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[edit] About the incomplete totality of the set of all prime natural numbers

A natural number n is prime if and only if it is divisible only by 1 and itself --- that is, whether n is prime or composite is determined via trial division by every natural number less than n (the tests for divisibility of n could be simply limited to all prime numbers <= square root of n). The “sieve of Erathosthenes” is usually used to determine all the prime numbers, as well as the factorizations of composite numbers, up to a finite natural number n by filtering first all the multiples of 2 greater than 2, then consecutively sifting out from what remains the multiples of the succeeding smallest residual (prime) numbers m (3, 5, 7, …) greater than m --- it suffices to do this for m <= square root of n.

Thus, the definition of a prime number is indeed self-referential or impredicative (that is, a definition of an object or a collection of objects that uses the total collection itself) or violates Russel’s vicious circle principle (“whatever involves all of a collection must not be one of the collection”). Hence, the set P of all prime natural numbers is an indeterminate infinite proper subset of the set N of all natural numbers --- that is, all of P’s elements could not be established as actually determined in some finite amount of time considering the inevitable cumulative divisibility tests to be performed on natural number elements that have inexhaustible successor natural number element.

Richard Dedekind and Georg Cantor introduced the concept of one-to-one correspondence (or bijection) to define an infinite set --- that is, a set is infinite if and only if it could be put in some 1-1 correspondence with a proper subset of itself. Galileo’s paradox concerns the 1-1 correspondence between the natural numbers and their respective squares (that is, a natural number multiplied by itself) despite the blatant fact that the latter set is a proper subset of the former set. Hence, there is an innate contradiction in comparing the sizes of two infinite sets with regard to “one being a proper subset of the other” (that is, one infinite set has ‘fewer’ elements than the other) and to “their elements being in 1-1 correspondence” (that is, the two infinite sets have ‘the same number’ of elements).

Georg Cantor also flaunted the viewpoint regarding the “actually completed totality of an infinite set” in opposition to the traditional Aristotelian perspective of “potential infinity” only. No logical contradiction arises if one perceives as “actually completed totality” the infinite set N of all natural numbers, as well as its infinite proper subsets of “all even” or “all odd” or “all square” natural numbers and many other subsets which have determinate defining 1-1 correspondence function (or rule or pattern) with the set of all natural numbers (indeed, they are all simply isomorphic progressions or infinite monotonic increasing sequences that conform to Peano axioms) just because the specified 1-1 correspondence function (or rule or pattern) induces a “completed mapping” of respective elements that absolutely transcends time.

However, a logical contradiction ensues with viewing as some “actually completed totality of an infinite set” Cantor’s theorem’s proof’s bone-of-contention set T = “set of all the elements of an infinite set which are not contained in their respective images” for a presupposed 1-1 correspondence between the elements of the given infinite set and the members of its power set. Georg Cantor had to posit the idea of “actually completed totality of an infinite set” in support of his alleged “proof” of his hierarchy-of-infinite-power-sets theorem. The cited “contradiction” with his bone-of-contention set T transpires only if one views T as an “actually completed totality of an infinite set” so that the question of T’s generator being included or not in T somehow “makes sense” --- that is, without T being perceived as an “actually completed totality” one could not know whether or not T’s generator is included in T simply because one does not know all the elements of T; on the other hand, with T perceived as an “actually completed totality”, it does not matter whether one knows every element of T but it does “justifies” that T entirely contains all its elements and has a generator g (whatever it is, “g exists” because “T exists”) and this allows for invoking the “contradiction” that g is both included and not included in T due to the pathological definition of T. In this case, the generally accepted position by the majority of current mathematicians (and philosophers) of accepting the “truth” of Cantor’s theorem (that the power set of any finite or infinite set always has greater cardinality than the generating set) is “rationalized” (via an untenable “proof by self-contradiction” argument) by denying the presupposed “1-1 correspondence” between an infinite set and its power set.

