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

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 :

>

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

>
They cannot evade if they are invariable.

>
Sets don't change.

>
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.

>
Circumstances like "there is no last element"?

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

>
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.
>

the set of denominators have no largest element to 'start' with.

>
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.
>
Is that too hard to understand?

>
Apparently, for you.

>
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.
>
You know who discovered mathematics? Philosophers.

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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
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