I have always thought that ω was the first transfinite number. The hyperfinite angle does take an interesting poke at the gap problem by kind of embracing the gaps.
I assume that the notation works by having more overlined numbers for each infinity level. That 25¯037¯0.¯021 is 25∗102ω+37∗10ω+21∗10−ω. For ωω start adding lines? 87¯¯025¯037¯0.¯021
Atleast for surreals they are “too continous” to count as continous on the same axiom as reals get defined. As reals {1/2,1/4,1/8,1/16,...} has a unique limit of 0. But for surreals {0|1/2,1/4,1/8,1/16,...}=ϵ gives birth to {0|ϵ}=ϵ/2 and with enough iteration you can get to {0|ϵ/2,ϵ/4,ϵ/8,ϵ/16,...}=ϵ/ω . So there are plenty of strong limits going on but they are not unique because a tighter limit is always possible but none of them is “the limit”.
I am not convinced that the limit procedure is the correct one. The “old” notion of smooth would only count up to real accuracy, so in surreals and I would imasgine in hypernumbers reals are actually disconnected. In what they try to build as surreal analysis they can get the real integral etc by only taking the standard part much like one takes the integer part from a real to get a result type compatible with N. I would expect a funtion that approaches 1+ϵ from the left and 1−ϵ from the right to be discontinous when one approaching 1 from left and right would have chances to be continous. I am not convinced that the approach treats the non-standard parts adequately. First times encountering this it hammered home that smoothness or “completeness” is a relative property. Natural numbers considered in themselfs are without gaps, it is only when we compare them to other systems that “unrepresented elements” are apparent.
The bit about diffusion being fundamental seem quite important but unfortunately I am not understanding that at all.
Reread this and this is awesome: I didn’t think of the case of multiple bars at all. As for the surreal stuff, I’ve read about them and while the lexicographic ordering is nice, the lack of transfer principle hurts. Are you in the bay?
I think there are quite a lot of transfer principles going on. Atleast real to surreal and the hypernumber transfers have exact analogs (but might not have to the surreals that are not that kind of hypernumber). Would not be surprised if “hyperfinites” would be limited to ωx where x is finite (so ωω is actually too big to be a hyperfinite).
I am not in the San Fransico bay area and it is not convenient for me to physically meet in America.
I have always thought that ω was the first transfinite number. The hyperfinite angle does take an interesting poke at the gap problem by kind of embracing the gaps.
I assume that the notation works by having more overlined numbers for each infinity level. That 25¯037¯0.¯021 is 25∗102ω+37∗10ω+21∗10−ω. For ωω start adding lines? 87¯¯025¯037¯0.¯021
Atleast for surreals they are “too continous” to count as continous on the same axiom as reals get defined. As reals {1/2,1/4,1/8,1/16,...} has a unique limit of 0. But for surreals {0|1/2,1/4,1/8,1/16,...}=ϵ gives birth to {0|ϵ}=ϵ/2 and with enough iteration you can get to {0|ϵ/2,ϵ/4,ϵ/8,ϵ/16,...}=ϵ/ω . So there are plenty of strong limits going on but they are not unique because a tighter limit is always possible but none of them is “the limit”.
I am not convinced that the limit procedure is the correct one. The “old” notion of smooth would only count up to real accuracy, so in surreals and I would imasgine in hypernumbers reals are actually disconnected. In what they try to build as surreal analysis they can get the real integral etc by only taking the standard part much like one takes the integer part from a real to get a result type compatible with N. I would expect a funtion that approaches 1+ϵ from the left and 1−ϵ from the right to be discontinous when one approaching 1 from left and right would have chances to be continous. I am not convinced that the approach treats the non-standard parts adequately. First times encountering this it hammered home that smoothness or “completeness” is a relative property. Natural numbers considered in themselfs are without gaps, it is only when we compare them to other systems that “unrepresented elements” are apparent.
The bit about diffusion being fundamental seem quite important but unfortunately I am not understanding that at all.
Reread this and this is awesome: I didn’t think of the case of multiple bars at all. As for the surreal stuff, I’ve read about them and while the lexicographic ordering is nice, the lack of transfer principle hurts. Are you in the bay?
I think there are quite a lot of transfer principles going on. Atleast real to surreal and the hypernumber transfers have exact analogs (but might not have to the surreals that are not that kind of hypernumber). Would not be surprised if “hyperfinites” would be limited to ωx where x is finite (so ωω is actually too big to be a hyperfinite).
I am not in the San Fransico bay area and it is not convenient for me to physically meet in America.
Think of diffusion as a micro random walk. A hyperfinite number of infinitesimal steps, sampled from a discrete set of vectors.