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


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