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

On 10/28/2024 3:27 PM, WM wrote:

On 28.10.2024 19:31, Jim Burns wrote:

On 10/26/2024 12:04 PM, WM wrote:

No, you falsely assume that
all natnumbers can be defined.

>
I assume that
natural numbers are finite ordinals, and
that a finite ordinal is predecessored or is 0  and
  each of its priors is predecessored or is 0, and
that an ordinal is successored and
   a set of ordinals is minimummed or is {}

>
That is irrelevant
with respect to the fact that
between every chosen unit fraction and zero
there are ℵo unit fractions which
cannot be chosen.

Your alleged fact depends upon
what exists among what's discussed.
I can always inject
  flying rainbow sparkle ponies
into what's discussed.
And, once I have injected them, then what?
Your dark numbers are
  flying rainbow sparkle ponies,
injected only in order for you to say
they're there.
Without flying rainbow sparkle ponies,
⎛ an ordinal has its successor,
⎜ a non.{} ordinal.set holds its minimum,
⎜ a finite ordinal and each prior ordinal
⎜  has its predecessor or is 0,
⎜ a natural number is a finite ordinal,
⎜ an integer is the difference of naturals,
⎜ a rational is the quotient (w/o 0.denom)
⎜  of integers,
⎜ a real is between fore and hind of
⎝  a split of rationals.

between every chosen unit fraction and zero
there are ℵo unit fractions which
cannot be chosen.

Without flying rainbow sparkle ponies,
⎛ each unit fraction is countable.down.to from.⅟1
⎜ between each unit fraction and zero,
⎜  there are ℵ₀ unit fractions which are
⎝  countable.down.to from.⅟1
Any other "unit fractions" are
  flying rainbow sparkle ponies.
'Infinite' does not mean what you want it to mean.