# X

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Sorry about this..I'm not so numerically eloquent as verbally. But doesn't 0 = 0 when Multiplied or Divided regardless? Rendering the entire proof slightly unneeded? Or does f(x) f(s) and f(t) intend another meaning besides multiplication..
Sorta lost me on that..

f(x) denotes the function f mapped onto the variable x. For example, f(x) could mean sin(2x), so that when you graph f(x), for every value x, the y-component would be the sin of x. So f(0)=0, etc. You many be more familiar with y=sin(x), but that's not used as much as function notation.

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Yeah, I'm afraid that if you haven't seen function notation yet, you may not be able to make much of Khoth's bogus proof. Apart from basically irrelevant issues, it purports to be a proof that the only continuous curve that you could draw in a plane, passing through (0,0), is the line along the x-axis. Which is, of course, nonsense, since there are all kinds of possible random squiggles, not to mention regular shapes and curves, that pass through the origin from any direction, and then veer off anywhere else you want. But the trick is to spot the fallacy in the proof.

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threads that stay on topic are no fun anyway

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Originally Posted By: Like Mana from the Pacific
Comparisons are by definition not equations. Equations require that equals sign. But your point is otherwise correct.

—Alorael, who wouldn't be too sure about horses. He's fairly sure that's how dogs' brains work.

His point is that the notion of comparing less than/ greater than (in the way that seems most obvious for real numbers) comes from mathematics, rather than from our external experience of real numbers. I don't find this to be accurate (though I'll readily admit that the reals look nice under that ordering--but I could well just be biased about it!).

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Yeah, pretty sure that inductive proofs don't work on functions whose domain is the real numbers. Nearly every inductive proof I've seen uses the natural numbers, and a few have the integers as the domain. This gets me thinking -- do inductive proofs work for the rational numbers? It should work, as they have the same cardinality as the natural numbers. Intuitively, I don't see it working if you go through them sequentially, but maybe you have to go through them using Cantor diagonalization. Has anyone seen/done anyth9ing like this before?

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I think it should be possible to do induction on the rationals, but I've never actually done it or seen it done, so really I've no idea...

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Induction works on well-ordered sets. Being well-ordered is completely different from being countable, although the natural numbers happen to be both. But uncountable well-ordered sets are much studied. The rational numbers in the usual ordering are not well-ordered.

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Originally Posted By: Micawber
Induction works on well-ordered sets. Being well-ordered is completely different from being countable, although the natural numbers happen to be both. But uncountable well-ordered sets are much studied. The rational numbers in the usual ordering are not well-ordered.
Oh, that makes sense. In fact, I should have known this -- I remember a calculus prof telling us that strong induction and the well-ordered principle were equivalent. Browsing on Wikipedia lead me to this, which states what you've already mentioned.

Originally Posted By: Khoth
I think Gödel's theorems limit thinking ability in approximately the same way that relativity limits running speed.
I like this analogy, and am totally going to steal it.

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Originally Posted By: Micawber
Induction works on well-ordered sets. Being well-ordered is completely different from being countable, although the natural numbers happen to be both. But uncountable well-ordered sets are much studied. The rational numbers in the usual ordering are not well-ordered.
But if I'm not mistaken, it's possible to well-order the rationals, isn't it? Perhaps using a lexicographical ordering? (ie 1, 1/2, 1/3, ..., 2, 2/3, 2/5, ..., 3, 3/5, 3/7, 3/8, ..., etc)

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Originally Posted By: Celtic Minstrel
Originally Posted By: Micawber
Induction works on well-ordered sets. Being well-ordered is completely different from being countable, although the natural numbers happen to be both. But uncountable well-ordered sets are much studied. The rational numbers in the usual ordering are not well-ordered.
But if I'm not mistaken, it's possible to well-order the rationals, isn't it? Perhaps using a lexicographical ordering? (ie 1, 1/2, 1/3, ..., 2, 2/3, 2/5, ..., 3, 3/5, 3/7, 3/8, ..., etc)

Every set can be well-ordered if you're down with the axiom of choice, so yes--but the "lexicographical" ordering doesn't really work. For a well-ordering, every nonempty subset of the set you want to well-order has to have a least element. What's the smallest negative rational?

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You don't need the axiom of choice to well-order the rationals. Any ordering you come up with to show they're countable will do (eg the one you get by writing them all down in an infinite plane and spiralling out from 0).

