Re: Incompleteness of Cantor's enumeration of the rational numbers (extra-ordinary)

On 12/12/2024 5:11 PM, WM wrote:

On 12.12.2024 17:15, Jim Burns wrote:

The existence of a bridge implies that,
somewhere we can't see,
a size which cannot change by 1
changes by 1 to
a size which can change by 1

>
Every set can change by one element.

A finite.cardinal is Original Cardinal.
⎛ When your sheep head out to the field to graze,
⎜ as each sheep passes, put a pebble in your pocket.
⎜ When they head in at end.of.day,
⎜ as each sheep passes, take a pebble out.
⎜ When your pocket is empty, all your sheep are in.
⎝ That's Original Cardinal.

Every set can change by one element.
No size is required and no size is possible
if this is forbidden.

All sets have one.emptier and one.fuller counterparts.
Some sets cannot match
their one.emptier and one.fuller counterparts.
We call them finite sets.
Other sets, which aren't finite, can match
their one.emptier and one.fuller counterparts.
We call them infinite sets.
Examples of finite sets include
the sheep in your pasture, the pebbles in your pocket.
Examples of infinite sets include
the set ⟦0,ω⦆ of all finite ordinals k = ⟦0,k⦆
⎛ finite == cannot match one.emptier or one.fuller
⎝ infinite == can match one.emptier or one.fuller
We know that ⟦0,ω⦆ is infinite because
⟦0,ω) can match one.emptier ⟦1,ω⦆
one.to.one f(k) = k∪{k} describes one such match
f: ⟦0,w⦆ ⇉ ⟦1,w⦆: one.to.one
Ignoring cardinality changes nothing
about f(k) = k∪{k} and ⟦0,w⦆ and ⟦1,w⦆
f(k) = ku{k} is still as much one.to.one
from ⟦0,ω⦆ to one.emptier ⟦1,ω⦆ as it ever was.