doesn´t define it and avoids that dilemma, by which here a category of quality ... a bijective depiction (look at the first diagonal argument of Cantor for N â> Q).
Assertion: `The grading of power-sets gets inconsistent by definition and can´t be a criterion of uncountability´
Proof: Cantor had formulated an assumption, by which there couldn´t exist a quantity which stays between the infinity of the naturals N and the continuum of the reals R. For to discover that problem, usually the formulation for power-sets is chosen. A power-set, by definition, should be that set, which is the set of all the subsets of an initial set. In relationship to N the power-set P is depicted as P(N). By the power set the graduation of the forms of increase get depicted as: P (P (N)), P (P (P (N))), … and by that the expression `cardinals´ (classes of quantities of numbers (of amount)) gets introduced. The first step of the graduation, P (N), already shall be more `powerful´ (higher in potency) as the initial set N. By N it should be possible to generate elements of the powerset, which number (of amount) should be beyond the number (of amount) of the elements of N, should be more powerful. How the expression `powerful´/ `potnency´ is to understand? The expert opinion doesn´t define it and avoids that dilemma, by which here a category of quality (infinity, infinite number (of amount)), should be distinguishable by a category of value (quantity, distinguishable number (of amount)). By a way out the hypothesis of potency depends on depicting. More precise, on bijective depiction between this infinite sets which should get distinguishable considering their so called potency. If we take away the unlogical attitude of `potency´ of a quantitative course of action, the criterion of bijective depiction will be left. An infinite set shall be the same in `potency´ in relationship to another, if it is able to show a bijective depiction (look at the first diagonal argument of Cantor for N –> Q). Thereby it is not valid, that, by a reverse-conclusion, a difference in potency has to be seen as proven, because a bijectivity wasn´t found so far. The usual refutation proofs with axiom of choice mustn´t be disproved in particular case, if one had understand Brouwer. One constructive counter-example would do. If it would be possible to show a depiction of N to P (N) that way, Cantor did it at his first diagonal argument, such a graduation gets broken at the beginning.
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In contrast to the development (of graduation from smaller to bigger values) of the set of the naturals (0, 1, 2, 3, …; short: n e N), which should be well known, the definition of the power-set of a set is worth to be declared (stated). 1. 2.
P (N) ? N All subsets of N form, as elements, P (N)
Normally at this point the proof would be complete, because by the chapter `Cantor´ no sets, even subsets, could be elements. Elements have to be identical among themselves and atomic (element-name: singular; set-name: plural)! Because an infinite set couldn´t be described complete by their elements, depictions of continnueable development (rule for to develope) do help. By that the ever continuing development of P(n) (with n –> `) stays for the sequence of the finite subsets of P(N). The first elements of the sequence are typical written as: P(0): P(1): P(2): P(3):
{Ø} [emty set] {Ø; {1} } [emty set and element 1 as subset] {Ø; {1}; {2}; {1,2}} [+ new element, + combination(s) as subsets] {Ø; {1}; {2}; {3}; {1,2}; {1,3}; {2,3}; {1,2,3}}
And so on. At P(n) with n = N then symbolic the complete powerset, the set of all subsets of the initial-set, is depicted. For to be unambiguous to the constructs of the second diagonal argument of Cantor, it should be said that the, separated by putting a comma, elements of N are only depictable naming. That didn’t mean value at special position. A second explanatory for an abort is given thereby that the power-set isn’t limited to attract only one set N, containing only one element 1, one element 2, … . But uses, to create its own elements, the elements of N infinite times. And, by that, is declaring that N realiter is coexisting. It seems to be possible to determine any amount of subsets inside N (which number (of amount) will stay in relationship to N at least). But the power-set will stay in relationship to infinitely sets N. Maybe one could see the possibilities to create a subset as to be countable. Then the elements of N, which are used more times, have to be understand symbolic only. A third explanatory for an abort is given thereby that it is argued on an empty set by concluding. Such a definition stays in contradiction to the first two definitions of the theory of sets. Isn’t there any element, so the majority of that, which gets real up to two elements, also isn´t existent as if there would be one element. That doesn´t has to do anything with the possibility of imaging a set as well as an element or to intent to use a nomenclature. The contradiction `emty set´ could be avoided by omission this construct. Inconsistancy is demonsrated by that. Of course it is valid for all graduations P(N). But we will leave aside all abort-reasons to comprehend exemplary, inside the non-logical axioms, that it is not possible to distinguish inside the quality `infinite´ by quantity, by counting out.
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A possibility of listing the development of N and P(N) looks like this for the first four lines of the finite subsets: 1. 2. 3. 4.
0 0, 1 0, 1, 2 0, 1, 2, 3
P(0): P(1): P(2): P(3):
{Ø} {Ø; {1}} {Ø; {1}; {2}; {1,2}} {Ø; {1}; {2}; {3}; {1,2}; {1,3}; {2,3}; {1,2,3}}
At the left side the development for N, at the right side that for P(N). A bijectivity is given, if an index-set could be established.
