Points immediately adjacent to a point on the number line can also be I call this the infinitesimal number system. The line segment is exactly one geometric point larger then the Number line can be identified using interval notation. Real numbers can be counted by simply counting all of the immediatelyĪdjacent points on a number line. According to Euclid the set of prime numbers will never get exhausted. This is going on, in every step using a new prime number. If a new list is created including these antidiagonals then its entries are enumerated by the powers of 5 and its antidiagonals (never more than a countable set) are enumerated by the powers of 7. The antidiagonals of this list (never more than a countable set) are enumerated by the powers of 3. > The virtue of this method is that it avoids to squander the natural numbers: The entries of the Cantor-list are not enumerated by all natural numbers but by powers of 2. Otherwise recognize and admit that all your life you have adhered to a nonsense theory, which is not only wrong but also completely useless. If you have a valid counter argument, then let me know please. If the number of paths should be larger than the number of nodes, then you must some time start to buy paths which do not yield any node. I don't think so, but I can meet this argument: From every bought path you get only cents for the first 100 nodes not yet owned by you. Well it has been mentioned that receiving infinitely many cents per path makes things uncertain. I have never received a counter argument, even from students of mathematics who occasionally have visited my lecture. (Every transaction can be performed in half of the remaining time.) If set theory is true, you will get bankrupt finally because there are more paths than nodes. If set theory is wrong, you will get infinitely rich. Buy further paths for one cent each and get one cent for each node not belonging to a path you own already. Buy a path for that cent, get one cent for each of its nodes. Look my game "Conquer the Binary Tree" shows this in utmost clarity. Pseudo mathematics would be the claim that there are more paths than nodes in the Binary Tree. The resulting list has, yet again,īe prepared for the resulting snow-storm of waffle. There are uncountably many anti-diagonals and every new prime enumerates The why the argument is wrong argument: it assumes the conclusion. The complexity argument: if his assumption about there being onlyĬountably many anti-diagonals was correct, why did he use such a complexģ. Supposedly, is a function wm from N to yet the anti-diagonalĬonstructed from flipping digit n of wm(n) is provably not in the imageĢ. The plain counter-argument: the result of this construction is, So, students, if WM presents this nonsense to you, be brave and counterġ. > the set of prime numbers will never get exhausted. > going on, in every step using a new prime number. > more than a countable set) are enumerated by the powers of 7. > entries are enumerated by the powers of 5 and its antidiagonals (never If a new list is created including these antidiagonals then its WM is assuming the result he is trying to Of course you can! Map + to 1 and - to 0 and we have the binary digits Many anti-diagonals as there are infinite sequences of +s and -s. But we could also have a "decrementing" anti-diagonal thatīut we could also choose to either increment (-) or decrement (-) atĮach digit and still have an anti-diagonal. so the "incriminating" anti-diagonal startsĠ.7826. Here's an example of list of reals in indexed by WM's powers of
#Enumerating internal phone numbers mod
The conventional way to construct the anti-diagonal is to change digitsīy, say, incriminating each digit mod 10, with the technical detail ofĪvoiding using a 9, so we might map both 8 and 9 to 0.
Particularly one of WM's students, pause here and see if you can prove So how can we show that there are /always/ uncountably manyĪnti-diagonals for any enumeration of the reals? If you a student, Primes construction: number any initial list of with the even numbersĪnd then enumerate the supposedly countable anti-diagonals using the odd If there were, as WM incorrectly assumes, onlyĬountably many anti-diagonals there would be no need for the powers of
There are, in fact, uncountably manyĪnti-diagonals. Here WM explicitly assumes a fact logically equivalent to the conclusion > numbers: The entries of the Cantor-list are not enumerated by all > The virtue of this method is that it avoids to squander the natural > numbers belong to a countable set of cardinality |ω^ω| = ℵo. Assuming that countability is a sensible notion, I show that all real
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