference between this fact and the assertion that the world of Mathematics is then incomplete, or inconsistent, because of such: It is, perhaps, missing quite a few axioms for the element infinity, what may mean simply changing the reference system, as suggested here, from now onwards, what we shall soon endeavor to do. 5

Thus, we are actually spelling out what Mathematics has left to be `implied': Every time someone creates a set, the mathematical steps involved are 1) Determine the elements of the set; 2) Choose a name or accept having to always repeat the whole set when referring to it; 3) (Fourdational Axiom) Once the previous items have been addressed, we hold name := (set; name), that is, every time the name of the set is used, we refer to a physically delimited area containing those elements from the Foundational Axiom and vice-versa, and only to that do we refer.

6

Our special assertion on the 5th dimension: It is actually true that if Mathematics holds anything close to human, that connection has to be made via the concept infinity, which will be the closest mathematical idea for numbers to that of transcendence for humans.

did, it would be there as an element, what means already defined by the 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 (as relevant as a nuclear bomb at our door!). A person is born little, they then grow: They keep their name only 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 the time the machine produces reading it is already something else, obviously and trivially. When a human eye, for instance, observes a fetus, the image arriving to mind differs substantially from the current image: It is already another image 7....

.And, to make it worse, the

machine time also differs from the other two. The same scale of mistake we find involving the baby in the tummy, the machine, the observer/reader of the scan and, finally, the doctor, we find involving the translation entities (writer, reader, translator, actual text, and etc.), as we have pointed out before). If we understand 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. Notwithstanding, 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 the 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 defend as an actual possibility: Whatever is not is obviously something else than whatever is! If they do not occur at the same point in time, how can that generate any problem in any deduction? 7 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 world is what the `readers of the world’ read through their own limited perception. And there is still the possible `Matrix’ effect (reference to the 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 the vast majority of the people (or the vast majority of its readers). I. M. R. Pinheiro (32 pages) 19

Logic is, once more, just like the well-written, or the well-posed, mathematical problems, or even logical: Attached to a context, a context which is human, and that is all the own humans may deal with… . It does include, obviously, minimum human environmental conditions (time, location, participants of relevance, and probably other conditions even, which we are currently unable to mention). A few mathematical entities may exist per se, of course, and so they 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. In this case, we must keep in our minds the `trigger rule’: If human matters are implied by the time of the 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 the 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, but both life and language are not accessible via Science, at most part of them is. 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 being the same in Mathematics). The confusion is always generated by some superficial understanding of either the English language or the basic logical principles: In English, we can do it, but not in Mathematics!

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 the 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, what 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). 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 in the second order and be not deductible from whatever existed in the first order because it is impossible to enumerate all natural numbers, for instance, in the clearest case. If we tie the application of the 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 the 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 second order: `For all’ may only be inferred from another `for all’ or from `there is not a single element which does I. M. R. Pinheiro

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not…’. If one defines Arithmetic to only be that of the first order then it is complete. Apparently, the problem held by many researchers in the area, in what regards completeness, is the rejection of the induction process as a foundational axiom to support conclusions. The other issue some have 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 (going from a small sample to `for all', in an inference, instead of testing one by one) is a better reason than the Russell’s paradox (again, is there any actual paradox? We start thinking they are all like The Parallax Mistake: Only an equivocated observation –superficial- of things), in terms of Science. However, it is pretty clear that induction does generate correct conclusions and one may easily go from conclusion to theorem if told there has been induction and viceversa, 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. Nonetheless, we had proof of the opposite and all examples, in the sense they would like to have seen, have been proven wrong here. 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’. The vast majority of the confusional thoughts regarding Arithmetic comes from superficial analysis of the concept Axiom.

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 a possible source). Interesting that this is part of the statements defining the operation of summing: It is trivially included there. Thus, if we take the statements of definition as axioms, 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 not provable 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 have written 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 them there. Included, however, as we propose, perhaps we get no inconsistency, or incompleteness, or thoughts going on about those issues. Inconsistency means we may infer two conclusions, fully contradictory, from the 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 a 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 a contradictory set of conclusions has never been mentioned in the literature. I. M. R. Pinheiro

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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, in a logical system, means that it is not the whole set of possible logical statements which may be proved inside of the 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 (induction should trivially ALSO be part of the axiomatic system for Arithmetic). 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, it must be included in 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 5 axioms (as for the literature here mentioned)! Peano has then performed a careless job presenting only a reduced number of axioms for the natural numbers when there is actually an infinite number of them.

