Steve Jackson Games - Site Navigation
Home General Info Follow Us Search Illuminator Store Forums What's New Other Games Ogre GURPS Munchkin Our Games: Home

Go Back   Steve Jackson Games Forums > Roleplaying > Roleplaying in General

Reply
 
Thread Tools Display Modes
Old 05-26-2022, 04:17 PM   #101
Anthony
 
Join Date: Feb 2005
Location: Berkeley, CA
Default Re: Gaming philosophy conundra

Quote:
Originally Posted by whswhs View Post
It seems as if it still must be the case that if you represent every number on the apeirohedron as a decimal (which is how Lovewyrm chose to represent them), there must be many numbers whose representation has infinitely many digits and therefore counts as a "sword" of infinite length.
You are guaranteed arbitrary length but not infinite length. I'm actually curious what the difference between a polyhedron with aleph-0 sides and a sphere (same cardinality as the reals; if the continuum hypothesis is true this would be aleph-1) would be.
__________________
My GURPS site and Blog.
Anthony is offline   Reply With Quote
Old 05-26-2022, 06:30 PM   #102
whswhs
 
Join Date: Jun 2005
Location: Lawrence, KS
Default Re: Gaming philosophy conundra

Quote:
Originally Posted by Anthony View Post
You are guaranteed arbitrary length but not infinite length. I'm actually curious what the difference between a polyhedron with aleph-0 sides and a sphere (same cardinality as the reals; if the continuum hypothesis is true this would be aleph-1) would be.
If you are using a decimal representation, then so simple a fraction as 1/3 equates to 0.333 ..., which will have infinite length.

As for the difference between an apeirohedron and a sphere, I was wondering that myself. The Wikipedia article on apeirohedra describes some really odd configurations, but doesn't even mention anything that sounds spherelike.
__________________
Bill Stoddard

I don't think we're in Oz any more.
whswhs is offline   Reply With Quote
Old 05-26-2022, 06:49 PM   #103
Anthony
 
Join Date: Feb 2005
Location: Berkeley, CA
Default Re: Gaming philosophy conundra

Quote:
Originally Posted by whswhs View Post
If you are using a decimal representation, then so simple a fraction as 1/3 equates to 0.333 ..., which will have infinite length.

As for the difference between an apeirohedron and a sphere, I was wondering that myself. The Wikipedia article on apeirohedra describes some really odd configurations, but doesn't even mention anything that sounds spherelike.
Well, an apeirohedron isn't a closed surface, so it can't be similar to a sphere. I'm not actually sure it's possible to define a fair (all sides equal) polyhedron with countably infinite sides (you can define an unfair one by selecting an infinite series that adds up to a finite value, such as (r^n) where r is a number between 0 and 1).
__________________
My GURPS site and Blog.
Anthony is offline   Reply With Quote
Old 05-26-2022, 08:05 PM   #104
whswhs
 
Join Date: Jun 2005
Location: Lawrence, KS
Default Re: Gaming philosophy conundra

Quote:
Originally Posted by Anthony View Post
Well, an apeirohedron isn't a closed surface, so it can't be similar to a sphere. I'm not actually sure it's possible to define a fair (all sides equal) polyhedron with countably infinite sides (you can define an unfair one by selecting an infinite series that adds up to a finite value, such as (r^n) where r is a number between 0 and 1).
My geometric intuition is that as the number of sides of the polyhedron goes to aleph sub null, the area of each side goes to 0, and so does the length of the boundary of each side. So it seems as if all the sides would be geometrically identical. In effect, each would be a geometric point. That seems to be a sort of regular polyhedron; you would lose the irregularities in taking the sidedness to aleph sub null.
__________________
Bill Stoddard

I don't think we're in Oz any more.
whswhs is offline   Reply With Quote
Old 05-26-2022, 09:48 PM   #105
khorboth
 
khorboth's Avatar
 
Join Date: Aug 2007
Location: Denver, CO
Default Re: Gaming philosophy conundra

The PCs are in a trolley on a railroad designed by the GM. The players can pull a lever which move it from track A to track B. The GM has carefully planned out the railroad on track A, but track B results in chaos.

On Track A, The GM will have a great time, and the players will have a little fun.

On Track B, everyone will have a random amount of fun, but the GM will likely have less fun than on track A and the players will likely have more fun than on track A. The GM will also have great stress because he really likes his trolley railroad.

Is it morally permissible to pull the lever?

Is it morally permissible to not pull the lever?

