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From: Mikko <mikko.levanto@iki.fi>
Newsgroups: sci.logic
Subject: Re: Replacement of Cardinality
Date: Wed, 31 Jul 2024 11:51:40 +0300
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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