Infinite Martingale

Quote: weaselman

No? Where else do you find it?

Loop quantum gravity also uses these concepts.

Though this is now way over my head, my understanding is mostly limited to classical physics and special relativity, so I won't get into a String Theory vs Loop Quantum Gravity boxing match.

Quote: weaselman

Paradoxically, an object of size R*Lp with R<1, making it smaller than Lp is predicted to exhibit the same properties as an object of size Lp/R, which would be larger than Lp. This opens some mind-blowing possibilities of entire universes inside black holes, forever separated from our own by their event horizons.

Then I overgeneralized, it seems. Didn't know that far into it.
The concept of Lp and finite minimum lengths for valid interactions still retains meaning, though. Following this, it seems that you do have a non-zero probability of hitting Pi with a dart, for physical purposes (at least as far as the dartboard and the dart are concerned), though the tolerance distance is not necessarily exactly Lp.

Yes, you are right I forgot about LQG (never paid very much interest to it since it does not offer the unification of forces).
But the earlier point still stands, since LQG formulation would not be possible without differential analysis and thus the notion of continuum, it is not logically correct to use it as a basis to refute the notion of continuum.

Quote: weaselman

But the earlier point still stands, since LQG formulation would not be possible without differential analysis and thus the notion of continuum, it is not logically correct to use it as a basis to refute the notion of continuum.

The way I see it, it does not need to be refuted fundamentally, merely behave in a quantized way for mundane physical objects. I.e. if it's impossible to determine whether you hit Pi exactly or within 1e-36m of it (with a dart, a perfect, but a classical mechanics one) and it and the Pi point interact as if they coincide, you have hit Pi. An engineering rather than strictly scientific approach, so to speak.

So, If I understand correctly, if you were to try an infinite Martingale, there would be a point where String Theory dictates a black hole would appear, swallowing up the planet shortly before the Universe collapses in on itself ending all existence.

Well, I knew Martingales were bad.....:-D