Re: DIS: Re: BUS: registration and CFJs
Ed Murphy wrote: YAFI, YGI. What does that expand to? The Clerk of the Courts may, without objection, unlink one or more of the linked CFJs from the others by announcement. If e unlinks more than one as a set, then those CFJs remain linked to each other. There are interacting time limits here. Taking an action Without Objection takes at least four days, and the Judge has to be assigned within a week. What if the CotC is away for three days? Without Objection and Linked CFJs should be capitalised. If the restriction on Barring passes -- it's the obvious fix -- then there's no need for preemptive unlinking. Unlinking is a matter to be decided on the content of the CFJs, so I think that's better left to a Judge rather than the CotC. -zefram
DIS: proto-proto: expiry of rules
I have a concept for real-life national law, which I think would work well in nomic too. The idea is that, based on the principle that no society can morally bind its successors, laws should not automatically last more than a generation. Thus every law should expire twenty years after enactment, unless extended by a positive legislative act. Extending a law for another twenty years would be a much simpler process than enacting it again from scratch, of course, provided that it's uncontroversial. So there wouldn't be any fuss about continuing the law against murder, but a law banning demonstrations in certain places would have a harder time of being extended for another generation. In Agora the effective generation length is much shorter, perhaps two years. I envision recording an expiry date for each Rule, which would be published in the Rulesets. Extension of the expiry date for a Rule would be by a vote resembling a Proposal, possibly actually a Proposal. Interested? If so, I'll draft a proto-Proposal. -zefram
DIS: Re: BUS: votes on 4877-4892
Zefram wrote: On the Proposals listed below I vote thus: I thought voting power was set at the beginning of the week (so you wouldn't have any votes on these proposals), but I can't seem to find that in the ruleset any longer. Did that requirement go away? Sherlock Never Miss an Email Stay connected with Yahoo! Mail on your mobile. Get started! http://mobile.yahoo.com/services?promote=mail
Re: DIS: Re: BUS: votes on 4877-4892
Jonathan Fry wrote: I thought voting power was set at the beginning of the week (so you wouldn't have any votes on these proposals), but I can't seem to find that in the ruleset any longer. Did that requirement go away? Looks like it. The phrase voting limit appears only in Rule 1950, which sets the defaults, and Rule 2126, which allows modifications by expenditure of VCs. Rule 106 says the eligible voters are the active players, but I don't see anything that says when that's determined, so my interpretation is that eligibility should be determined at the time of voting. -zefram
DIS: Re: BUS: Re: OFF: Distribution of Proposals 4877-4892
Sherlock wrote: 4879 | Dishonor Rolls, redux | Murphy| 1 | 16Dec06 | O I love this... FOR I feel obligated to remind folks that I didn't write this rule, just proposed to patch it in from a previously adopted proposal that accidentally didn't have its full intended effect. 4882 | The Lady, or the Tiger? | Murphy| 1 | 22Dec06 | O I don't know how to vote on this. I hate proposals to retcon the game state. Given that the relevant portion of the game state has been deliberately twisted into self-contradiction, there seems little alternative.
Re: DIS: Re: BUS: registration and CFJs
Zefram wrote: Ed Murphy wrote: YAFI, YGI. What does that expand to? You asked for it, you got it. The Clerk of the Courts may, without objection, unlink one or more of the linked CFJs from the others by announcement. If e unlinks more than one as a set, then those CFJs remain linked to each other. There are interacting time limits here. Taking an action Without Objection takes at least four days, and the Judge has to be assigned within a week. What if the CotC is away for three days? In theory, the CotC may be prosecuted for violating ASAP. In practice, no one bothers to prosecute much of anything these days, but of course it only takes one vigilant player to change that. Without Objection and Linked CFJs should be capitalised. Things have been drifting away from Wes's old practice of Capitalizing Words deemed to be Important. If the restriction on Barring passes -- it's the obvious fix -- then there's no need for preemptive unlinking. Unlinking is a matter to be decided on the content of the CFJs, so I think that's better left to a Judge rather than the CotC. Point. Even with ineligibility by request repealed, we still allow judges to do this. Actually, I see a couple more things to improve in that rule. Proto-Proposal: Clean up CFJ linkage Amend Rule 2024 (Linked Statements) to read: Linked CFJs are multiple Calls for Judgement deemed to be sufficiently similar that they should be have a single judge. Linkage is transitive. When a set of one or more linked CFJs change jurisdiction, they remain linked to each other, but become unlinked from any other CFJs; they may become linked to one or more CFJs within the new jurisdiction. Multiple CFJs, submitted in a single message and clearly labelled as Linked CFJs, become linked. The players (if any) barred by the caller from judging the first CFJ are the only players e may bar from judging the others. The Clerk of the Courts shall assign a judge to a set of Linked CFJs, as if they were a single CFJ. The judge must be eligible to judge each of the Linked CFJs, and is simultaneously assigned as judge of each of the Linked CFJs. The judge of a set of Linked CFJs shall submit eir judgement of each of those CFJs in a single message. A trial judge may remand one or more linked CFJs to the Clerk of the Courts by announcement. E ceases to be their judge. A trial judge may transfer one or more of eir CFJs to a second trial judge by announcement (identifying one of the second judge's CFJs), provided that the second judge consents, and is eligible to judge all of them. The transferred CFJs become linked to the second judge's CFJ (and any others to which it is already linked). Create a rule titled Excess CFJs with this text: A CFJ made by a person who has previously made five or more CFJs during the same Agoran Week is an Excess CFJ. The Clerk of the Courts may dismiss an Excess CFJ by announcement.
