05262022, 09:13 AM  #91  
Join Date: Jun 2005
Location: Lawrence, KS

Re: Gaming philosophy conundra
Quote:
I'm not sure what happens if your die has continuous sides. If you give the south pole a value of 0 and the north pole a value of 1, and every point in between has some fractional value, then for example if you land on the point whose fractional value is exactly 0.5, your opponent has a 50% chance of beating you, right? There are no more points between 0.5 and 1 than there are between 0 and 0.5. Can you treat the interval [0, 1] as isomorphic to the interval from 0 to infinity? I'm not sure you can. "Infinity" here probably means aleph sub null, the infinity of natural numbers, which can be visualized as a line; but I don't think the infinity of the continuum is on that line. So maybe there's a conceptual difference between a sphere and an infinitesided polyhedron, or an apeirohedron (I just looked on Wiktionary and they say that word already exists, so I'm not the first to make it up)? It seems as if you have said that when you throw an apeirohedral die, the number of sides with larger numbers must always exceed the number of sides with smaller numbers. Is that the same as saying that it must always land on the "bottom"?
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Bill Stoddard I don't think we're in Oz any more. 

05262022, 10:19 AM  #92 
Join Date: Feb 2005
Location: Berkeley, CA

Re: Gaming philosophy conundra
Unless your d∞ is infinite because it can produce noninteger values, in which case a range from 0 to 1 is still infinite.

05262022, 12:10 PM  #93  
Join Date: Apr 2022

Re: Gaming philosophy conundra
Quote:
But consider this number 0.034749592304805347502348023489574509683045803570 24803457045382304702357606035739045203485704596804 80345870345830674056 vs 0.555555554 First one's longer because it has more digits. Second is larger, numerically, but that's boring. In other words, I was proposing it from the perspective of "The die must be able to accomodate all the digits of ∞ on its surface so the roller can read the results." A result with sufficient length would possibly be suitable to bonk one another with. 

05262022, 01:04 PM  #94  
Join Date: Jun 2005
Location: Lawrence, KS

Re: Gaming philosophy conundra
Quote:
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Bill Stoddard I don't think we're in Oz any more. 

05262022, 01:48 PM  #95  
Join Date: Oct 2010
Location: earth....I think.

Re: Gaming philosophy conundra
Quote:
There are more real numbers between 0 and 1 than there are whole numbers from 0 to positive infinity. 

05262022, 02:07 PM  #96  
Join Date: Jun 2005
Location: Lawrence, KS

Re: Gaming philosophy conundra
Quote:
Long ago, when I was taking real analysis, I asked my instructor why Cantor's argument didn't apply equally to whole numbers. He answered that rational numbers, in decimal representation, could have an infinite number of digits, but whole numbers could not. So using an apeirohedral die would give you many numbers with infinite digits; in fact I think that statistically that would be the expected result, and perhaps getting a number with finite digits might be so unlikely as to have effective probability 0. A sword of infinite length would be unwieldly. But that problem doesn't need real numbers; you can get it already just with rational numbers.
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Bill Stoddard I don't think we're in Oz any more. 

05262022, 02:17 PM  #97 
Join Date: Feb 2005
Location: Berkeley, CA

Re: Gaming philosophy conundra
The issue isn't precisely infinite digits, the issue is infinite information content. If you limit yourself to real numbers that can be expressed by a finite equation, you wind up with a set that is no larger than the set of whole numbers.

05262022, 02:28 PM  #98  
Join Date: Jun 2005
Location: Lawrence, KS

Re: Gaming philosophy conundra
Quote:
But whole numbers cannot possibly have infinitely many digits. The number 0.111 ... is a rational number but the number ... 111. is not a whole number.
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Bill Stoddard I don't think we're in Oz any more. 

05262022, 02:42 PM  #99  
Join Date: Feb 2005
Location: Berkeley, CA

Re: Gaming philosophy conundra
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05262022, 03:03 PM  #100  
Join Date: Jun 2005
Location: Lawrence, KS

Re: Gaming philosophy conundra
Quote:
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Bill Stoddard I don't think we're in Oz any more. 

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philosophy, sisyphus, theseus, trolley problem 
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