Re: Contradiction of bijections as a measure for infinite sets

On 4/3/2024 10:13 PM, Ross Finlayson wrote:

On 04/03/2024 06:13 AM, Jim Burns wrote:

[...]

>
Iota-values:
the word "iota" means "smallest non-zero value".

That which you use iota to describe
is not the continuum.
...and you've been told this before,
but you think that those telling you are wrong.
I think the reason that you wrongly think we're wrong
is that what you think a limit is
is not what a limit is.
Consider an example we use when we explain
why an arc.length.integral is ∫ √​̅1​̅+​̅f​̅′​̅²​̅(​̅x​̅) 𝑑x
AKA
a proof that π = 4
Consider the n×n grid of points in [0,1]×[0,1]
and the unit.circle in [0,1]×[0,1]
x² + f²(x) = 1
The horizontal and vertical grid.to.grid segments
which intersect the unit circle form
a continuous but not.differentiable curve
from ⟨0,1⟩ to ⟨1,0⟩
The length of
those joined horizontal and vertical segments
is 2
As n ⟶ ∞  the length is 2
That limit is 2
As n ⟶ ∞
for dₙ = the maximum distance between the circle and
those joined horizontal and vertical segments
dₙ ⟶ 0
It's very reasonable to define 'limit' in such a way
that the limit of the horizontal and vertical segments
is the circle.
However,
unless π = 4
the arc.length of the limit (circle) is not
the limit of the arc.lengths (2)
----

Iota-values:
the word "iota" means "smallest non-zero value".

That which you use iota to describe
is not the continuum.
The continuum is not simply
points veryveryvery close.

Real-values:
 all the values between negative infinity and infinity.

There are several ambiguities in that description.
How about instead
Real.values:
least.upper.bounds of
bounded.non.empty.sets of
differences.of.ratios of
ordinals not.fitting.predecessors.