Re: DIS: Re: BUS: proposal: fix unanimity
On 1/12/07, Michael Slone [EMAIL PROTECTED] wrote: On 1/11/07, Taral [EMAIL PROTECTED] wrote: I *hate* the ultrapower construction, because nobody's been able to actually construct a free ultrafilter. Nobody's been able to construct a free ultrafilter because it's impossible to do so. Hope this helps. Yes, I know. Hence why I prefer the polynomial ratio construction. -- Taral [EMAIL PROTECTED] You can't prove anything. -- Gödel's Incompetence Theorem
Re: DIS: Re: BUS: proposal: fix unanimity
On 1/12/07, Taral [EMAIL PROTECTED] wrote: Yes, I know. Hence why I prefer the polynomial ratio construction. Are you talking about the Schmieden--Laugwitz construction (using a cofinite filter)? Their construction produces a ring with zero divisors, and it isn't even an ordered ring. -- Michael Slone
Re: DIS: Re: BUS: proposal: fix unanimity
On 1/12/07, Michael Slone [EMAIL PROTECTED] wrote: Are you talking about the Schmieden--Laugwitz construction (using a cofinite filter)? Their construction produces a ring with zero divisors, and it isn't even an ordered ring. Hm, it seems I was mistaken. -- Taral [EMAIL PROTECTED] You can't prove anything. -- Gödel's Incompetence Theorem
Re: DIS: Re: BUS: proposal: fix unanimity
Ian Kelly wrote: Come to think of it, it's also more correct. A voting index of aleph-null should properly only be used when infinitely many FOR votes are placed, Not at all. The voting index is not inherently a cardinal. It is not a count of FOR votes, but (mostly) a ratio between two cardinals. Provided that there's at least one AGAINST vote, the VI is a rational number, and it can well take values such as 1/3 or 7/2. If there is fewer than one AGAINST vote counted then it is to be expected that the VI would exceed the number of FOR votes. (Imagine if it were possible to cast a half vote.) If there are no AGAINST votes at all then we have no choice but to leave the realm of rational numbers. Calling it Unanimity just obscures its true behaviour as a transfinite hyperreal. -zefram
Re: DIS: Re: BUS: proposal: fix unanimity
On 1/11/07, Zefram [EMAIL PROTECTED] wrote: Ian Kelly wrote: Come to think of it, it's also more correct. A voting index of aleph-null should properly only be used when infinitely many FOR votes are placed, Not at all. The voting index is not inherently a cardinal. It is not a count of FOR votes, but (mostly) a ratio between two cardinals. Provided that there's at least one AGAINST vote, the VI is a rational number, and it can well take values such as 1/3 or 7/2. If there is fewer than one AGAINST vote counted then it is to be expected that the VI would exceed the number of FOR votes. (Imagine if it were possible to cast a half vote.) If there are no AGAINST votes at all then we have no choice but to leave the realm of rational numbers. Calling it Unanimity just obscures its true behaviour as a transfinite hyperreal. You're probably right that my usage is also incorrect. That was merely the product of trying to envision a voting index for which the value aleph-null does make sense. By your argument, aleph-null should never be used for voting index, since aleph-null is not a hyperreal (as far as I am aware -- my understanding is that an infinite hyperreal is defined as the inverse of an infinitesimal hyperreal, which is not how aleph-null is defined).
Re: DIS: Re: BUS: proposal: fix unanimity
On 1/11/07, Zefram wrote: (Imagine if it were possibleto cast a half vote.) I've always thought this was a good idea... now if only i can come up with a plausible way to implement it.
Re: DIS: Re: BUS: proposal: fix unanimity
Ian Kelly wrote: By your argument, aleph-null should never be used for voting index, since aleph-null is not a hyperreal (as far as I am aware -- my understanding is that an infinite hyperreal is defined as the inverse of an infinitesimal hyperreal, which is not how aleph-null is defined). Curious. I rather thought it was. Which infinite hyperreal do you suggest in its place? I must confess I have never formally studied nonstandard analysis. -zefram
Re: DIS: Re: BUS: proposal: fix unanimity
On 1/11/07, Zefram [EMAIL PROTECTED] wrote: Ian Kelly wrote: By your argument, aleph-null should never be used for voting index, since aleph-null is not a hyperreal (as far as I am aware -- my understanding is that an infinite hyperreal is defined as the inverse of an infinitesimal hyperreal, which is not how aleph-null is defined). Curious. I rather thought it was. Which infinite hyperreal do you suggest in its place? I must confess I have never formally studied nonstandard analysis. Neither have I. And I'm not convinced that n/0 (I don't know whether it has a name, let's call it h)is actually a hyperreal, since that seemingly would require 0 to be its inverse, which would require 0 * h to be 1 if the hyperreals are to be a field, which just strikes me as being a bit odd. I normally just satisfy myself with the usual definition than n/0 is undefined, which is why unanimity makes as much sense to me as anything.
