V 165
Infinity/material/Quine: if you need an infinite number of characters (e.g. for natural numbers) you cannot say, a sign is a physical object, because then you will soon come to an end. Also forms are not used as classes of inscriptions. These are again physical realizations of forms.
IX 64
Infinity/Quine: is only necessary for induction - x = {y}, y = {z}, z = {w} ... ad infinitum - this is the case if {,,,x}.
XIII 96
Infinite Numbers/Quine: For example, suppose we randomly assign items to any class, the only limitation is that no object can belong to more than one class.
Problem: then there will not be enough items for all classes! A class for which there is no correlate will be the class of all objects that do not belong to their correlated classes. Because its correlate should belong to it, iff it does not belong to it.
Cantor: proved in 1890 that the classes of items of any kind exceed the number of items.
XIII 97
The reason for this has to do with the paradoxes, if the relation, which is mentioned there, is specified correctly.
It turns out that there are infinitely many different infinities.
For example, there are more classes of integers than there are integers.
But since there are infinitely many integers, the infinity of infinitely many classes of integers must be of a higher kind.
For example, there are also more classes of classes of integers than there are classes of integers. This is an even higher infinity. This can be continued infinitely many times.
The argument here depended on the class of non-elements of their own correlated classes (nonmembers of own correlated classes).
Russell's Antinomy/Quine: depended on the class of nonelements of selves.
Cantor's Paradox/Quine: if one takes the correlation as self-correlation, Cantor's paradox amounts to Russell's Paradox. That is how Russell came up with it.
Cantor/Theorem/Quine: his theorem itself is not a paradox.
Russell's Antinomy/Solution/Quine: is prevented by excluding a special case from Cantor's theorem that leads to it. (See
Paradoxes)
Cantor Theorem/Corollar/unspecifiable classes/Quine: the existence of unspecifiable classes follows as a corollar from Cantor's theorem. I.e. classes for which we cannot specify the containment condition. There is no other identifying move either.
For example, the infinite totality of grammatically constructible expressions in a language. According to Cantor's theorem, the class of such expressions already exceeds the expressions themselves.
Classes/larger/smaller/criterion/Quine: our criterion for larger and smaller classes here was correlation.
Def greater/classes/quantities/Quine: one class is larger than another if not each of its elements can be paired with an element of the other class.
XIII 98
Problem: according to this criterion, no class can be larger than one of its real subclasses (subsets). For example, the class of positive integers is not larger than the class of even numbers. Because we can always form pairs between their elements. This simply shows that infinite sets behave unusually.
Infinite/larger/smaller/class/quantities/Quine: should we change our criterion because of this? We have the choice:
a) We can say that an infinite class need not be larger than its real subclasses, or
b) change the criterion and say that a class is always larger than its real parts, only that they can sometimes be exhausted by correlation with elements of a smaller class.
Pro a): is simpler and standard. This was also Dedekind's definition of infinity.
Infinite/false: a student once wrote that an infinite class would be "one that is a real part of itself". This is not true, but it is a class that is not larger than a (some) real part of itself. For example the positive integers are not more numerous than the even numbers. E.g. also not more numerous than the multiples of 3 (after the same consideration). And they are also not less numerous than the rational numbers!
Solution: any fraction (ratio) can be expressed by x/y, where x and y are positive integers, and this pair can be uniquely represented by a positive integer 2x times 3y.
Conversely, we get the fraction by seeing how often this integer is divisible by 2 or by 3.
Infinite/Quine: before we learned from Cantor that there are different infinities, we would not have been surprised that there are not more fractions than integers.
XIII 99
But now we are surprised!
Unspecifiable: since there are more real numbers than there are expressions (names), there are unspecifiable real numbers.
Names/Expressions/Quine: there are no more names (expressions) than there are positive integers.
Solution: simply arrange the names (expressions alphabetically within each length). Then you can number them with positive integers.
Real Numbers/Cantor/Quine: Cantor showed that there are as many real numbers as there are classes of positive integers. We have seen above (see decimals and dimidials above) that the real numbers between 0 and 1 are in correlation with the infinite class of positive integers.
>
Numbers/Quine.