Re: DIS: proto: broaden annotations
Zefram wrote: The Judge of any CFJ, the Statement of which alleges that a Rule should be interpreted in a certain way, which is judged TRUE or FALSE, may, at eir discretion, issue an Order requiring the Rulekeepor to annotate the Rule in question accordingly. If the CFJ was judged TRUE then the annotation shall be the Statement of the CFJ, and if the CFJ was judged FALSE then the annotation shall be the contrary of the Statement of the CFJ. Such an annotation, while it exists, shall guide application of that Rule. Expressing the contrary of a given statement is not always trivial, unless you resort to the form The statement 'foo bar' is false. Just require the rule to be annotated with the Statement and Judgement.
Re: DIS: proto: broaden annotations
Ed Murphy wrote: Expressing the contrary of a given statement is not always trivial, unless you resort to the form The statement 'foo bar' is false. I think it is sufficiently simple, generally a matter of inserting the word not. require the rule to be annotated with the Statement and Judgement. I think that would be confusing. I think the annotation should consist of the legal holding that is to be applied. Consider: potentially we could have annotations from other sources; it would be clearest to have all annotations expressed in the same form. -zefram
DIS: Re: BUS: two proposals relating to low AIs
Zefram wrote: H. Promotor, I hereby submit the following Proposal, entitled hoopy: --- Be it therefore resolved that a Rule be created with title Sass That Hoopy and text: When the Clerk of the Courts publishes a Judgement as required by Rule 591, e must accompany the publication with the statement Zefram is a frood.. Any publication of a Judgement that is not so accompanied does not satisfy the Clerk of the Courts' obligation to publish the Judgement. This Rule takes precedence over Rule 591. --- I hereby set the Adoption Index of the above Proposal to 0.01. Rule 955 still requires it to get more votes FOR than AGAINST, and even if it didn't, Rule 1482 would overrule its claim of precedence. I hereby submit the following Proposal, entitled fix Power and Rule Changes: --- Be it therefore resolved that one second after the adoption of this Proposal Rule 105 Rule Changes be modified by replacing the text Enact a rule with Enact a Rule with Power no greater than its own, and by replacing the text Repeal a rule with Repeal a Rule with Power no greater than its own. --- I hereby set the Adoption Index of the above Proposal to 3. I noticed this bug four days ago (while answering a question from Peter about how voting credits work), and was planning to fix it with more style. Can't surprise people with it any more, may as well present it right away...