The relation N to P(N) exists! Each power-set of a finite set (development of the lines) is finit by it´s own, could be indexed by a finite part of N . This means that every subset of N which could be seen as an element of the power-set of N , gets an index. Each further development, even in chain (as a full sequence), could be indexed by N . If by `0 –> {Ø}´ the principle of a relation (from a functional value of N , zero, to an element of P(N), the empty set) is evidenced, then up to line four there is depicted the following relationship: 1. 2. 3. 4.
0 -> {Ø} 1 -> {Ø} 2 -> {1} 3 -> {Ø} 4 -> {1} 7 -> {Ø} 8 -> {1}
5 -> {2} 6 -> {1,2} 9 -> {2} 10 -> {3} 11 -> {1,2} 12 -> {1,3} 13 -> {2,3} 14 -> {1,2,3}
While at the development of N each element gets used only one times (left-unambiguous and also left-total), at the development of P(N) all elements of the preceding lines get listed again. Because this relation also is right-total (by the way surjective), we got more than the goal as a bijective relation. The set of the arguments, the set which pretends the value of the function (N), indexes the values of the goal-set (P(N)), clearly, more times. Do we cancel that expressions of the line by line development of P(N), which are already listed, the origin development of the power-set of N , which is produced so far, will stay at rest. N as the index simply moves back by these positions. Bijectivity is established (violet: to be cancelled). 1. 2. 3. 4.
0 -> {Ø} 1 -> {Ø} 1 -> {1} 3 -> {Ø} 4 -> {1} 2 -> {2} 3 -> {1,2} 7 -> {Ø} 8 -> {1} 9 -> {2} 4 -> {3} 11 {1,2} 5 -> {1,3} 6 -> {2,3} 7 -> {1,2,3}
No doubt about, to be continued line by line (number (of amount) of the lines towards ∞). At each new line — inclusive that by color marked subsets — whole the graduate power-set (P(0), P(1), P(2), … P(n)) gets listed (number (of amount): 2n). In each powerset all subsets of the previous power-sets (number (of amount): 2n/2). By the cancelling of the surplus subsets we develop a one-line depiction, which represents the complete development of all subsets.
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For P(1) the empty set has to be canceled (P(1)\{Ø}), for to depict all subsets which are new developed at this line. For P(2): P(2)\{Ø}; {1}, for P(3): P(3)\{Ø}; {1}; {2}; {1,2} and so on. Because every previous line lists all subsets of all previous lines, we may depict the corresponding power-set instead of the separate subsets. For to generate subsets by P(1): P(1): P(1)\{Ø}, by P(2): P(2)\P(1), by P(3): P(3)\P(2) and so on. And finally, for all n>0: P(IN) = { Ø; {P(1)\Ø}; {P(2)\P(1)}; {P(3)\P(2)}; …; {P(n)\P(n - 1)}; {P(n+1)\P(n)}} This line shows the rising sequence of the graduation of the potency sets only symbolic. The explicit, by one line, depiction of the subsets, developed by rising graduation, should following be shown for the first steps. Graduated: P(0) = {Ø} P(1) = {Ø; {1}} P(2) = {Ø; {1}; {2}: {1,2}} P(3) = {Ø; {1}; {2}; {3}; {1,2}; {1,3}; {2,3}; {1,2,3} P(4) = {Ø; {1}; {2}; {3}; {4}; {1,2}; {1,3}; {1,4}; {2,3}; {2,4}; {3,4}; {1,2,3}; {1,2,4}; {1,3,4}; {2,3,4}; {1,2,3,4}} Linear: P(IN) = {Ø; {1}; {2}; {1,2}; {3}; {1,3}; {2,3}; {1,2,3}; {4}; {1,4}; {2,4}; {2,4}; {3,4}; {1,2,4}; {1,3,4}; {2,3,4}; {1,2,3,4}; … There isn’t just a general rule how to develop a relation. But another system should be introduced. This possibility to index gets estastablished by a general valid character for infinitely many sets with infinitely many elements. Index:
1 2 3 4 5 6 7 8 9 11 12 13 14 15 16 17 18 19 21 …
Sequence-link:
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 …
If the index gets created by the naturals, so it is established that an infinite sequence (set) — the naturals — would be indexed unambiguous (identically to relation) by N , in which every time after nine steps of the normally increase of the quantity of the number a reserve- possibility for to index gets produced. The shortened index, by this way, nevertheless at last will be infinite. All the `reservists´, 10, 20, 30, … are, for themselves, again infinite in number (of amount) and are be able to generate also infinite sequences by omission next higher potency:
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ξ1 = 1, 2, 3, ... 9, 11, ... 19, 21, … 99, 101, ... 999, 1001, … ξ2 = 10, 20, 30, ... 90, 110, ... 990, 1010, … ξ3 = 100, 200, ... 900, 1.100, ... 9.900, 10.100, … ξ4 = 1000, 2000, ... 9.000, 11.000, ... 99.000, 101.000, … and by that: ξ2 = ξ1 ⋅ 101 ; ξ3 = ξ1 ⋅ 102 ; ξ4 = ξ1 ⋅ 103 ; ... ; ξn = ξ1 ⋅ 10n-1 Do we combine all index-sequences on the one hand and on the other hand endless infinite sequences, which are generating the continuum, so a bijective relation between N and R gets possible. More precise and complete: Cantor 3 By the naturals endless sequences with endless elements are indexable. In priciple it is valid that only a single index has to be taken for every way to create subsets (by taking brackets for random chosen numbers of N). From an infinite set, no doubt about N is, endless many indexes could be generated, which could be shaked into each other. The so called power-set is, although absurd by definition, indexable.