That T is an indeterminate infinite subset is easily demonstrated: Suppose N is an infinite set (you may think of N as the set of all natural numbers), P(N) its power set, and f:N<-->P(N) is a 1-1 correspondence and let T = {n in N | n not in f(n)}. Because the empty set { } has no member, it cannot “contain” its own generator, say, n0 in N. Therefore, f(n0) = { } implies n0 is in T --- hence, T != { }. Since f is a 1-1 correspondence, then f-1({n0}) != n0 because we had already presumed f(n0) = { }. So, suppose f-1({n0}) = n1 in N --- that is, f(n1) = {n0}; thus. n1 is in T since n1 is not in {n0} and by the very definition of membership in T. Similarly, we could suppose that n2 in N is f-1({n1}) or f-1({n0,n1}) so that n2 is in T. We could go on ad infinitum and establish that T = { n0, n1, n2, ... } is an infinite proper subset of N (by its definition, T != N since N contains its generator). We stress that we could only conclude that T is an infinite set --- because of T’s incomplete totality, whether f-1(T) is in T or f-1(T) is not in T is undecidable. The Cantorians might argue that f-1(T) is not in T by the very way we constructed T in our preceding argument and, hence, conclude that the presumption of the existence of one-to-one correspondence f between N and P(N) must be denied because of the contradiction with the definition of T --- that is, f-1(T) must be in T. As a counterargument to the Cantorians, we could simply state that T is a finite set at each stage of its construction [that is, f-1(T) is not in T] and the contradiction only transpires when we deem T as an “actually completed totality of an infinite set”.

The same “proof by self-contradiction” folly in Cantor’s theorem’s proof is manifest with the alleged “proofs” of Godel’s incompleteness theorem, Turing’s unsolvability-of-the-halting-problem theorem, Church’s unsolvability-of-the- entscheidungsproblem theorem, Tarski’s indefinability of truth theorem, Matiyasevich’s unsolvability-of-general-Diophantine-equation theorem (unsolvability of Hilbert’s 10th problem), and many others that directly or indirectly invokes Cantor’s diagonal argument --- which claims the non-existence of one-to-one correspondence between N and P(N) --- all of their bone-of-contention subsets of N are indeed indeterminate infinite proper subsets of N; hence, they are not completely constructible. In all cases, the definitions of the cited subsets of N involved negating for some natural numbers some respective properties that references the entire still uncompleted (ongoing construction) subsets of N --- hence, the latter’s membership in the former is indeed undecidable. It is resolutely emphasized that the self-contradiction ensues only if the former is deemed completed --- that is, as “an actually existing totality of an infinite set”. The antinomies only arises in the intentional attempt to construct a set whose membership condition stipulates for at least one of its would-be-infinite-elements to be self-contradictory (that is, to possess and not to possess at the same time and in the same respect the defining attribute for the to-be-constructed bone-of-contention subset of N) upon consideration of its actually completed totality as an infinite set.

An indeterminate infinite proper subset S of N could be considered as an infinite sequence of binary digits with 1 as the nth term, for all n in N, if n is in S or else 0. Even with a prefixed binary expansion point, this infinite sequence of binary digits only identifies a variable or subinterval of the unit interval (0,1) and does not actually define a real number point [for examples, Cantor’s “anti-diagonal number” and Alan Turing’s “computable number” are not constant real numbers (that is, points) but merely variables (that is, subintervals of the unit interval)]. Similarly, the information available about S is sufficient only to identify S as a legitimate variable-subset of N but not as some definite (constant) subset of N --- hence, the undecidability issue on some natural number being an element or not of S. Consequently, this membership self-contradiction ensures the incompletable constructability of S and, thus, it cannot exist as a “completed totality of an infinite set”.

As with an algebraic equation (over the real number field domain) whose terms are composed of constants and a variable --- say, x^2 - pi + 1 = 0 or x^2 + pi + 1 = 0 (here pi, 1 and 0 are real numbers while x is a variable) --- the variable may have 0, 1, several or even an infinite count of solutions which are the constant real numbers --- that is, variables are merely placeholders for constant real numbers. Thus, without a defining function, rule or pattern for the digits ri (for all i in N), the expression “0.r1r2r3…” is a variable and does not identify a real number (which is a constant). Similarly, “0.1234567890…” is not a real number unless all its digits are understood. However, with mutual understanding of its information content, “3.14159…” represents the real number pi.

