Re: super-all-adics
- From: "Ross A. Finlayson" <raf@xxxxxxxxxxxxxxx>
- Date: 21 Dec 2005 10:29:35 -0800
Ken Quirici wrote:
> Ross A. Finlayson wrote:
> >
> > Hi Ken,
> >
> > Your progression reminds me of consideration about a "base infinity",
> > an infinite radix where instead of the symbols ranging form one to some
> > higher finite integer and being assembled as a sequence of integral
> > moduli, modulating the digit by the base to the sequence index, eg base
> > ten or decimal, having an infinite radix where each of the natural
> > integers has its own symbol or letter in the alphabet that is infinite.
> >
> > Wouldn't you just need one digit?
>
> Yes, it seems so. However addition (for example) ceases to have any
> kind of algorithm and becomes just an infinite table of triplets of
> n1, n2, n3 where n3 = n1 + n2. (And you can't cheat by making
> your symbols for the numbers have any internal structure).
>
> >
> > Then, there would only need be one digit, of sorts, but because there
> > would only be one of those, then it's not really a digit per se but the
> > integer portion of the number.
> >
> > With something along the lines of an infinite base, then the
> > non-integer part would require only a digit as well, yet then then it
> > is unclear, or indefinite, which real number that portion represents,
> > only that it ranges over all possible values.
> >
>
> Yes because you can't compute e.g. the arbitrary sum of two
> numbers w/o looking them up in a table - i.e. computation is
> undecidable w/o tables - I think?
>
> > Base one, or unary, would be similar.
> >
> > Then, in binary, trinary, etcetera, base b >= 2, there are varying
> > observations about the portion of the number as it is generally
> > represented in positional notation and the way that the unit interval
> > is divided by those numbers. It still takes infinitely many of those
> > digits to represent some numbers.
>
> Well this seems like an interesting problem. The finite integers
> are [easily] representable by a set of symbols one for each. But the
> right-of-the-parentheses is basically equinumerous with the reals.
> I think we differ over how many of these there are, but in any event,
> I think you were saying you can't represent them by the same
> infinite 'digits' you do for the natural numbers, or at least you
> can't calculate their 'value', and I think that's right.
>
> > Where on the side that the base is
> > unary or infinitary besides zero it is unclear which number is
> > represented except with some notion of a least positive real number and
> > succession of the real numbers of the unit interval in their linear,
> > sequential, total, normal, natural, well-ordering, on the other the
> > density of the reals leads to numbers for which there is no finite
> > representation.
> >
>
> I didn't quite follow this, sorry Ross, e.g. why does a notion of
> a least postive real number help?
>
> > This is discussing the continuum of the unit interval more than where
> > you are discussing the Spinozan continuum of sorts of the naturals and
> > then poly-adic integers, they're still the same numbers.
> >
> > About the continuous and discrete and the nature of the continuum of
> > real numbers and properties of continua and continuity, I think I
> > should reread this short treatise on the historical development of
> > these notions.
> >
> > http://plato.stanford.edu/entries/continuity/
> >
> > Then, I would recommend reading Boyer's _The History of the Calculus
> > and Its Conceptual Development_.
> >
> > They're still the same numbers.
> >
> > Ross
>
> Thanks (and for the refs).
>
> Ken
Hi Ken,
Yes, if each numeral, or integer, and there are various usages of the
word numeral, has its own symbol, then addition is defined as a + b =
c, for integers a, b, and c, or n_1, n_2, n_3. That still leaves it
being the ring of integers. Those integers still have the internal
structure 0, S0, SS0, ..., the successors of zero, with a sign bit,
which looks just like unary representation. The "axioms" of that
structure as the integers have the same form. The integral modulus
representation, with the positional notation and a_1 + a_2 < b mod b,
for digits a and base b, is an addended structure. It takes infinitely
many sequences, or symbols, to represent the integers. The positional
notation does use a structure that is very useful because of integral
moduli and modular arithmetic, on the ring and field of integers.
A least positive real, or iota, is as we discussed long ago this year
about well-ordering the reals in "On Well-Ordering(s) of Sets Dense in
the Reals, Infinity." You might read the recent posts to the thread
"Well Ordering the Reals", where I have been describing that a
variation of Cantor's first, or nested intervals, actually implies that
some least positive real or iota value exists, else there are
contradictions in well-ordering the reals, which as well I have said
for quite some time. Also, that is not finding disagreement. Where
Cantor's first, in variation or basically extension or via transfer,
doesn't apply to a well-ordering, then it doesn't apply to a bijection
between the naturals and reals, and one interpretation of nested
intervals is that the only way to address the real numbers of a line
segment is via their normal ordering. Otherwise, the nested intervals
converge yet can't be said to meet.
Ross
.
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- From: Ken Quirici
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