Re: How many different unit fractions are lessorequal than all unit fractions? (infinitary)

On 10/8/2024 2:21 PM, Richard Damon wrote:

On 10/8/24 3:55 PM, Chris M. Thomasson wrote:

On 10/8/2024 5:35 AM, Richard Damon wrote:

On 10/8/24 5:54 AM, WM wrote:

On 07.10.2024 15:19, Alan Mackenzie wrote:

WM <wolfgang.mueckenheim@tha.de> wrote:

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No, even an unbounded sequence does not get longer when shifted by one step.

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The concept of "length" appropriate for finite sets doesn't apply to
infinite sets.

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It is the concept of number of elements. It is appropriate in actual infinity.

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Which has been shown to not exist for us finite beings, as it is too big for us to see.
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infinite means "without end" - unendlich.

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Actual infinity means complete. That implies a fixed number.

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Which has been shown to not exist for us finite beings, as it is too big for us to see.
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You don't understand that actual infinity is a fixed quantity.

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It may be "fixed" whatever that might mean, but to regard it as a
"quantity" is more than questionable.

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Fixed means that no element can be added and no element can be lost. The number of nines is fixed. That is an assumption only, but necessary for bijections.

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Yes, it is fixed, at INFINITY, which means there is no end to it, and thus we can't add a zero "at the end" which doesn't exist.
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This is why finite beings can't use "actual infinity" because it is too big for us to handle.

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Ummm... Well, not sure what to think about that. Hummm... Any time you use a number it is in actual infinity. Think of the number four. It is in a pool of the infinitely many natural numbers, and we just used it...
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Fair enough, or weasel words?
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 The difference is that in "potential infinity", the number might not have been already existing in the concesness, but was "created" by the rules of the set when it was needed.
 For the actual infinity to exist, all the numbers need to be already created and enumerated, and there are no numbers "left" to be created.
 The problem with "actual infinity" is we can't actually "perceive" it, and when we try to think about it, we can only hold it in refernce to other finite concepts, and thus it won't actually be infinite.
 The only way we will be able to understand that actual infinity is to think about the way that "potential infinity" created it.

One way I can ponder on it is to say taking a gallon of water out of an infinite pool means the pool is still infinite as if nothing happened at all... Actually, it kind of reminds me of an old Disney cartoon where Merlin was storing things in a little bag that can hold infinitely many objects, no matter how large. Here is the scene:
(The sword in the stone - Higitus Figitus)
https://youtu.be/Tb75RjpvBIk
Magic indeed! ;^)