I think the lexicographical ordering works if you extend it to negative numbers (just shove in -x after every positive x).

Of course, if you want to do induction, you have to actually use the well-ordering rather than the normal ordering.

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Originally Posted By: everyday847
Every set can be well-ordered if you're down with the axiom of choice, so yes--but the "lexicographical" ordering doesn't really work. For a well-ordering, every nonempty subset of the set you want to well-order has to have a least element. What's the smallest negative rational?
Oh right, I forgot about negatives. Well, as Khoth says, that's easily solved by putting -x right after x for all x.

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Originally Posted By: Khoth
You don't need the axiom of choice to well-order the rationals. Any ordering you come up with to show they're countable will do (eg the one you get by writing them all down in an infinite plane and spiralling out from 0).

Right: I'm saying that the axiom of choice is equivalent to something that's called something like the well-ordering principle that states that you can well-order any (nonempty?) set.

Originally Posted By: Khoth

I think the lexicographical ordering works if you extend it to negative numbers (just shove in -x after every positive x).

No. What, by the lexicographical ordering, is the smallest negative rational number? If you can't answer that question, it's not a well-ordering.

Ohh, are you proposing that you do something like

1, 1/2, 1/3, 1/4, ..., -1, -1/2, ...

Yeah, I guess that works.

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Originally Posted By: everyday847
You don't need the axiom of choice to well-order the rationals. Any ordering you come up with to show they're countable will do (eg the one you get by writing them all down in an infinite plane and spiralling out from 0).

Right: I'm saying that the axiom of choice is equivalent to something that's called something like the well-ordering principle that states that you can well-order any (nonempty?) set.

Originally Posted By: Khoth

Ohh, are you proposing that you do something like

1, 1/2, 1/3, 1/4, ..., -1, -1/2, ...

Yeah, I guess that works.
Actually, I thought we were proposing something more like this:

1, -1, 1/2, -1/2, 1/3, -1/3, 1/4, -1/4, ... etc

But your version works just as well.

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Originally Posted By: everyday847

Right: I'm saying that the axiom of choice is equivalent to something that's called something like the well-ordering principle that states that you can well-order any (nonempty?) set.

Yeah, but I'm one of those annoying people who, despite happily using the axiom of choice when it's needed, avoids using it when not.

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And here I was hoping to see a heated debate about whether X should make an appearance in Avadon or not... :[ I feel woefully insignificant in terms of logic or math or whatever it is going on in the x number of posts above me.

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Originally Posted By: Velzan
"Anvil spell"

I stole it from X.

Oh yes i did.

Well thats all right, X did steal it from Tom and Jerry, after all.

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A note on threeve: Any point on the line defined by f(x)=ix on the complex plane is equidistant from 3 and 5, and all of them save for x=0 is different from 4.

For some reason I was so sure of that before I typed it out, now I think it's catastrophically wrong. xD

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You Know, When i started a thread about X, I didnt Feel like it would turn into a Math thread. I thought it would be more Geological. Or Physological. Somehting involving logical.

Eh. Cant get everything.

Now, how many of you searc hteh internet for math sites before posting replies? Be Honest.

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Also not to mention the vast space between some of those last few posts! Spam is bad. I should know...check out pages 21-23 of my post count! Man was I stupid back then.

Post # 463

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Originally Posted By: Dantius
Originally Posted By: CRISIS on INFINITE SLARTIES
frpwns

I was unaware that the Nine-Headed Cave Cow dished out pwnage. Although that was a hilarious tyop...

And no. If Jeff wanted crossover Avernum/Geneforge/that one game with the Scottish people in it/Avadon games, he'd make them. And it would be seen as the ravings of a madman, unless thee Ecksian Skull said it (To the best of my knowledge, that is the only other time there is a direct reference to the other series in-game)

While i dont think there have been any direct crossovers, there was a quasi-crossover in that one game with the scottish people in it (otherwise known as nethergate, ressurection) there was a very powerful fairy/elf man who lived in the fairy bazaar who was called Solberg. Nethergate also had the GIFTS in it.

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It's not so much a crossover as a shoutout: a dude by the name of Andrew Solberg is a friend of Jeff's.

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The GIFTS in Nethergate (and in Exile/Avernum, obviously) is the one bona fide crossover from any of Jeff's games. IIRC, there are one or two other shoutouts in Nethergate that were also in Exile -- Starcap was one, I think.