On the other hand, being an infinite proper subset of N, the set P of all prime numbers could be put in 1-1 correspondence with N. While Cantor’s theorem’s bone-of-contention set T is merely a concocted set, P is a real indeterminate infinite proper subset of N all of whose prime-number-elements could not really be viewed as an “actually completed totality of an infinite set” --- that is, without [and there could not possibly be one --- even if there is one, it must be guaranteed (for example, remember the flop with the Mersenne and Fermat numbers) that the function’s values are all prime numbers but the non-divisibility tests for each natural number by preceding prime numbers would take an infinite time to accomplish) a defining 1-1 correspondence function n <--> pn (the nth prime natural number), one does not (and could not possibly) know all the prime numbers.

It is easily known that the googol + 1, the googolplex + 1 or the googolplex ^ googolplex + 1 are all odd but are they prime or composite natural numbers? As a matter of fact, even if some god started the “sieve of Erathosthenes” from the alleged Big Bang a mere 20 billion years ago, he would still not been able today (or even in the extremely far future) to have segregated all the prime from all the composite natural numbers. Thus, at best, one could only identify (for some finite amount of time) all the prime numbers up to a certain natural number n but there is forever a successor-to-n natural number n + 1 whose primality or compositeness always remains to-be-decided. Also, Euclid’s proof guarantees that there are ever more prime numbers greater than pn, for all natural number n.

Let Pn = {p1, p2, p3, …, pn} be the set comprised of all the first n primes. Now, consider Euclid’s proof of the potential infinitude of all the prime numbers --- then, there exist m > 0 prime number(s) between pn + 1 and p1p2p3…pn + 1. Therefore, there are always numerous (the excluded count m increasing much faster than the actual count n) prime numbers not included in each Pn so there could not be a 1-1 correspondence function f : n <--> pn (brought on by the impredicative definition of a prime number) even though there indubitably exist trivial 1-1 correspondence functions g : n <--> Pn as well as h : Pn <--> pn (it is reiterated that pn and Pn are defined for finite n --- however, the infinite set {pn} of all prime numbers and the set {Pn} of all the sets of first n primes are not completable infinite totalities).

It might be counter-argued that the reasoning above could also apply to, say, the set of all even natural numbers --- that is, suppose En = {e1, e2, e3, …, en} is the set comprised of the first n even natural numnbers. Then, for example, the sum or product of all the elements of En is an even natural number 2m that is not included in En. We simply note that, granting arguendo Cantor’s viewpoint of “completed totality of an infinite set” of the set of all En, this sum-or-product-even-number 2m is readily (in no time at all) identified to be included in Em. On the other hand, Euclid’s proof only establishes that there are excluded prime numbers from Pn --- the natural numbers inclusive between pn + 1 and p1p2p3…pn + 1 still have to be respectively verified for primality by prime numbers some even greater than pn. For example, with P4 = {2,3,5,7}, the natural number p1p2p3p4 + 1 = 211 requires the prime number 13 (also excluded from P4) for its primality divisibility tests.

Therefore, the set P of all the prime natural numbers (which is a real set) could be invoked in lieu of T (which is merely a concocted set) in the “proof” of Cantor’s theorem (applied to the infinite set N of all natural numbers and its power set) --- that is, the natural number generator of P could not in reality be ascertained to belong or not to belong to P simply because one does not know all the elements of P due to its indeterminate “enumeration” that would take eternity (an infinite time) to be “actually completed”. The principal issue that I am raising is that P is a simpler (than T) indeterminate infinite proper subset of N which could not be viewed as an “actually completed totality of an infinite set” just because the primality testing of each natural number would take an infinite amount of time [this concern is not apparent (or not believable) with T whose existence in itself is merely fabricated].