8. Conclusion

Arithmetic is both complete and consistent, but it misses having an exhaustive listing of axioms of real value. Any claim opposing either the completeness or the consistency of Arithmetic must be accompanied by other counter-examples, different from the ones so far, all of those having been proven to be equivocated, mostly for the same reason (shortage of understanding of the human factors by those doing Mathematics or Logic). The work performed by people like Frege and Russell is of primordial importance for any piece of Science: The right language for communications, or the adequate technical lingo. Unless a scientist is eternally on Earth, how can Science ever progress without the most objective lingo of all? What matters is communicating things to the level Science demands, that is, so that any person, simply reading the paper of someone else, will understand everything to the maximum detail, no matter who they are, as long as they have been adequately introduced to the rigors of Science. A paper may only be good, as minimum condition, if the majority of the people in Science, or Science literate people, 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, which are as repulsive as torture, brain-washing, and slavery: They cannot, ever, be considered scientists at all. Prizes to scientists must consider a primary rule: Simplicity and ponderability (as well as accessibility). Why? Because if even with a whole editorial board we get papers like those we mention (see our work on S-convexity), imagine if we intentionally limit the amount of people who are able to criticize/read them? The vast majority of the scientists does not have time to spare criticizing the research of others, that is, to contribute to another person’s research (actually also deserving another remark: For free?), imagine how rare criticism of good quality will I. M. R. Pinheiro

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become if we make it all difficult intentionally? It is obviously a huge favor if a paper of someone gets to be well criticized before its acceptance, for imagine finding out, in our 500th published paper, that the logical schemata we have taken for granted to be correct is actually wrong, and we have been using it in almost all our mathematical proofs? 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 will then be coherent with the democratically chosen principles (Science is definitely not a place for everyone, or it should not be, starting with those who wish to make it inaccessible intentionally). In short: Let’s do the right thing, not mattering how historically relevant (how do we actually measure that is another issue: Amount of papers produced? Number of non-thinkers copying?) the author is. If something is blatantly inconsistent 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 the most modern fabric ever, originating in top designers' facilities. 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!!

9. 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, (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 . Accessed on the 27th of April of 2008, (2005).

[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, Issue 1991, pp. 225 – 233, (1991).

[George Cantor 1890] George Cantor. Uber ein elementare Frage der Mannigfaltigkeitslehre. Journal of the German Mathematical Union (Deutsche Mathematiker-Vereinigung) (Bd. I, S. 7578 (1890-1)), (1890).

[Henry

2003]

Informal

source:

Gödel

numbering.

Planetmath,

accessible

via

http://planetmath.org/encyclopedia/GodelNumbering.html, seen on the 27th April 2008. Formal source: S. G. Shanker. Routledge History of Philosophy, V. IX, p. 26, ISBN: 978-0-415-05776-9, (1996).

[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, (1986).

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

[Pinheiro 2007] M. R. Pinheiro. A List of Statements Contained in A Solution to the Sorites and Further Details on the Solution. Submitted, preprint located at www.geocities.com/illmrpinheiro2, as seen on the 24th of July of 2009, (2007).

[Pinheiro 2009a] M. R. Pinheiro. Promoting the Sorites Paradox to a scientific level. Submitted, preprint located at

www.pdfcoke.com/illmrpinheiro2. Accessed on the 22nd of July of 2009, (2009).

[Pinheiro

2009b]

M.

R.

Pinheiro.

Note

on

Paradoxes.

Submitted,

preprint

located

at

www.pdfcoke.com/illmrpinheiro2. Accessed on the 22nd of July of 2009, (2009).

[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 of 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/. Accessed on the 20th of April of 2008, (2003).

[P.

Schwarz

2008]

Informal

source:

P.

Schwarz.

Looking

for

Extra

Dimensions,

http://www.superstringtheory.com/experm/exper5.html, as seen on the 01st of May of 2008, (2008). Formal source: D. Derbes, L. R. Lieber and H. G. Lieber. The Einstein Theory of Relativity: A Trip to the Fourth Dimension, Paul Dry Books, ISBN-10: 1589880447, (2008).

[Dale Group

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

http://www.encyclopedia.com/beta/doc/1G2-2897000064.html. Accessed on the 27th of April of 2007 (2001).

[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. Accessed on the 1st of May of 2008, (2008).

[Zimmerman 2007] B. Zimmerman, E. Hagedorn, V. Calder, R. Avakian and S. Smith. Number of Dimensions. Ask a Scientist,

http://www.newton.dep.anl.gov/askasci/math99/math99265.htm. Accessed on the 22nd of

July of 2009, (2007).

[Zwiebach 2004] B. Zwiebach. A first course in string theory. Cambridge University Press, ISBN: 0521831431, (2004).

[Rucker 1985] R. Rucker. Fourth dimension: A guided tour of the higher universes. Houghton Mifflin. ISBN: 9780395393888, (1985).

[Jacquette 2002] D. Jacquette. A companion to philosophical logic. Blackwell, ISBN: 0631216715, (2002).

[Darling 2004] D. J. Darling. The Universal Book of Mathematics: From Abracadabra to Zeno's paradoxes. John Wiley & Sons, ISBN: 0471270474, (2004). I. M. R. Pinheiro

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[Sorensen

2006]

R.

Sorensen.

Vagueness.

The

Stanford

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of

Philosophy.

http://plato.stanford.edu/entries/vagueness/ Accessed on the 22nd of July of 2009, ISSN: 10955054, (2006).

[Causey 2006] R. L. Causey. Logic, sets, and recursion. Jones and Barlett Publishers. ISBN: 0763737844, 2nd edition, (2006).