DIS: Re: BUS: Proposal: Red Tape Scam
Ed Murphy wrote: Amend the rule titled Fantasy Rule Changes to have number 105, Please don't renumber Rules. It screws up the amendment numbers (though I see they're no longer formally defined). -zefram
DIS: Re: BUS: CFJ on Unanimity
On 1/11/07, Zefram [EMAIL PROTECTED] wrote: H. Clerk of the Courts, I hereby Call for Judgement on the statement: Rule 955 should be interpreted such a voting index of Unanimity cannot meet or exceed any numerical adoption index. Unanimity is no longer defined anywhere in the Rules, so I think it no longer has any numerical properties. I concur. The definition of unanimity was removed when references to switches were culled from the ruleset. -- Michael Slone
DIS: Re: BUS: votes on 4877-4892
Zefram wrote: Rule 106 says the eligible voters are the active players, but I don't see anything that says when that's determined, so my interpretation is that eligibility should be determined at the time of voting. From the last paragraph of R1950: up to a number equal to that person's voting limit on that decision as determined when the voting period for that decision began, ^^ ^^ Easy to miss (I did until Maud pointed it out last month). -Goethe
DIS: Paradox resolution without retcon proposal
Zefram wrote: In a Platonist system this wouldn't be a problem. In the Platonist model, Judgements don't actually change the state of anything, they're just meant to point out what the state actually is. A contradiction of Judgements isn't a problem then: one of the purported Judgements is wrong and the other right, neither changes anything, and the game goes merrily on. I realized upon Zefram's post above that our Judgement system is Platonist by eir definition. Unless a Judge issues an Order, e merely opines on the game state at the time of the CFJ and we abide by the precedent and opinion of what the game state was (or we use the appropriate appeals mechanism if we disagree). In fact, judgements only carry weight as long as we all agree to abide by the precedents set in them. There's nothing to stop a player for continually calling a CFJ on the same statement until e gets the judgement e wants. If we all respect precedent, we would DISMISS or defer later CFJs to the first in such a series, but there's no legal requirement to do so. In the case of two conflicting CFJs which invalidate each others' authority, there is no precedent and the CFJ (arguably) can't be appealed, so the right solution (unless the whole judgement system is broken, which it isn't) is to call the CFJ again and have the new judge, hopefully someone who's authority isn't in question due to the paradox, judge the truth of the CFJ and thus resolve the conflict. (Note that this does not take away Murphy's win: Murphy attempted an action that required resolution of the paradox to determine, and the paradox wasn't resolved at that time). Learn something new every day. Thanks, Zefram!
DIS: Re: BUS: Proposal: Red Tape Scam
Zefram wrote: Please don't renumber Rules. It screws up the amendment numbers (though I see they're no longer formally defined). Things like amendment numbers were explicitly left to game custom as part of increasing Officers' discretion in their recordkeeping, but also in anticipation of the judiciary game picking up (e.g. more authority of judges to regulate items like this based on custom). Didn't mean that to apply to repeals tho, that there's a bug :). Agree with the nervousness of renumbering... how does it interact with R1586, for example? (both in continuity and same names). -Goethe
DIS: Re: BUS: Proposal: Agoran Arms
Grey Knight wrote: The coat of arms of Agora is defined by the following blazon: Nice. I was just thinking about proposing a rule specifying symbols of the game. I noticed that the rule specifying that the name of the game is Agora has gone, and I thought it ought to come back. But if it does, then why not generalise the concept? In addition to a name I thought the rules could also specify a flag, an anthem, a motto, and so on. A coat of arms is a good one to have too. -zefram
Re: DIS: Re: BUS: Proposal: Agoran Arms
On 1/11/07, Zefram [EMAIL PROTECTED] wrote: Grey Knight wrote: The coat of arms of Agora is defined by the following blazon: Nice. I was just thinking about proposing a rule specifying symbols of the game. I noticed that the rule specifying that the name of the game is Agora has gone, and I thought it ought to come back. But if it does, then why not generalise the concept? In addition to a name I thought the rules could also specify a flag, an anthem, a motto, and so on. A coat of arms is a good one to have too. -zefram I feel things of this nature might be changeable upon a win, popular vote, or some other period of time. Would allow for some theming or strange fun.
DIS: Re: BUS: CFJ on Unanimity
On 1/11/07, Zefram [EMAIL PROTECTED] wrote: H. Clerk of the Courts, I hereby Call for Judgement on the statement: Rule 955 should be interpreted such a voting index of Unanimity cannot meet or exceed any numerical adoption index. Unanimity is no longer defined anywhere in the Rules, so I think it no longer has any numerical properties. It retains the properties it had when it was last defined, no? -- Taral [EMAIL PROTECTED] You can't prove anything. -- Gödel's Incompetence Theorem
Re: DIS: Re: BUS: CFJ on Unanimity
On 1/11/07, Taral [EMAIL PROTECTED] wrote: On 1/11/07, Zefram [EMAIL PROTECTED] wrote: H. Clerk of the Courts, I hereby Call for Judgement on the statement: Rule 955 should be interpreted such a voting index of Unanimity cannot meet or exceed any numerical adoption index. Unanimity is no longer defined anywhere in the Rules, so I think it no longer has any numerical properties. It retains the properties it had when it was last defined, no? How could you justify that? If something loses its definition then its properties are unknown. In programming if you have a pointer pointing to an object, and you delete that object, the pointer now points to who knows what. In our case here the pointer - Unanimity - pointed to a definition - exceeding any numerical adoption index - yet when that was deleted any references to Unanimity refer to something with now undefined properties.