One might argue that some known natural number z could be arbitrarily assigned as the generator of P so that even if P is an indeterminate infinite proper subset of N there is no undecidability issue about P’s inclusion of its generator because the prime natural numbers are in standard monotonic increasing enumeration. But any arbitrary rearrangement of an enumeration f1, f2, f3, …, fz, … also results in a 1-1 correspondence --- say f97, f31, f101, … (that is, the f97 subset-of-N is now generated by 1, f31 subset-of-N is now generated by 2, f101 subset-of-N is now generated by 3, and so on --- the listing of fz = P could be put off indefinitely --- hence, the question of an undecidable generator for P would still come up. Furthermore, the inclusion of its generator in the set Qn of all prime numbers greater than the nth prime number pn would be also still undecidable.

This simple argument clearly supports the contention that, in fact, it is not the supposed 1-1 correspondence between an infinite set and its power set that must be denied in the so-called “proof by self-contradiction” of Cantor’s theorem but the flawed outlook of “actually completed totality of every infinite set”.

The ramifications of the “incomplete totality of the infinite set of all prime natural numbers” are manifold --- foremost of which is the affirmation of Aristotle’s potential infinity only.

1. Dedekind and Cantor’s view of “actually completed totality of every infinite set” based on the concept of 1-1 correspondence is utterly defective (it is emphasized that it is not the concept of 1-1 correspondence that is flawed but the viewpoint of “actually completed totality” of even an indeterminate infinite set whose “exhaustive” enumeration would take infinite time) --- therefore, Cantor’s theory of transfinite sets as well as alleged “proofs” of theorems that encompass “all prime numbers” are untenable.
2. The fundamental theorem of arithmetic [every natural number greater than 1 is the unique product of powers of prime numbers] and the fundamental prime number theorem [the count of prime numbers less than a given real number x is approximately x / ln x] as well as its many variants are valid because only the prime numbers less than a specified natural number are involved.
3. Goldbach’s conjecture [as modified by Euler --- every even natural number greater than 2 is the sum of 2 prime numbers] may be disproved by a counterexample and could be proved only up to some finite even natural number n (however arbitrarily large when given sufficient computing power and time) since the two prime number addends which need to be verified for primality are less than n. In other words, the conjecture could be proved for an arbitrary even natural number greater than 2 and then one invokes the generalization principle to claim the result for all the potentially infinite even natural numbers greater than 2.
4. The twin-prime conjecture [there exists infinitely many prime natural number pairs p and p + 2] and the more general problem whether the linear diophantine equation ax + by + c = 0 (with given integral coefficients a, b, and c each prime to the others) is always solvable in prime numbers x and y could not really be decided (just because there are potentially infinitely many prime numbers whose individual primality still need to be proved by the established prime numbers respectively less than them) and could be proved only by finitistic argument similar to Euclid’s proof of the potential infinity of the prime numbers. . Following the latter, one could readily assert that there exists infinitely many natural number n such that, for the first n prime numbers, both p1p2p3…pn - 1 and p1p2p3…pn + 1 are prime numbers. For examples --- for n = 2: (2)(3) – 1 = 5 and (2)(3) + 1 = 7; for n = 3: (2)(3)(5) – 1 = 29 and (2)(3)(5) + 1 = 31; for n = 5: (2)(3)(5)(7)(11) – 1 = 2309 and (2)(3)(5)(7)(11) + 1 = 2311; etc. The “arguable issue” is whether this contention is already a valid proof of the twin prime conjecture or just another “conjecture” considering that it could not also be actually completely verified nor disproven by some counterexample.
5. If Fermat’s last theorem is restated for all integers x, y, z and all odd prime natural numbers exponent p --- xp + yp  zp [since xn = (xk)p where k is a positive natural number and p is an odd prime number (every natural number n > 2 is either a multiple of 4 or of an odd prime and Fermat’s claim had been proven true for exponent 4 ( hence, for all multiples of 4 exponent)] --- then it is undecidable. The following salient differences are emphasized: (a) if the claim is for “all natural number exponent n > 2” then it could be proved true or disproved by a counterexample; (b) if the claim is for “up to some finite odd prime number exponent”, then it could be proved true or disproved by a counterexample; (c) if the claim is for “all odd prime natural number exponents”, then it could not be proved true (because the set of all odd prime numbers do not form an “actually completed totality”) but it could be disproved by a counterexample. Also, any proposed proof must not invoke “all the prime numbers” or “all the prime numbers greater than some natural number” in its argument. In Professor Andrew Wiles’ Annals of Mathematics article “Modular Elliptic Curves and Fermat’s Last Theorem”, the infinite “product of the principal local units at the primes above p of L(fp<infinity>)” was brought into play. In Professor Andrew Wiles’ Internet article “The Birch and Swinnerton-Dyer Conjecture” (Official Problem Description --- Clay Mathematics Institute), the “incomplete L-series of (an elliptic curve) C (incomplete because we omit the Euler factors for primes p|2<delta>)” was invoked.
6. The Riemann hypothesis [all the non-trivial zeroes or roots of the Riemann zeta function have real part 1/2] as it relates to “all prime numbers” is unsolvable or undecidable. Indeed, its precursor Euler zeta sum-function (for real s > 1) could not be actually split into two infinite sums of reciprocal powers of “all” prime and “all” composite natural numbers (because this separation would take infinite time to accomplish) to derive the Euler zeta product-function involving “all” the prime natural numbers pn. The convergence of the Euler zeta function does not involve “all” the prime natural numbers since the contributions to the limit value of all prime numbers > pn for some n are discarded.
7. Many other conjectures or hypothesis that are generalized “for all prime natural numbers” are undecidable. [BenCawaling@Yahoo.com] BenCawaling 10:03, 12 April 2006 (UTC)

