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General Discussion / Re: Random Stuff Part IV
« on: August 08, 2021, 12:31:33 am »
How am I still the 25th most prolific poster in these forums? I've barely even been here in the past seven years.
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General Discussion / Re: Maths thread.
« on: August 07, 2021, 11:19:39 pm »
I came up with a proof that 5 is a prime number.
5x5 = 25 = 4! +1, so 5 is not divisible by 2, 3, or 4.
5x5 = 25 = 4! +1, so 5 is not divisible by 2, 3, or 4.
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General Discussion / Re: Maths thread.
« on: March 19, 2017, 10:30:07 am »
An example that I've found useful for understanding the distinctions around axioms and models is to think about group theory. The definition of a group is essentially a list of axioms. A model of the group axioms is any collection of things which satisfy the group axioms -- in other words, a group! There are lots of statements which are undecidable from the group axioms. For instance, the statement "there is a nonidentity element which is its own inverse" is an undecidable statement, since there are some models where it is true (Z mod 2) and some where it is false (Z mod 3).
Similarly for rings. The definition of a ring lays out the axioms, and each ring is a model of those axioms. The statements "multiplication is commutative" and "each nonzero element has a multiplicative inverse" are independent of the ring axioms. If we choose to make these new axioms, then the models we are left with are called fields.
In a real analysis class, you probably laid out the axioms for the real numbers, namely that they form a complete ordered field. Any two such fields are isomorphic, but there are still different models. Two of the most common are Dedekind Cuts and equivalence classes of Cauchy sequences.
And of course no discussion of axioms is complete without mentioning Euclid's postulates. For a long time people wonder if his fifth postulate, known as the parallel postulate, could be deduced from the first four. This was finally settled a few centuries ago by producing alternative models of the first four axioms: flat, spherical, and hyperbolic geometry. The parallel postulate holds in flat but fails in the others, demonstrating that it is independent of the other axioms.
Similarly for rings. The definition of a ring lays out the axioms, and each ring is a model of those axioms. The statements "multiplication is commutative" and "each nonzero element has a multiplicative inverse" are independent of the ring axioms. If we choose to make these new axioms, then the models we are left with are called fields.
In a real analysis class, you probably laid out the axioms for the real numbers, namely that they form a complete ordered field. Any two such fields are isomorphic, but there are still different models. Two of the most common are Dedekind Cuts and equivalence classes of Cauchy sequences.
And of course no discussion of axioms is complete without mentioning Euclid's postulates. For a long time people wonder if his fifth postulate, known as the parallel postulate, could be deduced from the first four. This was finally settled a few centuries ago by producing alternative models of the first four axioms: flat, spherical, and hyperbolic geometry. The parallel postulate holds in flat but fails in the others, demonstrating that it is independent of the other axioms.
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Solo Challenges / Re: Embargo x100
« on: February 25, 2017, 01:46:01 pm »
Wouldn't there be zero Curses in a one player game?
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General Discussion / Re: Maths thread.
« on: February 25, 2017, 11:52:08 am »I've just been reading a bunch of Doron Zeilberger's opinions and other writings on ultrafinitism and I legitimately can't tell whether I just find them hilariously snarky or if they're actually starting to sway me.
I was going to read his opinions, but man, his list goes on forever.
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Dominion Articles / Re: Challenge: write a Moat article
« on: February 11, 2017, 07:46:04 pm »Against junking attacks Moat often isn't a defence in itself. Sooner or later your opponent is going play their attack when you don't have your Moat in hand.
You might not be able to block all of the curses, but so long as your opponents put more curses into their decks than yours you've come out ahead.
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Dominion Articles / Re: Challenge: write a Moat article
« on: February 10, 2017, 08:24:32 am »
Filling your deck with cantrips will dilute your Moats making it harder to defend with them.
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Dominion Articles / Re: Sentry
« on: January 31, 2017, 10:59:12 am »
I made a pretty nice little mega-turn engine yesterday out of Sentry, Herald, Apothecary, and Bridge. It was so satisfying.
