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

On 11/02/2024 11:41 AM, FromTheRafters wrote:

Ross Finlayson brought next idea :

On 11/02/2024 11:11 AM, FromTheRafters wrote:

After serious thinking Ross Finlayson wrote :

On 11/02/2024 06:54 AM, FromTheRafters wrote:

WM was thinking very hard :

On 01.11.2024 22:53, FromTheRafters wrote:

WM explained on 11/1/2024 :

On 01.11.2024 19:39, FromTheRafters wrote:

WM formulated the question :

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Infinite subsets don't do that for you, even if you wish really
hard.

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They cannot evade if they are invariable.

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Sets don't change.

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Therefore the elements do not depend on us and our knowledge. "If I
find x, then I can find x + 1" is not relevant. "For every x
(that I
find) there is x + 1" is no relevant. All elements are there,
independent of what we know or do. Therefore the first and the last
are also there independent of us. If they weren't, their existence
would depend on some circumstances and could change.

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Circumstances like "there is no last element"?

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That means, there is always another element. Potential infinity.

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Sets don't change. Forget about amplifying 'not finite' with such as
'actual' and potential' -- infinite simply means not finite and
'actual/potential' is a distinction without a difference. A useless
concept outside of math philosophy.
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the set of denominators have no largest element to 'start' with.

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If all unit fractions are existing, then a smallest unit fraction is
existing. If NUF(x) has grown to ℵ₀ at x₀, then ℵ₀ unit fractions
must
be between 0 and x₀. Hence at least ℵ₀ points with ℵ₀ intervals of
uncountably many points must be between 0 and x₀. That cannot happen
at x₀ = 0.
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Is that too hard to understand?

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Apparently, for you.

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Au contraire, there are multiple law(s) of large numbers,
and in mathematics like emergence after convergence,
the potential / practical / effective / actual distinction,
of "infinity", is a thing.
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You know who discovered mathematics? Philosophers.

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AI sez:
>
https://www.google.com/search?client=firefox-b-1-e&q=did+philosophers+discover+math
>
>
>
AI Overview
Learn more…
While philosophers like Pythagoras played a significant role in the
early development of mathematics by exploring its concepts and
principles, it's not accurate to say that philosophers "discovered"
math; rather, mathematics evolved from practical applications across
different civilizations, with philosophers contributing to its
theoretical understanding and philosophical implications.
Key points to consider:
>
    Historical context:
    Ancient civilizations like the Egyptians and Babylonians used
mathematics for practical purposes like measuring land and calculating
taxes before the emergence of formal philosophical inquiry.
>
Philosophical contributions:
Thinkers like Pythagoras and Plato later examined the nature of
mathematical concepts, leading to philosophical discussions about the
existence and universality of mathematical truths.
>
Debate on discovery vs. invention:
Today, philosophers of mathematics still debate whether mathematics is
"discovered" (existing independently of humans) or "invented" (a human
construct).
>
    Philosophy of mathematics - Wikipedia
    Pythagoras is considered the father of mathematics and geometry as
he set the foundation for Euclid and Euclidean geometry. Pythag...
    Wikipedia
>
Platonism in the Philosophy of Mathematics
Jul 18, 2009 — Platonism about mathematics (or mathematical platonism)
is the metaphysical view that there are abstract mathematical ...
Stanford Encyclopedia of Philosophy
Is Math a Human Invention or a Natural Phenomenon?
Jan 8, 2023 — There is ongoing debate among philosophers and
mathematicians about whether mathematics is discovered or invented.
Some...
Medium
>
    Show all
>
Generative AI is experimental.

>
Oh, what does it say about "multiple law(s) of large numbers"?

>
AI Overview
Learn more…
The "law of large numbers" in statistics essentially states that when
you repeat an experiment a large number of times, the average of the
results will tend to get closer and closer to the expected value; there
are two primary versions of this law: the weak law of large numbers and
the strong law of large numbers which differ in how precisely they
describe this convergence towards the expected value, with the strong
law being more rigorous.
Key points about the law of large numbers:
>
    Basic principle:
    As the number of trials increases, the sample average (the average
of the results from the trials) will get closer to the population mean
(the theoretical average).
>
Weak Law of Large Numbers:
This version states that the probability of the sample average being
significantly different from the population mean becomes very small as
the number of trials increases.
Strong Law of Large Numbers:
This version states that with probability 1, the sample average will
eventually converge to the population mean as the number of trials
approaches infinity.
>
Important considerations:
>
    Does not guarantee individual outcomes:
    The law of large numbers does not imply that any single trial will
necessarily reflect the expected value, only that the average of many
trials will be close to it.
>
Independent trials:
The law applies best when the trials are independent of each other,
meaning the outcome of one trial does not influence the outcome of another.
>
    7.1.1 Law of Large Numbers - Probability Course
    The law of large numbers has a very central role in probability and
statistics. It states that if you repeat an experiment indepen...
    Probability Course
>
Law of Large Numbers - Definition, Example, How to Use
Corporate Finance Institute
What Is the Law of Large Numbers? (Definition) | Built In
Jan 12, 2023 — The strong law of large numbers states that, with
probability one, the average of the results of a large number of tri...
Built In
>
    Show all
>
Generative AI is experimental.
>

There certainly is a divide between a strong mathematical platonism,
as some kind of perceived objectivist realism,
and a usual sort of nominalist fictionalism,
as with regards to an ontological commitment,
to a theory.
>
Yet, mathematical platonism and strong mathematical platonism,
that mathematics is discovered not invented,
is widely held since antiquity by the academicians,
and is most people's theory about it today.
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That they have one, a theory, ....
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Curious, though, I'm curious what your new guru considers
as with regards to "multiple law(s) of large numbers", ...,
would you paste it?

>
I have no guru, don't be insulting.
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I just did.

My humble amendment, for a moment I wondered that you might
be as one of those "Phone-y" cultists who I so often see
praying to their icon what appears to be a pet rock of some sort.
Not that there's anything wrong with a stone, for example,
a smooth stream-worn stone in a concrete world of disconnection
and disassociation from the greater world.
Then, what might you find for, "nonstandard probability theory"?
The "nonstandard analysis" of a usual sort of query will
mostly provide nothing but a useless conceit.