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And if Fermat's Theorem wouldn't be proved till now, you would like to say that this is also not provable. What do you want to say ? That all attempts to proof Riemann's Hypothesis should be stopped ? If you mean that you can prove that RH is not provable then do it and ask some math professors to check it. If not then the whole text is only your opinion without any new cognition and therefore dispensable. 15 April 2006 (UTC) Aschauer

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[edit] Discussion moved from Talk:Prime number

I don't buy it. The primes are computable (i.e., the nth prime is a primitive recursive function of n, and should be considered complete.) — Arthur Rubin | (talk) 01:12, 25 March 2006 (UTC)

Seconded. The primes are as complete as the natural numbers (which are also defined recursively) and only the most extreme type of finitist refuses to acknowledge the validity of any statements about all natural numbers. Elroch 23:22, 25 March 2006 (UTC)
Perhaps you can enlighten me by writing out your "primitive recursive function that completely enumerates the set of all prime natural numbers"? What must be emphasized is that the defining 1-1 correspondence functions, rules or patterns, betwenn N and some of its determinate infinite proper subsets allow for the viewpoint of a "completed totality of an infinite set" since the completion induced by the mapping function, rule or pattern transcends time while an indeterminate "enumeration" must account for the infinitude of time to be completed. The incomplete totality of the set of all prime numbers simply means that the over 2,300 years old traditional Aristotlian potential infinity view is correct (or, at least, not wrong as is contemporarily widely held). In the discussion section for "Cantor's theorem", it was claimed that “An object does not have to be "completely constructible" to be used in a valid proof.” to which I made the following response ==> then you have a self-contradictory (or, at least, selective) understanding of the soundness of the “proof” of Cantor’s theorem (I am assuming that you also believe this) which I am assailing here. As I already noted above, if Cantor’s bone-of-contention set, T = {all subsets of N that does not include its generator in a presupposed 1-1 correspondence between N and its power set P(N)}, is admitted to be not completely constructible then there is _no_ contradiction with the undecidability of its generator being included or not included in it simply because you don’t know _all_ of T’s elements. What I am pointing out in my rebuttal here is that the set P of all prime natural numbers is a simpler indeterminate infinite proper subset of N (just like T but P is not merely a concocted set that T is) all of whose elements could not be viewed as a completed totality of an infinite set — that is, you don’t actually know all the prime natural numbers [at best, you could only know (for some finite amount of time) all the prime and composite numbers less than or equal to a certain finite natural number n (however arbitrarily large) but there is always a successor natural number n+1 to n whose primeness or compositeness you still need to establish]. What is sad to note here is the apparent eagerness with which you accept Cantor’s theorem and the likes that I mentioned above with undecidability issues despite the very elementary counterarguments (understandable by even honor high school students) that I have proffered. I have posted my arguments here to let the readers decide!
I guess you are the extreme finitist which Elroch referred to. In any case, the primes as as complete as the even numbers or the squares. — Arthur Rubin | (talk) 21:27, 30 March 2006 (UTC)
You've edited the section to which I replied. In any case, all extreme finitists other than yourself recognize that the primes are as complete as the even numbers or squares. There is a (provably) finite rule for determining whether a natural number x is prime and for determining the nth prime. The complexity of the rule should be irrelevant to whether the set is a completed or potential infinity. As for the twin prime conjecture, Goldbach's conjecture, etc -- many finitists agree with you, but that discussion probably doesn't belong in this article.
Oh, and by the way, you're wrong about Godel's second incompleteness theorem. The proof is finitary, although it does use the law of the excluded middle, and non-intuitionist constructivists believe there are much simpler unprovable sentences. — Arthur Rubin | (talk) 17:14, 8 April 2006 (UTC)
You're absolutely wrong about Fermat's last theorem. The results for n > 2 and for n > 2 (and prime or 4) are provably equivalent, even by extreme finitists. — Arthur Rubin | (talk) 17:17, 8 April 2006 (UTC)