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Dominion General Discussion / Re: trashing two hovels on turn 2
« on: January 28, 2017, 01:22:35 pm »The one time when buying an Estate to trash Hovel t2 is actually a crazy strong play.
Eh, that still depends a lot on what else is available. Regardless, this is such an absurd corner case that I'd buy the Estate too just for the story.
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Dominion General Discussion / Re: trashing two hovels on turn 2
« on: January 28, 2017, 01:16:15 pm »
Phew, I'm glad the abbreviations are case sensitive
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General Discussion / Re: Maths thread.
« on: January 28, 2017, 11:28:01 am »What is the best thing to read if I want to learn set theory? If there is free access to it online, that's a bonus.
I started here ... I don't think I can judge yet whether that's a good script. a few chapters in I took a peek forward and saw that they define the natural numbers as
0 = ∅
1 = {0} = {∅}
2 = {0, 1} = {∅, {∅}}
3 = {0, 1, 2} = {∅, {∅}, {∅, {∅}}}
etc.
Is this how proper set theory works – constructing everything through empty sets and sets of empty sets?
Well, there are really two things going on here. First we write down the rules for what arithmetic ought to obey, which we call axioms, e.g. the Peano axioms.
Next, we go about constructing a model of these axioms, that is a set of objects and operations that satisfy the axioms. There are usually lots and lots of different models, and they might not all behave the same way. One model is this one involving nested sets and empty sets.
For another example, in real analysis you usually write down a few axioms for the real numbers, namely those of being a complete ordered field. One model is given by Dedekind cuts, another is given by equivalence classes of Cauchy sequences. Neither model can be said to be the "true" real numbers.
When people say that AC is independent of ZF, they mean that there is both a model of ZF which satisfies AC and a model of ZF where AC fails.
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Variants and Fan Cards / Re: New Mechanic - Delay
« on: January 28, 2017, 10:38:05 am »Just for clarity: Are cards that go through the Delay mat ever gained, and if so, when?
It seems that all are gained, except via Byway.
Also, is the Time mat the same as the Delay mat?
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Dominion General Discussion / Re: Lurker
« on: January 17, 2017, 01:24:44 am »What has not been mentioned yet is end-game pile control Lurkers can drain a pile when you want to trigger a 3 pile but lack the coins or buys to empty it in the buy phase.
It can help empty out the Scout pile without contaminating your deck
If only they exsited in the same universe
Donald had probably avoided making Lurker previously because the interaction with Scout was too OP.
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General Discussion / Re: Maths thread.
« on: January 16, 2017, 06:34:28 pm »
sudgy, I haven't read your request too closely, but here is an example to watch out for:
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 ...
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 ...
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 ...
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 ...
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 ...
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 ...
and so forth. Each natural number eventually turns red except for 1.
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 ...
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 ...
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 ...
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 ...
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 ...
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 ...
and so forth. Each natural number eventually turns red except for 1.
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General Discussion / Re: Maths thread.
« on: January 16, 2017, 01:41:13 pm »However, if you have a sequence of subsets of of N that covers N, and you prove a statement for each such subset, it holds for all natural numbers.
Even if it requires an infinite number of subsets?
Yes. Let n be an arbitrary natural number. Since our collection subsets covers N, we know that n belongs to one of our subsets. But we've proved that every element of the subset has our property. So n has the property.
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General Discussion / Re: Maths thread.
« on: January 16, 2017, 01:31:19 pm »
This might be a more useful response. It seems that what you really want to do is put what it called a "measure" on the set of natural numbers. A measure is a way of assigning a size to each subset of numbers in such a way that the size of a disjoint union is just the sum of the sizes of of the individual subsets. In your example, you would want the measure of the full set of natural numbers to equal one, which is sometimes called a probability measure.