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Originally Posted By: Lilith
It's not so much a crossover as a shoutout: a dude by the name of Andrew Solberg is a friend of Jeff's.

Wasn't Rone a character of one of Jeff's DnD buddies? I wonder how many characters there are like that...

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Why, there they are both, baked in that pie;

Whereof their mother daintily hath fed,

Eating the flesh that she herself hath bred.

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Originally Posted By: Velzan
You Know, When i started a thread about X, I didnt Feel like it would turn into a Math thread. I thought it would be more Geological. Or Physological. Somehting involving logical.

Logic is, among other things, a mathematical discipline. It's not so formally a part of sciences.

While i dont think there have been any direct crossovers, there was a quasi-crossover in that one game with the scottish people in it (otherwise known as nethergate, ressurection)

The game takes place in England, not Scotland.

—Alorael, who supposes it's something of a hair-splitting activity. Nethergate takes place in Brigantes territory, which was in what would become England. Scotland didn't exist yet, so "Scottish" doesn't really work.

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Originally Posted By: Focus on Particulars
While i dont think there have been any direct crossovers, there was a quasi-crossover in that one game with the scottish people in it (otherwise known as nethergate, ressurection)

The game takes place in England, not Scotland.

—Alorael, who supposes it's something of a hair-splitting activity. Nethergate takes place in Brigantes territory, which was in what would become England. Scotland didn't exist yet, so "Scottish" doesn't really work.

I suppose that just using the descriptor Celtic would work best, since the concept of "England" or even the concept of nation-states didn't emerge until much later.

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The concept of England predates the nation-state by quite a few centuries. Celts would include tribes from Spain to Turkey. British might be the most accurate descriptor. While Great Britain didn't become an entity for centuries either, Britannia was a recognized term.

—Alorael, who still wants a Nethergate sequel, spiritual or otherwise. (And you should have voted for Alcritas.)

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Someone could go start a poll on wether we should have voted for him/her/it or not. I know I would have voted for maybe. And thats on a poll with only yes or no answers.

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Originally Posted By: Aldark
—Alorael, who still wants a Nethergate sequel, spiritual or otherwise. (And you should have voted for Alcritas.)

I agree with this on a discursive level. I think there was even the discussion of it being an Aztec versus Spanish type of sequel.

I think that would probably be cool, if Jeff did it. I'd buy that game.

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Originally Posted By: Goldenking
I agree with this on a discursive level. I think there was even the discussion of it being an Aztec versus Spanish type of sequel.

I think that would probably be cool, if Jeff did it. I'd buy that game.

So basically a game where you slaughter indigenous people with superior technology and viruses. I agree with you; I'd totally buy that game.

(Incidentally, Avatar would also have been my new favorite movie if it had adopted that stance to Human/Navi relations, i.e. we kill them all accidentally with the common cold/Smallpox/the Flu)

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So killing them accidentally with the cold is more fun than with explosives?! It seems like you dont know the difference between a good movie and a great movie.

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This is a Nethergate-style game. Conquistadores kill with guns, germs, and steel; Aztecs retaliate with obsidian and blood-fueled magic.

—Alorael, who would play that game. Especially if he could play as the diseases. Pandemic the RPG?

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the word "macahuitl" is surprisingly difficult to turn into a bad pun

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That's why the Aztecs lost. Their language constrained them from making important intellectual advances like bad puns.

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Originally Posted By: Dantius
So basically a game where you slaughter indigenous people with superior technology and viruses. I agree with you; I'd totally buy that game.

*bottomburp*

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Originally Posted By: Student of Trinity
That's why the Aztecs lost. Their language constrained them from making important intellectual advances like bad puns.
For the record, that is totally not true of Nahuatl.

Also, remember how good slings were in Nethergate? Just imagine the power of the atlatl.

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Language is not necessarily a limit on conquest. The Romans conquered the Mediterranean world, yet couldn't say "the," (at least not in Latin) a fact that still astounds me.

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The atlatl is just a better way to lob javelins. A big advantage of slings is not having to lug around all that ammo or worry about it running out.

—Alorael, who puts puns in the same category as bagpipes in their utility to the war effort.

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So both puns and bagpipes destroy morale? And here I thought it was only puns that did that.

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Ach, the tunes! The tunes of glory!

