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

Am 12.10.2024 um 05:51 schrieb Chris M. Thomasson:

On 10/11/2024 3:15 PM, Richard Damon wrote:

Right, but it means we lack some of the ability to actually PERCEIVE the reality of the infinite.

A truism. :-)
Peter Suber:
"Do We Experience Anything Infinite?
So we agree with Descartes that we do possess a positive idea of infinity. If Descartes is correct in his second thesis that we could not have obtained the idea from our finite experience and creative resources, then we feel the pressure he felt to posit an infinite being. So let us face directly the question whether we experience anything infinite.
The words "infinite" and "infinity" are often used loosely in street English to suggest that we do experience infinites. For example, we may say that a film is infinitely clever, a coral reef has an infinite variety of wildlife, a spouse has infinite patience, or that a vinyl upholstery cleaner has infinitely many applications. (That's why it's called a miracle product.) Before cameras were automated, they had a focal-length setting called "infinity", presumably for photographing the arrow Lucretius shot into the edge of space. In these cases we speak loosely, and "infinity" means very many or very large, perhaps indefinitely many or large.[Note 20] On a clear day the sky may seem infinitely deep, but it's really just a wild blue yonder —an indefinitely deep 'out there'.[Note 21]
Do we ever experience something which is literally infinite? If time, space, or matter are infinitely divisible, then to experience a finite chunk of any one of them is to experience its infinity of parts. Having said this, I would like to put to one side the question whether time, space, or matter really are infinitely divisible. Not only is it very thorny, it is unnecessary to answer the question on the table. For even if time, space, and matter are infinitely divisible, we experience their infinite parts bundled into chunks most of whose parts are indiscernible to us. When a movie runs at 24 frames per second, it appears continuous, its separate frames indiscernible to us. We certainly experience 24 chunked frames, but not the 24-ness, or even the finitude, of the chunking. Once the eye is fooled into seeing continuity, the number of frames per second could increase to a billion, or to an infinite number, and we would not notice the difference.[Note 22] This is the sense in which we could experience something infinite without experiencing its infinitude. Similarly, if time, space, and matter were continuous and infinitely divisible, then the spectacle of life would be like a movie run at an (uncountably) infinite number of frames per second; but while we would experience expanses, durations, and objects with infinitely many parts, but we would not experience the infinitude of those parts.
As the movie shows, the same is true of finite divisibility. If my car has (say) 5,000 parts, I experience it as an object with many parts; but I don't experience the 5,000-ness of the parts."
BUTE note this thought:
"While we do not experience the infinitude of time, space, or matter, even if they are infinite in extent or divisibility, neither do we experience large finite magnitudes. I've seen estimates of the number of sub-atomic particles in the universe ranging from 10^65 to 10^85. But to be conservative, let's say that nothing in the universe, including the universe itself, has more than 10^100 parts. The name for 10^100, or 1 followed by 100 zeroes, is a googol. So even if there are more than googol of ultimate particles, it's fair to say that no collection of physical objects that we have ever experienced —grains of sand on a beach, snowflakes in a storm, stars in the sky— has more than a googol of members.[Note 33] If true, then we did not obtain our idea of a googol from experience. But it does not follow that we must posit a very large finite being —Googolzilla— to be the source of our idea. We know exactly what a googol is as a concept, even if we have never experienced it manifest in a sensation or image. We can list the million natural numbers which are its closest neighbors, we can do arithmetic with it, and we know infallibly whether an arbitrary natural number is larger or smaller than it. If we may export the lesson of this to the infinite, then we may suggest that while we have no experience of the infinitude of anything, we have a perfectly good concept of infinity, and that the ultimate explanation of this fact lies not so much in anything special about infinity as in the distinction between concepts and images."

That's interesting to me because it kind of reminds me of 2d vs 3d vs 4d observers. Us 3d beings can look down at a 2d beings world and see all of it. A 4d being can look at our 3d world and see all of it. Think of a 4d being at a location in n-ary space that has a non-zero 4d component to its vectors. We would not be able to see it because is off axis in a higher dimension. However, it can see us, right through us, all the way down...

Yeah, nice ideas.
You certaily now "Flatland. A Romance of Many Dimensions" written in 1884 by Edwin Abbott Abbott.
See: https://en.wikipedia.org/wiki/Flatland

When one tries to do that, it is too easy to presume characteristics of the infinite are just like the finite, when they aren't (like having "ends").

Right. One has to be careful here. [...]