In this case, if you have a disjoint union of subsets whose measures sum up to one, then your subsets will indeed include all but a subset of measure zero. Now, it depends on the measure you use, but you can indeed have a nonempty set with measure zero. However, the phrase "almost all" is frequently used by mathematicians in a very precise sense to mean "all but a set of measure zero".
tl;dr
rephrasing your question in measure theoretic terms, almost all natural numbers will have your property.
In this case, if you have a disjoint union of subsets whose measures sum up to one, then your subsets will indeed include all but a subset of measure zero. Now, it depends on the measure you use, but you can indeed have a nonempty set with measure zero. However, the phrase "almost all" is frequently used by mathematicians in a very precise sense to mean "all but a set of measure zero".
tl;dr
rephrasing your question in measure theoretic terms, almost all natural numbers will have your property.
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General Discussion / Re: Maths thread.
« on: January 16, 2017, 01:23:37 pm »Okay, for a really simple example, say you find that something gives half the numbers the property, then half of the remaining numbers that property, then half of those remaining numbers, and so on. In this case it seems pretty obvious that all numbers have the property, so I'm wondering if that works for any infinite sum that sums to one.
This would still not imply that all natural numbers have the property. Mostly because there isn't really a useful definition of "half" of a countably infinite set.
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Let's Discuss ... / Re: Let's discuss Alchemy cards: Transmute
« on: January 10, 2017, 02:12:24 am »Transmuted Transmute
Action - $4
Trash a card from your hand. If it is:
An Action, gain a Victory card costing up to $5
A Victory card, gain a Treasure costing up to $6
A Treasure, gain an Action costing up to $4.
I don't like the last line since there might not even be any Action cards costing up to $4 in the kingdom. This is especially true if Alchemy is the only expansion you're using.
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Dominion Online at Shuffle iT / Re: Buttons in the log was a serious mistake
« on: January 10, 2017, 01:55:45 am »I categorize a serious mistake as one that actually causes damages like say, leaks passwords in plain text, or deletes bought cards, or charges a user's account for another user's purchases.
Interface annoyances are therefore categorically not serious mistakes.
I agree that those are more serious. Hopefully they're missing from the to-do list because they are not existing problems. My point was that the problems I mentioned are more serious than any of the ones listed.
weird placement of buttons is annoying, but is it really more urgent than game freezing bugs or the lack of matching for 3+ player games?
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Dominion Online at Shuffle iT / Re: Lord Rattington
« on: January 10, 2017, 01:12:36 am »If you discard your 6th Vault in a turn, he'll slow play you while he tries to figure out how to discard when he has no cards in his hand, and you are forced to resign.
I'm not apologizing for the slow play when you're the one playing that ridiculous vault engine against me.
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Dominion Online at Shuffle iT / Re: The Dominion online 2017 thread
« on: January 08, 2017, 06:13:40 pm »How do I change the number of players for quick/good match?
You don't, and it isn't planned either (at least not short term).
Our plan is to improve the usefulness of the 'tables' to make it easier to find 3+ player matches.
Why the distinction? Is matching with 3+ players significantly more difficult?
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Dominion Online at Shuffle iT / Re: The Dominion online 2017 thread
« on: January 08, 2017, 11:42:29 am »
How do I change the number of players for quick/good match?
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Feedback / Re: Username change?
« on: January 07, 2017, 09:56:16 pm »I strive for a respect ratio of 1.69 :-)
I'm at 1.64; if you all give me 153 respect for this post then I'll be there.
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Dominion Online at Shuffle iT / Re: here's a link to https://www.dominion.games/
« on: January 06, 2017, 11:17:14 pm »hello, yes, i am an SEO guru
http://dominion.games dominion online 1080p steam unboxing mukbang review remastered ASMR tutorial walkthrough google remix frog fractions 2 xbox one VR barbara striesand
I didn't know there was a sequel until now
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Feedback / Re: Username change?
« on: January 06, 2017, 04:51:00 pm »
Unless the name is taken, you could always just make a new account.