Re: The set of necessary FISONs

On 1/26/2025 2:31 PM, Richard Damon wrote:

On 1/26/25 9:28 AM, WM wrote:

On 26.01.2025 13:38, Richard Damon wrote:

On 1/26/25 3:51 AM, WM wrote:

On 25.01.2025 15:16, Richard Damon wrote:
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I said:
Sure it does, you just need to take the union of an infinite number of them.

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But that is impossible because there are not two consecutive actually infinite sets in ℕ. Since every FISON is followed by an actually infinite set, ∀n ∈ U(F(n)): |ℕ \ {1, 2, 3, ..., n}| = ℵo, there is no actually infinite set of FISONs.

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Why do we need "consecutive" infinite sets,

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They would not exist even if they were needed.
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FISONs enumerate themselves. There is no infinite FISON and hence no infinite number of them.

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Then, what is the highest FISON?

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That depends on the system. All we know is that it is finite.

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No, it doesn't exist.
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{1}
{2, 1}
{3, 2, 1}
...
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The first column never gets larger than a FISON.

 Sure it does, when you COMPLETE the process in infinity.

For some reason I think of completing infinity simply by writing this:
Say:
N = all of the natural numbers
So, N is completed. It's all of them. Now, why does WM seem to think that saying completed implies a largest natural number?

 Of course, your logic can't complete the action, so can't actually have an infinite set, and thus can't process one.
 

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If there is only a finite number of them, THEN there is a maximum

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A variable maximum, "something becoming, emerging, produced, i.e., as we put it, the potential infinite." [Hilbert]

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Which is about KNOWLEDGE, not the actual existance.

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FISONs are about knowledge.

 So, you agree that they don't tell you about the actual existance of the number, just what you can know about them.
 At best, you are proving that with your logic, you can't know the full properties of infinite sets, as thus LIE when you make statements about such things.
 

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Regards, WM
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