••• For previous arguing and depicting deliberately the usual nomenclature as well the axioms of the expert opinion are used. Strict about this, by the part `Cantor´ it wouldn´t be allowed, as previous remarked. The constructs, the symbols used for each number (of amount) greater than one are already subsets which are constructed by the combination of corresponding number (of amount) of elements (more times one). To ensslave the speech in that way and leading to absurd, couldn’t bring stable results for fundamental axioms. Axiom of extensionality: If the elements of the second view (B) are identical to that at the first view (A), then the sets are identical not similar. Mathematics does this by a triple-line. A ≡ B is not logical. At this place we are at terminology not at the technic of the calculation of an equation (3 + 4 = 7 e. g.), by which the equivalence between before and after executing the operation `+ ´ should be depicted. In which the form before the execution only is different, not the quantity of the expression. Axiom of empty set: The existence of the elements as well as the existence of the sets causes one another, are intrinsic one to another. Out of the context giving a name to each of the classifications, the nomenclature, this is valid. By using the empty set as well as the potential endless number (of amount) by the graduation of the power-sets, the expert opinion ZFC borrows an intuitionistic principle without using it in the meaning of the intuitionism.
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Because by the axiom of regularity (foundation) it seemed to be known that a set could not contain itself as an element, it must be a new creation of a set (P(N)) which should be used for the supposed graduation in power in the sense of uncountability. At least there is the quantitative question about the number (of amount) (of the possible subsets), camouflaged by the look of bijection. By the demonstration of the relation the obstacle gets overcome, even inside the illogical definitions. A one to one (definite) connection is presented. The subsets have to be developed by the introduced logic and are staying in bijective relationship to the index which has to be developed linear increasing. The development never ends. There couldn’t be a finit step by which the subset N is listed complete; when N depicts the subset symbolic. By that the axiom of regularity gets supported; independent of the contradiction which results by the definition of a difference from elements among themselves at identical set.
••• No more relation to other axioms here. P (N) isn´t anything else then a suborder of N (among all the pears there are even fouls) because the subsets couldn´t create new sets (sets just aren’t elements). The suborder of N as an index … Above it was pointed out that it couldn’t bring more realization if at the examination the so called graduation of power also would be expanded. N, P (N), P (P (N)), P (P (P (N))), … Only for to show that the fault of the first power-set was repeated, by originating the more times availability of the by N solitary represented, different numbers, the beginning of the second graduation of power should be implied here. We will have a look at the development of the equivalent constructed power-set at second graduation (exclusively Ø). P(P(N)):
{ {{1}, {2}}; {{1}, {1, 2}}; {{1}, {3}}; {{2}, {3}}; {{1}, {2, 3}}; {{1, 2}, {2, 3}}; {{1}, {1,2, 3}}; {{2}, {1,2,3}}; {{3}, {1, 2, 3}}; {{1, 2}, {1, 2, 3}}; {{1, 3}, {1, 2, 3}}; {{2, 3}, {1, 2, 3}}; {{1}, {4}}; …; {{…}, {N}} }
To use a sequentially link of N more times couldn’t be the right approach / beginning for to develop a subset of second order. By calling N all links of the sequence 1; 2; 3; are applicable as existent and uniquely in presence. There couldn’t be a subset-enclosure of a sequentially link of N, which takes a link twice to determine such a subset. `Choice´ (use) and `putting back´ (for to be used again) gets performed in discussion, but doesn’t need to be refuted explicitly, does it? By reversal (reverse-conclusion) we would handle with infinite much sets N, which already are creating the first increase of the graduation of the power-sets. Consequently by the indexes to the members of N there would be endless many index-sets.
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If one has fun in doing `nonsense´ one could index just this set by N too.
Copyright 2010 / 2013 Peter Kepp, Germany (translated November 2018, corrected, extended 2019) [4th part of the der paper-series `Logic of Formalism´] Each propagation, by word, picture, tone or rhetoric (teaching-event) is protected by copyright and needs, if interested, the approval by contract. Electronic transmission is equal to `each propagation´.
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