Does the answer change when taking into account future campaigns and less need of rails?
khorboth is offline   Reply With Quote
Old 05-26-2022, 11:10 PM   #106
Anthony
 
Join Date: Feb 2005
Location: Berkeley, CA
Default Re: Gaming philosophy conundra

Quote:
Originally Posted by whswhs View Post
My geometric intuition is that as the number of sides of the polyhedron goes to aleph sub null, the area of each side goes to 0, and so does the length of the boundary of each side. So it seems as if all the sides would be geometrically identical. In effect, each would be a geometric point. That seems to be a sort of regular polyhedron; you would lose the irregularities in taking the sidedness to aleph sub null.
The degenerate case of a point is not aleph-null -- the number of points on a surface is the same cardinality as the set of real numbers.
__________________
My GURPS site and Blog.
Anthony is offline   Reply With Quote
Old 05-26-2022, 11:20 PM   #107
Lovewyrm
 
Lovewyrm's Avatar
 
Join Date: Apr 2022
Default Re: Gaming philosophy conundra

Quote:
Originally Posted by whswhs View Post
The number 1/9, in decimal representation, is 0.11111 ..., with an infinite repetition of the digit one. So it seems it would be infinitely long. That might be a challenge to wield as a sword.
Yes, if you can't wield it, you lose the contest.

Just comparing numbers when you have a magical d∞ seems a bit mundane to me.

But yeah a real die like this might very well just a potential grenade that impales things.
P.S.:
I assume the d∞ to start as a small die that has the infinity symbol on it, and only shows the result when rolled.
I can't even imagine a die that actually has the sides needed to accomodate all the faces 'as is', otherwise.
Even my simple one kind of boggles me. Lol.
Lovewyrm is offline   Reply With Quote
Old 05-27-2022, 12:11 AM   #108
Daigoro
 
Daigoro's Avatar
 
Join Date: Dec 2006
Location: Meifumado
Default Re: Gaming philosophy conundra

Quote:
Originally Posted by Lovewyrm View Post
Just comparing numbers when you have a magical d∞ seems a bit mundane to me.
That is the point of rolling dice though. You could just as well throw d6's at your GM if you wanted that as a game mechanic.
Quote:
P.S.:
I assume the d∞ to start as a small die that has the infinity symbol on it, and only shows the result when rolled.
I can't even imagine a die that actually has the sides needed to accomodate all the faces 'as is', otherwise.
Even my simple one kind of boggles me. Lol.
It might be a sphere which holographically displays its result above it when it comes to rest. I would also presume it to give integer results, so that two very close results could be compared in finite time.

But my original point was that rolling d∞'s would have the paradoxical result that the probability of an outcome was determined by the order of the dice rolls, whereas for finitely sided dice the probability is independent of the order.
__________________
Collaborative Settings:
Cyberpunk: Duopoly Nation
Space Opera: Behind the King's Eclipse
And heaps of forum collabs, 30+ and counting!
Daigoro is offline   Reply With Quote
Old 05-27-2022, 03:13 AM   #109
whswhs
 
Join Date: Jun 2005
Location: Lawrence, KS
Default Re: Gaming philosophy conundra

Quote:
Originally Posted by Anthony View Post
The degenerate case of a point is not aleph-null -- the number of points on a surface is the same cardinality as the set of real numbers.
That's true, but I'm not sure it's relevant.

Here, for example, is the interval [0, 1]. Now we look selectively at only the points in that interval that correspond to rational numbers (it could be algebraic numbers, but they're no more numerous than rational numbers, so let's stay with the simpler case). As we go through the ordered list of rational numbers that correspond to proper fractions (x/y, where x < y and both x and y are positive), the length of the line segment occupied by each point seems to go to zero as the number of points goes to infinity.

Similarly, on a spherical surface, the area occupied by each rationally numbered point seems to go to zero. The fact that those points do not make up the entirety of the spherical surface may not make a difference.
__________________
Bill Stoddard

I don't think we're in Oz any more.
whswhs is offline   Reply With Quote
Old 05-27-2022, 06:25 AM   #110
Agemegos
 
Agemegos's Avatar
 
Join Date: May 2005
Location: Oz
Default Re: Gaming philosophy conundra

Quote:
Originally Posted by whswhs View Post
Can you treat the interval [0, 1] as isomorphic to the interval from 0 to infinity?
Yes. Use a tan function. Or 1/x - 1
__________________

Decay is inherent in all composite things.
Nod head. Get treat.

Last edited by Agemegos; 05-27-2022 at 06:31 AM.
Agemegos is offline   Reply With Quote
Reply

Tags
philosophy, sisyphus, theseus, trolley problem

Thread Tools
Display Modes

Posting Rules
You may not post new threads
You may not post replies
You may not post attachments
You may not edit your posts

BB code is On
Fnords are Off
[IMG] code is Off
HTML code is Off

Forum Jump


All times are GMT -6. The time now is 02:20 AM.


Powered by vBulletin® Version 3.8.9
Copyright ©2000 - 2024, vBulletin Solutions, Inc.