Not at all -- the cardinality of the points of the Koch curve is identical to the cardinality of the real line segment [0,1] which is equal to the cardinality of $$\mathbb{R}$$.

I challenge you to cite and quote this supposed source that claims the number of points is impossible to well define -- for I believe that is another unreliable implication you have assumed because you don't know math.

Equinumerosity is an equivalence relation between sets where it follows that there exists a one-to-one onto mapping between one set and another, like pairing off dance partners in the 1950's as a proof that there are the same number of girls and boys in the class. Thus it follows that if you have a one-to-one mapping of set A into a set B (where the image of A under the mapping is a subset of B ) then the cardinality of B is greater than or equal to A.

Because I can establish a procedure which maps a set with the cardinality of the continuum [0,1] one-to-one into the Koch curve K it follows that $$ [0,1] \preceq K $$.

And because I know that the Koch curve is a subset of plane, it follows that $$ [0,1] \preceq K \preceq \mathbb{R}^2 $$.

But because the plane also has the cardinality of the continuum it follows that all of these sets are equinumerous $$ [0,1] \approx K \approx \mathbb{R}^2 $$.

Thanks for your response, rpenner. Have a bad cold and going back to bed, so can't engage in detailed discussions, even if I wanted to. back in a few days, I hope. But while I am here, I will again give some indication of where I am coming from and going to wards on this and other non-conventional maths perspectives of mine arising from my from scratch review of current maths as it is (from which you are observing, and thanks for that, but I am not in the same construct' as you are). Briefly then...

The essence and intents of my further-seeing approach to this review is encapsulated by the comment made by some mathematician (expounding on Fractals or Fracta calculus or some such topic) whom I can't recall at present, to the effect:

If you will recall way back when, I made the observation that my perspective takes into account the surrounding dynamical environment of the otherwise static and sterile number line construct/points treatment. That is why I spoke then of a new CONTEXTUAL MATHS system with re-jigged axioms which will cater for all cases, not just notional infinite/dimensionless 'points/geometries, but also for the real meanings of the maths system elements per se which will re-tool all the current conventional treatments/definitions (which currently output 'undefined/undetermined' etc etc) to better prepare for the needs of the final complete reality based ToE physics modeling.To use a metaphor, the fractional derivative requires someperipheral vision.

Anyhow, sorry but I must go. That's all I am at liberty to say at this stage. Back whenever. Thanks again, rpenner, for your time and trouble and polite response. Cheers and stay well as can be, everyone!