DIS: Unanimity issue
With the definition of Unanimity in question, would someone like to cast AGAINST votes as needed to prevent any of the proposals currently in their voting period from passing unanimously? I'd propose a fix to the definition, but I need to head offline now (and it's time someone else had a chance at some fix proposals, anyway).
Re: DIS: Re: BUS: CFJ on Unanimity
It retains the properties it had when it was last defined, no? How could you justify that? If something loses its definition then its properties are unknown. In programming if you have a pointer pointing to an object, and you delete that object, the pointer now points to who knows what. In our case here the pointer - Unanimity - pointed to a definition - exceeding any numerical adoption index - yet when that was deleted any references to Unanimity refer to something with now undefined properties. The rules aren't a computer program. Clearly, the definition isn't simply lost to the world as your metaphor would suggest. After all, I can still see it in the archives. -root
Re: DIS: Re: BUS: CFJ on Unanimity
Taral wrote: It retains the properties it had when it was last defined, no? Certainly not. Game custom has never supported definitions outliving their repeal. The meaning of the formerly-defined term reverts to whatever it would be if the definition had never existed. In this case, Unanimity has a meaning from ordinary English, which I believe now prevails, but that meaning has no numerical aspect. -zefram
Re: DIS: proto-proto: expiry of rules
Zefram wrote: I have a concept for real-life national law, which I think would work well in nomic too. The idea is that, based on the principle that no society can morally bind its successors, laws should not automatically last more than a generation. Thus every law should expire twenty years after enactment, unless extended by a positive legislative act. Extending a law for another twenty years would be a much simpler process than enacting it again from scratch, of course, provided that it's uncontroversial. So there wouldn't be any fuss about continuing the law against murder, but a law banning demonstrations in certain places would have a harder time of being extended for another generation. In Agora the effective generation length is much shorter, perhaps two years. I envision recording an expiry date for each Rule, which would be published in the Rulesets. Extension of the expiry date for a Rule would be by a vote resembling a Proposal, possibly actually a Proposal. How would the expiry date change in the rule was amended? I'd guess that there are now very few rules in the ruleset that haven't been amended in the last two years. But because of the way we tend to re-use rules (amendments often completely replace the old text of a rule with something completely different), many rules were originally created a good while ago. Michael.
Re: DIS: proto-proto: expiry of rules
Michael Norrish wrote: How would the expiry date change in the rule was amended? With indivisible Rules as we have, I think any major change would want to extend the expiry date just as a new enactment would. Small amendments, however, such as a change of responsible officer due to repeal of an office, should not. Probably the way to handle it is that an amendment by default does not extend the expiry date, but the proposal doing the amendment has the option to explicitly extend it. Thanks for pointing out the issue. -zefram
DIS: Re: BUS: proposal: fix unanimity
What do we need to introduce cardinality concepts for? For voting purposes, unanimity is much more intuitive. 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, which I don't believe has ever happened.
DIS: Re: BUS: CFJ on Unanimity
Zefram wrote: Taral wrote: It retains the properties it had when it was last defined, no? Certainly not. Game custom has never supported definitions outliving their repeal. The meaning of the formerly-defined term reverts to whatever it would be if the definition had never existed. In this case, Unanimity has a meaning from ordinary English, which I believe now prevails, but that meaning has no numerical aspect. In this case, I would say that since it became undefined, we are required to use its ordinary English meaning, but if that English meaning doesn't make sense in context (no numerical meaning), a judge would be allowed to fall back on game custom and precedent, i.e. use its old definition. -Goethe
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
DIS: Re: BUS: proposal: fix unanimity
root wrote: A voting index of aleph-null should properly only be used when infinitely many FOR votes are placed, which I don't believe has ever happened. I tried once, but according to Kelly I only said I did. -G.
