Sujet : Re: Simple enough for every reader?
De : mikko.levanto (at) *nospam* iki.fi (Mikko)
Groupes : sci.logicDate : 11. Jun 2025, 09:01:23
Autres entêtes
Organisation : -
Message-ID : <102bd4j$1s22l$1@dont-email.me>
References : 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27
User-Agent : Unison/2.2
On 2025-06-03 13:26:11 +0000, WM said:
On 03.06.2025 10:11, Mikko wrote:
On 2025-06-01 14:09:23 +0000, WM said:
On 01.06.2025 13:58, Mikko wrote:
On 2025-05-31 13:47:51 +0000, WM said:
On 31.05.2025 12:11, Mikko wrote:
On 2025-05-30 14:46:55 +0000, WM said:
{1, 2, 3, ..., n, n+1} has infinitely many (ℵo) successors is a consequence of {1, 2, 3, ..., n} has infinitely many (ℵo) successors, and ℵo - 1 = ℵo.
Still no proof.
Proof that every definable natural number has more successors than predecessors.
You have not shown that proof, either.
I have above. You cannot understand it. That is a different thing.
No proof of anything above. Besides, nothing that requires any
understanding beyond ordinary proof checking cannot be a proof.
There is no further understanding required.
That contradicts your above "You cannot understand it".
No, ordinary proof checking and your understanding are two different things.
Of course they are. Ordinary proof checking is a skill more than just
understanding, and to the exent it is understanding it is only a small
part of my understanding, most of which is irrelevant to any discussions
here.
Every proof checker can confirm that every definable natural number has finitely many predecessors.
Probably, but that is not a part of proof checking. The skill of proof
checking is to determine wheter a sentence is an axiom or follows by
an inference rule from the sentences it is claimed to follow.
If there are more than finitely many numbers following upon every definable number as successors, then every definable number has more successors than predecessors.
Provable.
All can be removed collectively. ℕ \ {1, 2, 3, ...} = { }.
The word "collectively" is not a well defined mathematical term. It can
be used in informal presentation but not in a rigorous proof.
If all could be removed as individuals, then a last one would be removed. Contradiction.
Likewise, "as individuals" has no well defined mathematical meaning
so cannot be used in a rigorous proofs.
This is a proof of dark natural numbers.
It is not a proof of anything.
-- Mikko
Simple enough for every reader?
Re: Simple enough for every reader?