A 'tune of glory' is a bagpipe tune that has been played in battle. There are not so many of these, since battle makes people conservative. Once a few tunes have established a reputation as good fighting tunes, few people will risk trying something new. So it's a considerably higher bar than getting radio play. Worth a bit more than a Grammy, I'd say. 'And the award for song you would most want to get shot to, goes to ... "The Black Bear".'

I suppose it doesn't help that nobody plays bagpipes in battle anymore. Ach, weel. Just as well, on the whole, I suppose.

Also, musical tastes were different in the days when people did prefer to walk into fast lead to music. The tunes most drenched in blood are gentle, lilting ditties like the Grenadiers' March, Lilliburlero, and Garryowen. Or The Black Bear. Not one of them would be likely to get you off your barstool these days, let alone make you march ahead to Victory or Death.

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Originally Posted By: Triumph
Language is not necessarily a limit on conquest. The Romans conquered the Mediterranean world, yet couldn't say "the," (at least not in Latin) a fact that still astounds me.
This is not entirely accurate. The Romans could say "the" in Latin -- they just didn't use a word to communicate the ideas we communicate with the word "the."

The English articles "a" and "the" are examples of determiners, a broad group of words and particles that tell you where its attached noun sits within a particular semantic domain that deals with existence, quantification, and so on. However, in splitting that domain, they crisscross it in arbitrary and lurching ways. Determiners are one of the trickiest bits of grammar to master, which is why you will often hear otherwise proficient English learners who grew up speaking a language that does not use articles the way English does (like Russian or, if it were still spoken as a primary language, Latin) leave out articles altogether. Different languages put different delimitations around the particular meanings of articles and other determiners. But which concepts are and are not conveniently packaged in a given language is quite arbitrary, and there are always times when a non-packaged element is important and has to be described in some other way -- that's true in any language.

In other words, Latin can certainly express the various characteristics specified by the definite article in English, despite the fact that it does not use a parallel article. Since we translate words based on parallel meaning and not based on precisely parallel grammatical structure, we can indeed translate "the" to and from Latin, and it is not quite true to say that the Romans couldn't say "the."

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Okay restating a question:Do you guys look this up or do you just know it already from school?

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Originally Posted By: Shaper Cirikci
Okay restating a question:Do you guys look this up or do you just know it already from school?

Most of them know it from school. Slarty used to be a linguistics major and there are plenty of physicists and computer scientists here.

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Originally Posted By: Shaper Cirikci
Okay restating a question:Do you guys look this up or do you just know it already from school?

Yes, although replace "school" with "school, work, and the vagaries of life" for accuracy. But a lot of knowledge now isn't actually knowing things, it's knowing what things you almost know that you can look up. Or, sometimes, which search queries to try.

—Alorael, who has now encountered the term "metacortex" for computer- and internet-assisted cognition. He thinks it's both cute and useful.

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Originally Posted By: Randomizer
Originally Posted By: Shaper Cirikci
Okay restating a question:Do you guys look this up or do you just know it already from school?

Wikipedia.
Originally Posted By: Not the #\$^@* Mother-Being!
Originally Posted By: Shaper Cirikci
Okay restating a question:Do you guys look this up or do you just know it already from school?

Wikipedia.

I believe this is the answer you are looking for, Cirikci.

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Wikipedia is not the answer to everything, just most things.

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I don't normally post things here if I only know them because I read them on Wikipedia. I'm old enough that I know some things the old fashioned way.

You know. Somewhere there are experts on certain topics. You worry that they might not totally approve of that Wikipedia article, for instance. Well, on a few topics, I'm one of those experts. It's not that I'm so special. Put in your time and you could be one, too, if you're not already.

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Funny story. I'm not a regular Wikipedian, but I saw one article with a single-letter typo that led to a rather substantial error that I had to correct. My edit was almost immediately reversed, as I had not cited a source. The problem quickly became that most sources that I could cite were not available to the public, while a great deal of Google-indexed material was either Wikipedia mirrors or clearly Wikipedia-based. Wikipedia created a consensus based on a typo that was difficult to overcome.

—Alorael, who doesn't worry much about stumbling across Upper Peninsular Wars on Wikipedia. It's the little errors that can trip you up.

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Remember how there used to be PGP keys? Pretty good privacy. We should revive that acronym. PGP. Pretty good pedia. Not unbreakable, but good enough for industrial use.

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