Re: DIS: Re: BUS: proposal: fix unanimity
Ian Kelly wrote: And I'm not convinced that n/0 (I don't know whether it has a name, let's call it h)is actually a hyperreal, It's not. It's undefined. That's why the Rule determining the voting index needs a special case for where there are no AGAINST votes. 0/0 is also undefined, which is why that's got a special case too. Strictly, what we want for the case of zero AGAINST votes and more than zero FOR votes is the limit of 1/n as n approaches zero from above. I think that's aleph-0, but I'm not 100% sure. This does seem to match your definition of an infinite hyperreal. Applying the same logic to the case of zero votes both ways, we could reasonably pick either the limit of n/n as n approaches zero, which is 1, or the limit of 0/n as n approaches zero, which is 0. The present Rule chooses the latter of these. -zefram
Re: DIS: Re: BUS: proposal: fix unanimity
Zefram wrote: Strictly, what we want for the case of zero AGAINST votes and more than zero FOR votes is the limit of 1/n as n approaches zero from above. I think that's aleph-0, but I'm not 100% sure. This does seem to match your definition of an infinite hyperreal. Applying the same logic to the case of zero votes both ways, we could reasonably pick either the limit of n/n as n approaches zero, which is 1, or the limit of 0/n as n approaches zero, which is 0. The present Rule chooses the latter of these. Life would be a lot simpler if we dispensed with ratios entirely, and framed AIs in terms of differences. An AI=n might require that the number of FORs exceed the number of AGAINSTs by n. If we were worried about this making things too easy (particularly if we ever had a larger population), we could make the formula n * ceil(numplayers / 10) (with ceil the ceiling or round-up function). Michael.
Re: DIS: Re: BUS: proposal: fix unanimity
Michael Norrish wrote: Life would be a lot simpler if we dispensed with ratios entirely, and framed AIs in terms of differences. Interesting. I have a concept for quorum which might interact with that idea. There is a problem with the usual way quorum works, that there are situations where voting AGAINST a proposal actually makes it more likely to pass. (For example, if there have been four votes FOR and there is a quorum of five.) I'd like to avoid such perverse situations. So instead of requiring that a certain proportion of the electorate vote on a proposal for it to pass, there should be a requirement that a certain (smaller) proportion of the electorate vote in favour. This is in addition to the requirement that a certain proportion of those expressing an opinion vote in favour. For example, an ordinary proposal might require that at least 10% of the electorate and over 50% of those expressing an opinion vote in favour. A very significant proposal might then require 20% of the electorate and 75% of those expressing an opinion. Working with differences instead of ratios, my quorum replacement would require a proposal to get at least N more votes in favour than against, and at least M votes in favour absolutely. Both of these numbers could be selected as (modified) proportions of the size of the electorate. Another approach to the whole thing would be to express the voting result as the ratio of votes in favour to the total number of votes cast. The maximum voting index would then be 1 (i.e., 100%). We can translate VIs and Powers: 1 - 50%, 2 - 67%, 3 - 75%, 4 - 80%, Unanimity - 100%. This still needs a special case for where there are no votes cast, but we don't need infinite numbers there. -zefram
Re: DIS: Re: BUS: proposal: fix unanimity
On 1/11/07, Zefram [EMAIL PROTECTED] wrote:
Strictly, what we want for the case of zero AGAINST votes and more than
zero FOR votes is the limit of 1/n as n approaches zero from above.
I think that's aleph-0, but I'm not 100% sure. This does seem to match
your definition of an infinite hyperreal.
YDAFI, BYGIA.
Let R denote the real numbers and N the natural numbers. One way to
construct the hyperreal numbers is as follows. Consider the structure
R^N, the collection of all sequences of real numbers. This has a
natural ring structure as a product of copies of R, but unfortunately
it isn't a field, since it is full of zero divisors. The hyperreals
are a special quotient ring of R^N. In other words we say that two
sequences are ``the same'' if they satisfy a particular rule.
The next paragraph is quite technical, so skip it if you want.
The rule we choose is the following. First select a nonprincipal
ultrafilter on the natural numbers. An ultrafilter on N is a maximal
family of subsets of N which doesn't include the empty set, is closed
under taking supersets, and has the finite intersection property. It
is nonprincipal if the intersection of all the subsets is the empty
set. Zorn's lemma must be used to show the existence of such an
ultrafilter. With the nonprincipal ultrafilter selected, we now say
that two sequences (x_n), (y_n) are to be identified provided that the
set of natural numbers n such that x_n = y_n is exactly an element of
the ultrafilter.
Okay, so we've got a rule ~ for identifying sequences.
The structure *R := R^N/~ we get by identifying sequences of real
numbers according to this rule is called an ultrapower of R. There is
a theorem by Los (in TeX, \L o\'s) that says that the new structure *R
satisfies all the *first-order* axioms satisfied by the original
structure R. Since the axioms for a field are first-order, this
implies that *R is a field. Moreover, there is an embedding of R into
*R; a real number r is mapped to the (equivalence class of the)
constant sequence (r, r, r, ...).
The only axiom of R that is not first-order is the greatest lower
bound axiom. This is the axiom that says that any nonempty subset of
R which has a lower bound has a greatest lower bound. And in fact *R
doesn't satisfy this axiom. (If it did, it would be isomorphic to R.)
Unfortunately, Los's theorem doesn't guarantee this, it has to be
shown explicitly.
Luckily, this is not difficult to do. All we have to do is find a
number which is positive but smaller than any positive real number.
Here is one: the (equivalence class of the) sequence
(1, 1/2, 1/3, 1/4, 1/5, ..., 1/n, ...).
Since the sequence is decreasing, it's smaller than any constant
sequence of the form
(r, r, r, ...)
where r is positive. But since all its terms are positive, it's
bigger than zero. Hence it defines an infinitesimal, which we call h.
Since h is not zero and *R is a field, we can then define a rather
large hyperreal H by H = h^{-1}.
The original construction of *R was performed in the 1960s by Abraham
Robinson. The definition of Aleph-nought is classical and is due to
Georg Cantor. Briefly, Aleph-nought is the smallest infinite
cardinal. In general, a cardinal is the smallest ordinal of a
particular cardinality, although for Aleph-nought there's really only
one obvious choice anyway.
--
Michael Slone
