I 219
Not all abstract objects are properties: numbers, classes, functions, geometric shapes, ideas, possibilities - give up or re-trace abstract objects - can be distinguished by the faithful use of "-ness" from concrete objects.
- - -
II 26
Numbers: quantification is objectification, numeric name - diagonals: are irrational, scope: is transcendental.
Measure: measuring scale: is a multidigit general term, puts physical objects in relation to pure numbers - counting: measuring a class.
>
Measuring/Quine.
II 28
Numbers/Ontology: Numbers are merely a "facon de parler". - Higher classes needed to replace numbers - otherwise there would only be physical objects.
- - -
IX 54
Numbers/Frege/Quine: like predecessors (ancestor): Definition Predecessor/Frege: the common elements of all classes for which the initial condition was fulfilled: "y ε z" and the seclusion condition: which resulted in "a" "z 0 ε z]}. - Problem: the successor relation could also lead to things that are not numbers.
Numbers/Quine: we will mainly use them as a measure of multiplicities (that is how Frege had defined them). - a has x elements"- the scheme goes back to Frege: a has 0 elements ↔ a = Λ. - s has S°x elements ↔ Ey(y ε a n _{y} has x elements.
IX 59
Numbers/Zermelo: (1908)
(1) takes Λ as 0, then {x} as S°x for each x. (i.e. "{x}" always one more than x! - {x} successor of x! - As numbers we then receive Λ, {Λ}, {{Λ}}.. etc.
IX 59ff
Numbers/Von Neumann (1923)
(2) regards every natural number as the class of the previous numbers: 0 becomes Λ again, - but successor S°x does not become {x}, but x U {x}. (Combined with) - 1: as in Zermelo: equal {Λ} - but 2: {0,1} or {∧,{∧}}. - 3: {0,1,2} or {Λ,{Λ},{Λ,{Λ,{Λ}}}.
For von Neuman this says that a has x elements, that a ~ x. (number, equipotent) - that’s just the "a ~ {y: y < x}" from chapter 11, because for von Neumann is x = {y: y ‹ x}.
Decimal numbers/dimidial numbers/decimal system/Quine: Example en gros: comes from "large quantities" = 1 dozen times 1 dozen.
Score: = 20.
Decimal system: Example
365 = 3 x 10² + 6 x 10
1 + 5 x 10
0.
1. Zermelo, E. (1908). Untersuchungen über die Grundlagen der Mengenlehre I. Mathematische Annalen, 65, 261-281. http://dx.doi.org/10.1007/BF01449999
2. Neumann, John von. (1923) Zur Einführung der transfiniten Zahlen; in: Acta Scientiarum Mathematicarum (Szeged); Band: 1; Nummer: 4; Seite(n): 199-208;
- - -
XIII 41
Exponent/high numbers/high zero/high 0: why is n
0 = 1 and not 0? Because we want that
n
m + n is always n
m x n. Example m = 0: then n
1 is = n
0, or n = n
0 x
n; therefore n
0 must be = 1.
Decimal system: the positions correspond to a built-in abacus.
Comma/Decimal Point: was inspired by negative exponents:
Example 3.65 = 3 x 100 + 6 10-1 + 5 x 10-2.
Counting/Division: had little to do with each other before this breakthrough. Because division happened on the basis of division by 2, while at the same time already in the decimal system was counted.
Real numbers: some are finite, e.g. ½ = 0.5.
Decimal Numbers: their correspondence with real numbers is not perfect: each finite decimal number is equivalent to an infinite: Example 5 to .4999...
Solution: the correspondence can simply be made perfect by forgetting the ".5" and sticking to the ",4.999".
Infinite/infinite extension/decimal number/Quine: Example a six-digit decimal number like 4.237251 is the fraction (ratio) 4,237,251
1 million
Infinite decimal number: is then approximated as a limit value by the series of fractions, which of ever longer fractions, is represented by sections of this decimal number.
Limit value: can be here again a fraction Example .333..., or .1428428... or irrational e.g. in the case of 3,14159 ((s) N.B.: here for the first time a number before the decimal point, because the concrete number is π).
XIII 42
Infinite decimal numbers/Quine: we must not regard them as expressions! This is because real numbers that exceed any means of expression are ((s) temporarily) written as infinite decimal numbers. ((s) So one (necessarily finitely written decimal number) can correspond to several real numbers).
Decimal system/Quine: each number >= 2 could function instead of the 10 as basis of a number system. The larger the base, the more compact the notation of the multiplication table.
Dual System/binary/"dimidial"/binary numbers/binary system/Quine: from "0" and "1", i.e. numbers are divided by halves (partes dimidiae): Example
365 = 2
8 + 2
2 + 2
5 + 2
3 + 2
2 + 2
0.
N.B.: Law: every positive integer is a sum of distinct multiples of 2. This is only possible with 2 as a base, no other number! I.e. at 365 the 10² does not occur once, but three times.
Decimal Comma/binary: in binary notation: the places on the right are then negative powers of 2. Example ,0001 is a 16th.
Real numbers/binary notation: nice consequence: if we consider the series of real numbers between 0 and 1 (without the 0), we have a 1:1 correspondence between these real numbers and the infinite classes of positive integers.
Solution: each binary represented real number is identified with a binary extension which is infinite in the sense that there is no last "1".
XIII 43
Integers: the corresponding class of integers is then that of the integers that count the places where the "1" occurs. For example, suppose the binary representation of the real number in question begins with "001011001": the corresponding class of integers will then begin with 3,5,6 and 9. Because "1" occurs at the third, fifth, sixth and ninth digit of the binary expansion.
N.B.: the class thus determined is therefore infinite! Because there is no last occurrence of "1" in the binary expansion.
And vice versa:
Real numbers: every infinite class of positive integers defines a real number by specifying all places where "1" occurs instead of "0".
