On Thursday, 31 July 2014 18:33:27 UTC+2, Jürgen R. wrote:
> No you are not. You have simply reached the point where > Mueckenheim will play the trump card: The "complete" sets N > and Q don't exist, arbitrarily large intervals don't exist.
Wrong. Arbitrarily large intervals of not indexed rationals do exist. My sequence is increasing without threshold. That will convince every body without brain damage to recognize the truth. > > Every sober mind understands this immediately.
An infinitely increasing sequence will not have limit 0. Why should sequences of cardinals be discontinuous in the limit?
> If you like, > he will say that the "theorem" is both true and false at the > same time und that the reason is that "set theory" is > fatally flawed.
That is the reason. Without finished infinity there is no set limit that contradicts the cardinality limit. Then both are improper limits. No contradition. >
> Mueckenheims silly "theorem" depends upon the particular > enumeration q_n.
Of course. I chose it bbecause this particular enumeration is expected to cover all rationals. > > What he is apparently attempting to express is that on the > real line there are arbitrarily large intervals not > containing any of the points q_1, q_2, ... , q_n. This > obvious fact is true for any enumeration of Q.
If you enumerate always n/1 by n, then my argument fails. But in that case nobody claims completeness.