[edit] Discussion moved from Talk:Cantor's theorem

So what you're saying is that all these professional mathematicians are wrong because of a simple error that only you see? An object does not have to be "completely constructible" to be used in a valid proof. —Keenan Pepper 13:27, 24 March 2006 (UTC)
"_all_ these professional mathematicians are wrong" is a self-contradiction (impredicative!) considering that I am also a professional mathematician who is making the assailment of much of modern mathematics. Besides, there were Henri Poincare, Leopold Kronecker, Luitzen Brouwer, and many others who had foretold of "Cantor's transfinite paradise" as a "pathological disease that would one day be cured". What must be emphasized is that the defining 1-1 correspondence functions, rules or patterns, betwenn N and some of its infinite proper subsets allow for the viewpoint of a "completed totality of an infinite set" since the completion induced by the mapping function, rule or pattern transcends time while an indeterminate "enumeration" must account for the infinitude of time to be completed. The incomplete totality of the set of all prime numbers simply means that the over 2,300 years old traditional Aristotlian potential infinity view is correct (or, at least, not wrong as is contemporarily widely held). BenCawaling 02:48, 27 March 2006 (UTC)
I didn't say "all professional mathematicians", I said "all these professional mathematicians", by which I meant those who disagree with you.
Good come back --- for a 19 year old, you've got an excellent future ahead of you! ... - BenCawaling
I am confused by your use of the phrase "improper subset". What do you mean by it? —Keenan Pepper 03:32, 27 March 2006 (UTC)
"improper subset" should be without the "im" --- a misspelled word (Thanks, my mistake --- I already edited the erroneous word.) ... - BenCawaling
If you truly believe what you wrote --- “An object does not have to be "completely constructible" to be used in a valid proof.” --- then you have a self-contradictory (or, at least, selective) understanding of the soundness of the “proof” of Cantor’s theorem (I am assuming that you also believe this) which I am assailing here. As I already noted above, if Cantor’s bone-of-contention set, T = {all subsets of N that does not include its generator in a presupposed 1-1 correspondence between N and its power set P(N)}, is admitted to be not completely constructible then there is _no_ contradiction with the undecidability of its generator being included or not included in it simply because you don’t know _all_ of T’s elements. What I am pointing out in my rebuttal here is that the set P of all prime natural numbers is a simpler indeterminate infinite proper subset of N (just like T but P is not merely a concocted set that T is) all of whose elements could not be viewed as a completed totality of an infinite set — that is, you don’t actually know all the prime natural numbers [at best, you could only know (for some finite amount of time) up to a certain finite number n (however arbitrarily large)whether or not n is prime or composite but there is always a successor n+1 to n whose primeness or compositeness you still need to establish]. What is sad to note here is the apparent eagerness with which you accept Cantor’s theorem and the likes that I mentioned above with undecidability issues despite the very elementary counterarguments (understandable by even honor high school students) that I have proffered. I have posted my arguments here to let the readers decide!
One might argue that you could arbitrarily assign a known natural number generator for P so even if P is an indeterminate infinite proper subset of N there is no undecidability issue about its generator because the prime natural numbers are in standard monotonic increasing enumeration. But arbitrary rearrangements of a 1-1 correspondence f1, f2, f3, ... … also results in a 1-1 correspondence – say, f97, f31, f101, ... (that is, f97 is now generated by 1, f31 is now generated by 2, f101 is now generated by 3, etc. — hence, the question of an undecidable generator for P would still come up since we don’t know (and we could not possibly know in reality or otherwise) all the prime natural number elements of P.

