Re: Replacement of Cardinality

On 8/25/24 3:47 PM, WM wrote:

Le 24/08/2024 à 19:06, Richard Damon a écrit :
 

I guess by your finite logic, Achilles can't pass the tortoise,

      11.1 Achilles and the tortoise [From my 5th book appearing 2025]
 Achilles and the tortoise run a race. The tortoise gets a start and the race begins (state 0). When Achilles reaches this point, the tortoise has advanced further already (state 1). When Achilles reaches that point, the tortoise has advanced again (state 2). And so on (states 3, 4, 5, ...). Since Achilles runs much faster than the tortoise, he will overtake (state ), but only after infinitely many finitely indexed states of the described kind. Their number must be completed. Otherwise Achilles will not overtake. But there must not be a last visible finitely indexed state. (The last 1000 states Achilles remembers have indices much smaller than .) This can only be realized by means of dark states.
According to set theory, all states can be put in bijection with all natural numbers. This is impossible as completeness and well-order require a last mark. The three notions "all" and "infinite" and "well-ordered" do not match. This dilemma can only be solved by refraining from well-order of the set Y of dark-numbered states.
 Regards, WM

So, your logic is contradictory.
You say the number IS completed to allow it to reach the final state, but it also must not be complete.
Why do you say that it is "1000" states that are dark?
Either your sum of "defined" steps is finite, and thus stops before getting to the Tortoise, and then we can do the operation your 1000 more times, and find Achilles still short of the Tortoise, or your "defined" steps are infinite without an "max value" that destroys your theory.
Thus, it seems your logic just "punts" when it needs to deal with infinity and just admits that it can't handle it.
Your final answer is basically just admitting that your logic can't supply the needed properties of the Natural Numbers.