Re: Replacement of Cardinality

On 2024-07-30 14:41:15 +0000, WM said:

Le 30/07/2024 à 09:26, Mikko a écrit :

On 2024-07-29 13:15:36 +0000, WM said:
 

Le 29/07/2024 à 11:01, Mikko a écrit :
 

He also notes that what we have learned from finite quatities does
not apply to infinity.

 Unit fractions are isolated. They have distances. The function "unit fractions between 0 and x" can nowhere grow by more than 1.

 The number of unit fractions between 0 and x is infinite

 Not for every x.

The context where you mentioned unit fractions was a discussion about
Galileis opinions. At that time the concept of quantity did not include
absurd quantities and usually not zero, either. Therefore the expression
"unit fractions between 0 and x" should be interpreted to assume that
x > 0. Of course if x is zero there is nothing between 0 and x.

Growth by more than 1 is contradicting mathematics.
∀n ∈ ℕ: 1/n - 1/(n+1) > 0. Note the universal quantifier.

Irrelevant. But do you mean that 0 is not in ℕ?

and the number of unit fractions between 0 and y is infinite. Whether one
infinity is bigger or smaller than another infinity is not answerable merely
on the basis of our experinece about finite quantities but only after a carful analysis if at all.

 If (0, x) is a subset of (0, y), the relative quantities are clear.

As Galilei pointed out, it is not. That one is a subset of the other
makes one look bigger but the existence of a bijection between the two
makes them look equal. Bot appearences cannot be true, so the clarity
is a hallusination.
However, although Galilei was happy with all infinities being equalt
there is a way out, and in fact more than one way: at least cardinals,
ordinals, and measures, perhaps something else.
--
Mikko