[BenCawaling@Yahoo.com]BenCawaling 06:16, 28 March 2006 (UTC)

Besides, the primes are "completly constructible". Just because there isn't a simple formula (like 2*n) for them, doesn't mean there isn't a simple algorithm. — Arthur Rubin | (talk) 08:05, 15 April 2006 (UTC)

[edit] Unsolvability of the Riemann Hypothesis

Essay moved to here from Talk:Riemann hypothesis linas 13:14, 18 April 2006 (UTC)


Leonard Euler is known to have contended that the “sum” of the divergent infinite series of alternating 1 and -1 is ½.

I excerpted the following text about a so-called “Euler’s proof” for the infinitude of prime numbers from Paulo Ribenboim’ book “The Little Book of Big Primes” (New York: Springer-Verlag, 1991):

This is a rather indirect proof, which, in some sense is unnatural; but, on the other hand, as I shall indicate, it leads to the most important developments.
Euler showed that there must exist infinitely many primes because a certain expression formed with all the primes is infinite.
If p is any prime, then 1/p < 1; hence, the sum of the geometric series is
\sum_{k=0}^\infty \frac{1}{p^k} = \frac{1}{1-(1/p)}
Similarly, if q is another prime, then
\sum_{k=0}^\infty \frac{1}{q^k} = \frac{1}{1-(1/q)}
Multiplying these equations:
1 + 1/p + 1/q + 1/p2 + 1/pq + 1/q2 + … = 1/(1-1/p) x 1/(1-1/q)
Explicitly, the left-hand side is the sum of the inverses of all natural numbers of the form phqk (h &ge 0, k &ge 0), each counted only once, because every natural number has a unique factorization as a product of primes. This simple idea is the basis of the proof.
Euler’s Proof. Suppose that p_1, p_2, …, p_n are all the primes. For each i = 1, …, n
\sum_{k=0}^\infty \frac{1}{p_i^k} = \frac{1}{1-(1/ p_i)}
Multiplying these n equalities, one obtains
\prod_{i=1}^n \sum_{k=0}^\infty \frac{1}{p_i^k}=\prod_{i=1}^n \frac{1}{1-(1/ p_i)}
and the left-hand side is the sum of the inverses of all natural numbers each counted once &mdash this follows from the fundamental theorem that every natural number is equal, in a unique way, to the product of primes.
But the series
\sum_{n=1}^\infty \frac{1}{n}
is divergent; being a series of positive terms, the order of summation is irrelevant, so the left-hand side is infinite, while the right-hand side is clearly finite. This is absurd.
Why is it clear to you that the right hand side is finite? You have just sketched the proof that it is infinite. Elroch 20:39, 19 April 2006 (UTC)

