Re: Replacement of Cardinality

On 7/28/24 7:55 AM, WM wrote:

Le 27/07/2024 à 14:55, Richard Damon a écrit :

On 7/27/24 8:16 AM, WM wrote:

Le 27/07/2024 à 13:27, Richard Damon a écrit :

On 7/27/24 7:13 AM, WM wrote:

Le 27/07/2024 à 04:23, Richard Damon a écrit :
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By your logic, if you take a set and replace every element with a number that is twice that value, it would by the rule of construction say they must be the same size.

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That is true in potential infinity. But I assume actual infinity.

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So, what part is not true?

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In potential infinity there is no ω.
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Are you stating that replacing every element with another unique distinct element something that make the set change size?

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In actual infinity the number of elements of any infinite set is fixed.
Doubling all elements of the set ℕ U ω = {2, 4, 6, ..., ω}

 Mistake! ℕ U ω = {1, 2, 3, ..., ω}

And who was using that set?

 

yields the set
{2, 4, 6, ..., ω, ω+2, ω+4, ..., ω*2}.
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Why?

 See the correction.

But what number became ω when doubled?
Every natural number when doubled is a Natural Number.
so the result should be { 2, 4, 6, ... 2*ω}.

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Note, ω is NOT a member of the Natural Numbers, it is just the "least upper bound" that isn't in the set.

 I know. Therefore I wrote ℕ U ω, or better ℕ U {ω}.

But why? we were talking about the infinite set of the Naturals.

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There is no Natural Number that is ω/2 so that doubling it get you to ω, as every Natural Number when doubled gets you another Natural Number.

 There is no definable natural number ω/2. But if there are all elements, then there is no gap before ω but ω-1.

Which isn't a Natural Number, as if it was, then that set of Natural Numbers would have a maximun member and be finite.
Note, ω-1 doesn't exist in the base transfinite numbers, just as -1 doesn't exist in the Natural Numbers, you can't go below the first element.
But, just as we can expand the Natural Numbers to the Integers, and get negative numbers, we also might be able to define an extention to the transfinite numbers that can have a ω-1 element.

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Your "logic" just seems to be that ω is just some very big, an perhaps unexpressed, value of a Natural Number,

 No, it is the first transfinite number like 0 is the first non-positive number.

And thus you can't have ω-1, just like you can't have -1 in the Natural Numbers.

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The fact that you can't understand this is deplorable but does not make my theory wrong.

The fact that your theory is inconsistant makes it wrong.

Using the unit fractions itelligent readers understand that there must be a first one after zero. Others must believe in the magical appearance of infinitely many unit fractions.
 

Nope, since that implies there is a highest Natural Number, which breaks their definition,
It just shows you mind can't handle proper logic of unbounded sets, but is stuck with bounded logic that just breaks when used with unbounded sets.

Regards, WM