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

On 12/13/2024 6:25 AM, WM wrote:

On 13.12.2024 06:06, Jim Burns wrote:

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.
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.

>

Ignoring cardinality changes nothing
about f(k) = k∪{k} and ⟦0,w⦆ and ⟦1,w⦆

⎛ ∀k ∈ ⟦0,ω⦆:
⎜ ∃k′ ∈ ⟦1,ω⦆: k′ = k∪{k}  ∧
⎝ ∀j ∈ ⟦0,ω⦆: j≠k ⇒ j′≠k′

Ignoring that Cantor's claim requires to
empty the endsegments from all natural numbers
in order to use them as indices in mappings

Each finite.cardinal
⎛ is first in an end.segment.
⎝ is used as a index of an end.segment.
Each finite.cardinal
⎛ is absent from an end segment.
⎝ is emptied from the end.segments.
Require at will, sir.
On the other hand,
you (WM) could continue to amuse your students
with tales of infinite.finite.cardinals.

(ℕ to the set of endsegments, ℕ to ℚ,
ℕ to the lines of the Cantor list
for "proving" the uncountability of ℝ)
shows inconsistency.

...inconsistency which
your consistentᵂᴹ infinite.finite.cardinals
don't show, presumably.
----
Each set has
its emptier.by.one and fuller.by.one counterparts.
Each set which cannot match
its emptier.by.one and fuller.by.one counterparts
can match ⟦0,k⦆ for some finite.cardinal k.
⟦0,k⦆ cannot match ⟦0,k-1⦆ or ⟦0,k+1⦆
⟦0,k-1⦆ can match emptier.by.one counterparts.
⟦0,k+1⦆ can match fuller.by.one counterparts.
The set ⟦0,ℵ₀⦆ holds
each and only finite.cardinals k
those for which ⟦0,k⦆
cannot match its ⟦0,k-1⦆ or ⟦0,k+1⦆
The set ⟦0,ℵ₀⦆
cannot match any ⟦0,k⦆ (k ∈ ⟦0,ℵ₀⦆)
which cannot match its ⟦0,k-1⦆ or ⟦0,k+1⦆
⟦0,ℵ₀⦆ is NOT any set which cannot match
its emptier.by.one and fuller.by.one counterparts.
⟦0,ℵ₀⦆ CAN match
its emptier.by.one and fuller.by.one counterparts,
but, for each k ∈ ⟦0,ℵ₀⦆
⟦0,k⦆ CANNOT match
its emptier.by.one and fuller.by.one counterparts.
That is inconsistent with
your theory of infinite.finite.cardinals,
which claims
⛔⎛ each set, without exception, cannot match
⛔⎝ its emptier.by.one and fuller.by.one counterparts.