Bruno Marchal wrote: > > OK. And you are right, I could have done this without mentioning the > constructive ordinal. But it is worth mentioning it, even at this early > stages, because they will reappear again and again. > Note that all those infinite but constructive ordinal are all countable > (in bijection with N), and even constructively so. Note also, if you > haven't already done so, that omega is just N, the set of natural > numbers. I will soon give a more set-theoretical motivation for those > ordinals.
OK. I feel power sets coming. > > Actually there is a cute theorem about constructive ordinal. Indeed > they are equivalent to the recursive (programmable) linear > well-ordering on the natural numbers. Examples: > > An order of type omega: the usual order on N (0<1<2<3<4<5<6<...) > > An order of type omega+1 : just decide that 0 is bigger than any non > null natural numbers: > > 1<2<3<4<5<6< .... <0 > > It is recursive in the sense that you can write a program FORTRAN (say) > capable of deciding it. For example such a program would stop on "yes" > when asked if 4<8, and "no" if you ask 0<8, etc. > > An order of type omega+omega: just decide that all odd numbers are > bigger than the even ones, and take the usual order in case the two > numbers which are compared have the same parity: > > 0<2<4<6<8<10< ..... 1<3<5<7<9<... > > An order of type omega+omega+omega: just decide that multiple of 3 are > bigger than the multiple of two, themselves bigger than the remaining > numbers: > > 1<5<7<11<13<14<17<... 0<2<4<6<8<10<... 3<6<9<12<15<... > > Again it should be easy to write a Fortran program capable of deciding > that order (that is to decide for any x and y if x < y with that > (unusual) order. > > Exercise: could you find an order of type omega*omega? (Hint: use the > prime numbers). > You could use the Sieve of Eratosthenes (spelling?): 2*1<2*2<2*3<... 3*1<3*3<3*5<(all multiples of 3 that are not multiples of 2)... 5*1<5*5<5*7<(all multiples of 5 that are not multiples of 3 or 2)... It sounds like the cute theorem says that you can keep dividing up the natural numbers like this forever. > Those omega names are quite standard. > > > > > OK. So we haven't left the finite behind yet. It makes intuitive > > sense to me that you can diagonalize till the cows come home, staying > > within countability, and still not be done. Otherwise infinity > > wouldn't be infinite. > > > > On the tricky question, it also makes intuitive sense that you can > > sequence effectively on all computable growing functions. This is > > because the larger the growing function gets, the more uncovered space > > (gaps) there are between the computable functions. Any scheme for > > generating growing functions will also leave behind every-growing > > uncomputed gaps. Very unmathematical of me to be so vague, but you've > > already given us the answer, and I know you will fill in the gaps. :) > > I will. Unfortunately this week is full of duty charges. Meanwhile, I > would like to ask George and the others if they have a good > understanding of the present thread, that is on the fact that growing > functions has been well defined, that each sequence of such functions > are well defined, and each diagonalisation defines quite well a precise > programmable growing function (growing faster than the one in the > sequence it comes from). > Just a tiny effort, and I think we will have all we need to go into the > "heart of the matter", and to understand why comp makes our "universe" > a godelized one in the Smullyan sense. > > > I meant that it makes intuitive sense that you *cannot* sequence > > effectively on all computable growing functions. > > You are right. But would that mean we cannot dovetail on all growing > computable functions? I let you ponder this "not so easy" question. > > Bruno > > PS About Parfit, I have already said some time ago in this list that I > appreciate very much its "Reasons and Persons" book, but, in the middle > of the book he makes the statement that we are "token", where it > follows---[easily? not really: you need the movie graph or some strong > form of Occam]--- that comp makes us type, even abstract type. It just > happens that from a first person point of view we cannot take ourselves > as type because we just cannot distinguish between our many instances > generated by the Universal Dovetailer. A similar phenomenon occur > already with the quantum. But from the point of view of this thread, > this is an anticipation. The things which look the more like token, > with the comp hyp, are the numbers. This makes the second half part of > Parfit's book rather inconsistent with comp, but, still, his analysis > of personal identity remains largely genuine. (I don't like at all his > use of the name "reductionism" in that context, also, it's quite > misleading). > > http://iridia.ulb.ac.be/~marchal/ >From what you've said about dovetailing before, you don't have to have just a single sequence in order to dovetail. You can jump among multiple sequences. I have yet to understand how you could dovetail on something that is not effective. That seems to be where you're going. I guess we have to get to non-effective diagonalization first. Tom --~--~---------~--~----~------------~-------~--~----~ You received this message because you are subscribed to the Google Groups "Everything List" group. To post to this group, send email to [email protected] To unsubscribe from this group, send email to [EMAIL PROTECTED] For more options, visit this group at http://groups.google.com/group/everything-list -~----------~----~----~----~------~----~------~--~---