Re: DIS: Re: BUS: CFJ on Unanimity
Kerim Aydin wrote: judge would be allowed to fall back on game custom and precedent, i.e. use its old definition. Its old definition was not game custom, it was a Rule. The precedent of using that definition also does not apply, because a highly relevant aspect of the preceding situation has changed (i.e., the definition has been repealed). If you want to ressurect the old definition then reenact it. You can't ignore a repeal just because it's turned out to be inconvenient. -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
DIS: Re: BUS: registration and CFJs
Welcome to the game, Zefram! Heck of a way to jump right in. Make yourself at home. - Benjamin Schultz KE3OM OscarMeyr
DIS: Re: BUS: proposal: switch off the fountain
On Jan 11, 2007, at 10:39 AM, Zefram wrote: H. Promotor, I hereby submit the following Proposal, entitled switch off the fountain: --- WHEREAS the Town Fountain is an ugly triviality of ASCII art, unlike the fine Map of Agora, and whereas the scamming spirit of the game is adequately honoured by the Patent Title of Scamster held by each of the erectors of the Town Fountain, be it therefore resolved that Rule 2029 be repealed. --- The Town Fountain was created through a scam. You'll have to repeal it through another scam, if you want my vote. - Benjamin Schultz KE3OM Scamster OscarMeyr
DIS: Re: BUS: proposal: switch off the fountain
OscarMeyr wrote: The Town Fountain was created through a scam. You'll have to repeal it through another scam, if you want my vote. This is a scam, it repeals a power-4 Rule with an AI 1 proposal. Works because the Great Repeals deleted a part of 105(c). It's a scam that needs a majority, tho. -Goethe
Re: DIS: Re: BUS: registration and CFJs
On 1/11/07, Benjamin Schultz [EMAIL PROTECTED] wrote: Welcome to the game, Zefram! Heck of a way to jump right in. Make yourself at home. Technically, this ought to be s#Welcome#Welcome back# ... After all, Zefram was a Fugitive from Justice for over nine years. -- Michael Slone
DIS: a sentence from rule 105
The following sentence appears in rule 105: However, a proposal cannot cause a rule to have power greater than its own. This doesn't appear to affect Murphy's elegant scam, but it does imply that Zefram's scam wouldn't work. Briefly, rule 105 says that a proposal can *generally* enact a rule, and that such a rule would have power defaulting to one; since this is greater than the adoption index of Zefram's proposal, the proposal cannot *actually* do this. -- C. Maud Image (Michael Slone) I understand this. That frightens me. -- Murphy, in agora-discussion
DIS: Re: BUS: Doing stuff
On 1/11/07, Quazie [EMAIL PROTECTED] wrote: I declare PRESENCE on all proposals between 4877-4892 unless doing so does not help meet quorum. I move we repeal the conditional voting rule. -- Michael Slone
Re: BUS: Re: DIS: Unanimity issue
On 1/11/07, Michael Norrish [EMAIL PROTECTED] wrote: I hereby vote AGAINST all proposals currently up for vote. Which ones are those? -- Michael Slone
DIS: Re: BUS: proposal: fix proposal efficacy
Zefram wrote: Even after the fixes proposed for Rule Changes, it's still possible for a Proposal to be adopted with AI 1 and take effect with Power=0. That allows anything *except* Rule Changes to be done by an unpopular Proposal. Nope, Rule 955 prevents it: If the voting index exceeds one and meets or exceeds the adoption index of the decision, and if further quorum was achieved, then the option selected by Agora is ADOPTED. Otherwise, the option selected by Agora is REJECTED.
Re: DIS: Re: BUS: registration and CFJs
On Jan 11, 2007, at 9:01 PM, Michael Slone wrote: On 1/11/07, Benjamin Schultz [EMAIL PROTECTED] wrote: Welcome to the game, Zefram! Heck of a way to jump right in. Make yourself at home. Technically, this ought to be s#Welcome#Welcome back# ... After all, Zefram was a Fugitive from Justice for over nine years. -- Michael Slone Zefram 29 Oct 97 10:36:44 You're right! He's been on the lam for a very long time! - Benjamin Schultz KE3OM OscarMeyr
Re: DIS: Re: BUS: Doing stuff
On Jan 11, 2007, at 9:36 PM, Michael Slone wrote: On 1/11/07, Quazie [EMAIL PROTECTED] wrote: I declare PRESENCE on all proposals between 4877-4892 unless doing so does not help meet quorum. I move we repeal the conditional voting rule. Presence isn't a vote, is it? - Benjamin Schultz KE3OM OscarMeyr