Now, what is absurd are the ridiculous claims made in this so-called “proof”. Let P be the potentially infinite set of Peano natural numbers {0,1,2,3,…} defined by the Peano axioms for arithmetic where each natural number n has a successor natural number n+1. The fundamental theorem of arithmetic (that is, the unique prime factorization of every n > 1) &mdash we emphasize here the finiteness of every natural number (although there are potentially infinitely many of them) &mdash guarantees the finiteness of the count of all distinct prime factors (however, there are potentially infinitely many of them because there are potentially infinitely many natural numbers). Conversely, if there are only finitely many prime numbers &mdash p_1, p_2, p_3, …, p_n &mdash then the combination-product-of-their-powers could only generate finitely many “natural numbers” [let G be this potentially infinite set of generated natural numbers (that is, multiples of) from a finite set of prime numbers &mdash in addition to 0 and 1). Clearly, G is not equal to P because p_1p_2p_3…p_n + 1 is not a multiple of any of the finite prime factors. It must be emphasized that G is a true finite set while P is a potentially infinite set.

In the above so-called Euler’s “proof” of the infinitude of the prime numbers, the infinite-sum-index (“k”) values of a geometric series involves P not G. If we appropriately apply G as the domain of the infinite-sum index in the so-called “proof”, any geometric series still converges. However, the infinite-sum-index (“n”) values of a harmonic series involves P-{0} not G — hence, it diverges. If we appropriately apply G as the domain of the infinite-sum index “n”, then the reduced “harmonic” series is convergent. Therefore, the cited absurdity in the argument is non-existent.

What must be emphasized in my counterargument to this so-called “Euler’s proof” is the sage realization about the domain of the index of an infinite series or an infinite sum that is the set of all natural numbers each element of which is comprised of prime factors (any natural number itself could be a new prime number different from those previously known). Paulo Ribenboim also cited the so-called “Willans formula for the n^{th} prime” which involves the sum of a finite series whose index domain are the natural numbers from 1 up to 2^n. Clearly, aside from the computational complexity issue, this formula could not be a 1-1 correspondence since, for n > 2, the index values include natural numbers whose prime factors involves not only p_n.

The separation into the two distinct sets of “all prime” and “all composite” natural numbers require that every natural number greater than 1 has to be verified for primality so that the potentially infinite “set of all prime numbers” could never be totally completed.

The “most important developments” referred to by Paulo Ribenboim is Euler’s derivation of his zeta product-function and the related “progress” on the “distribution of primes”. The Riemann hypothesis [all the non-trivial zeroes or roots of the Riemann zeta function have real part 1/2] as it relates to “all prime numbers” is unsolvable or undecidable. Indeed, its precursor Euler zeta sum-function (for real s > 1) could not be actually split into two infinite sums of reciprocal powers of “all prime” and “all composite” natural numbers (because this separation would take infinite time to accomplish since every natural number has to be tested for primality or compositeness) to derive the Euler zeta product-function involving “all” the prime natural numbers. In particular, even though all the infinite sum’s term-values are positive, the rearrangement of these terms in the infinite sum could not be effected since the prime factors of greater natural numbers depend (because of the impredicative definition of a prime number) on the establishment of the prime factors of the lesser natural numbers On the other hand, the convergence of the Euler zeta function does not involve “all” the prime natural numbers since the contributions to the limit value of all prime numbers greater than p_n for some n are discarded (say, by the epsilon-delta definition of limit).

Please read my discussion text Prime number: About the incomplete totality of the set of all prime numbers in the Wikipedia article [[Prime number|]]. [BenCawaling@Yahoo.com] BenCawaling 10:38, 16 April 2006 (UTC)

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Comment: The left side of the equation above is only on par with sum(n=1 to infinite)(1/n) if ALL prime numbers are used exactly one time in the whole product. But than the right side is also infinite. Therefore: What a bosh that the left side is infinite and the right side is finite ? Please learn first basic mathematik ! ... Aschauer 16.April 2006

If you are referring to me with this command "Please learn first basic mathematik!" --- I would rather have you learn to read first. BenCawaling 08:12, 18 April 2006 (UTC)
Your whole comment ?! What a bosh. Your are wasting your time - and the time of others. ... Aschauer 20.April 2006

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