Re: Contradiction of bijections as a measure for infinite sets

On 4/10/2024 8:39 PM, Ross Finlayson wrote:

On 04/10/2024 01:13 PM, Jim Burns wrote:

ℕ = ⟦0,ω⦆
>
ω is after Avogadroᴬᵛᵒᵍᵃᵈʳᵒ
Avogadroᴬᵛᵒᵍᵃᵈʳᵒ = 6.02214076E23⁶ᐧ⁰²²¹⁴⁰⁷⁶ᴱ²³
and after any other number such that
it and all non.0 numbers.before.it
have predecessors.
>
Infiniteⁿᵒᵗᐧᵂᴹ is different.

>
You know, some people have that
Avogadro's number, is sort of a _running_, constant.
>
What I mean by that is that NIST CODATA every few
years arrives at the current values according to
the energy and configuration of experiment, and,
some of the constants are _running_ in the energy
and configuration of experiment, it results they
get not only more precise, even actually, _smaller_.
 Or, you know, larger.

Since 2019,
Avogadro's constant has been defined as
exactly 6.02214076×10²³ =
602,214,076,000,000,000,000,000
Given the declared intentions and past behavior
of the constant.definers, I can't imagine them
changing it ever again.
You might say that it has stopped running.
I was shown, some time ago, something like
what you describe maybe: a history of
measurements of the speed of light.
The reported values wander, no surprise,
as they are measurements, not definitions.
But they wander _less than_ their error bars
indicated that they should wander.
Except, sometimes there is a leap well.outside
the "random" cluster to a _new_ cluster
of measurements wandering less.than.expected.
These are measurements made in different decades,
in different centuries, with different equipment,
by different methods, with different technology,
by different people, for different institutions.
And yet, there are these signs of coordination.
The explanation I was offered is that these
scientific humans, presented with a value
kind.of.far from that day's consensus,
will look for a reason that it's that far.
And these are all clever people, remember.
They will find a reason.
Except for sometimes, when the technology has
improved to the point that there will be
no reconciling of the result with the consensus,
and a new consensus is born.