Turning 'limitless' into 'infinite'

[squeekenator]

If this has been brought up before, then whoops, never mind me.

I was brushing up on my understanding of Pun-Pun when I noticed that many of his attributes are limitless or near-infinite, rather than actually infinity, because reaching infinity requires some sort of loop, rather than just repeatedly buffing yourself. In order for Pun-Pun (or any other similar character) to reach truly infinite stats he would need to perform infinite actions, which he can't do, even with a limitless action loop such as the synchronicity shenanigans he uses. Now, correct me if I'm wrong, but can we work around that problem with Dominate Person?

1. Pun-Pun dominates himself and his familiar.
2. He orders himself and his familiar to perform the synchronicity loop and raise everything to infinity in their next turn.
3. Because the dominate effect decides their actions rather than the player, he is forced to continue with the action loop indefinitely, increasing every relevant attribute a truly infinite number of times in a single round.
4. Badass.

Does it work?

[kharmakazy]

I'm not sure I follow your logic... why would ordering yourself to do things... be any different than deciding to do them yourself...

and that is the only portion I am addressing at this time...

[Big Fau]

There is really nothing stopping Pun-Pun from using Manipulate Form to just set his ability scores to infinite anyway, since he can literally create a special ability that says:


"Infinite [Ex]: Your ability scores are infinite in value."

[squeekenator]

Essentially, anything you have to do manually can only be done a limitless number of times, not infinite. As long as you're in control of your character, no matter how many times you say 'I do x' you'll never do it an infinite number of times - there's nothing in the rules that allows you to say 'I do x an infinite number of times', you take your actions one at a time. The only way to achieve that is to have an automatic loop that functions without any input from the player, which, if I'm not mistaken, you can achieve by dominating yourself, as that gives the rules total control over your actions.

EDIT:

There is really nothing stopping Pun-Pun from using Manipulate Form to just set his ability scores to infinite anyway, since he can literally create a special ability that says SNIP

True, but this works for builds other than Pun-Pun, plus there's the interpretation that Manipulate Form can only grant abilities that actually exist (personally I support that interpretation but I'm not particularly interested in arguing about that).

[kharmakazy]

Those words mean the same thing anyhow. Infinite goes on forever. So do things without limits.

[kardar233]

Mmmmm, no.

Say you're at an all-you-can-eat buffet. You could take as much food as you want. The amount of food you can take is limitless.

No matter how much food you take, you will never have an infinite amount of food.

See the difference?

Yeah, I like infinities. See the guy walking around with a sig of mine with Cantorian Set theory. I'm crazy already, what can it do?

[kharmakazy]

But since we are talking about abstract numbers, a limitless amount of numbers is an amount of numbers that will never end. Infinite just mean not finite... Finite means limited, not limited means without limits, limitless.

You can take most synonyms and change the context to make them sound like different things.

The amount of food you can take, is in fact, very LIMITED. It is limited to the amount you can eat without dieing.

[OracleofSilence]

yeah, limitless is very much a different thing then infinite.

think about it like this: numbers are limitless. there is always a bigger (or smaller) number. there is a number bigger then Grahams number, and a number smaller then its negative equivalent. and there is a number bigger, and smaller then those numbers, ad infinitum. never the less, without breaking math (divide by zero) you can never get infinity, since those numbers, no matter how ludicrously high they are, are still numbers, and still defined. Pun Pun is hit by this. His stats are just some arbitrarily high number.

Infinity however, is truly infinite not only is it incomprehensible, but it cannot be definied by numbers. only very unusual combo''s, like that used by Lord of Procrastination's Omniscifier can give actually infinite stats. So Pun Pun can theoretically give himself infinite stats, but the thing is, he can only do this to a few things, largely those that are affected by Masochism and Synchronicity Field (which isn't everything)

[Salanmander]

But since we are talking about abstract numbers, a limitless amount of numbers is an amount of numbers that will never end. Infinite just mean not finite... Finite means limited, not limited means without limits, limitless.

You can take most synonyms and change the context to make them sound like different things.

The amount of food you can take, is in fact, very LIMITED. It is limited to the amount you can eat without dieing.

So, the loops that punpun exploits all raise stats by a finite amount per iteration. No matter how many times he uses the loop, he will still have a finite ability score.

Finite does not mean limited. If I tell you "pick an integer", there are no limits on the integer you pick. However, it will still be finite. This is basically the buffet example, but more abstract to get around your objection about realism on the buffet.

So yes, limitless and infinite are very different, even when talking about numbers.

As for turning limitless into infinite via the dominate trick, it doesn't work. Even if you can increase your stats as a free action, the DMG explicitly calls out that there is a limit on the number of free actions you can take in a round. This gets handwaved for the peasant railgun and pony express and things like that, which is kinda okay because a DM could, within RAW, allow any number of free actions on a turn. However, a DM /can't/ actually allow an /infinite/ number of free actions on a turn and stay within RAW.

There are few truly infinite quantities possible in D&D, but the 1d2 crusader is one of them. Isn't there a thing that increases your stats based on the damage you take? We could team that person up with a 1d2 crusader (or get similar abilities on pun-pun) to get truly infinite stats.

edit:
OracleofSilence, I agree with most of what you said (got ninja'd by a lot of it, actually), but i have one minor nitpick.

never the less, without breaking math (divide by zero) you can never get infinity, since those numbers, no matter how ludicrously high they are, are still numbers, and still defined.

You can very much get infinity without breaking math, and infinity is very much defined. You just can't pick a number and have it be infinite.

For example "The number of integers" is perfectly well defined, and is infinite. Infinities are, in fact, quite present and useful in a lot of math. It's just that arithmetic with them is pretty much meaningless, and you can't pick the value "infinity".

[kharmakazy]

Sure... ok.



Limitless
Definition: never-ending, infinite

That's funny.. what about


Infinite
Definition: limitless, without end


That's so weird. I wonder why the dictionary would tell such a bald-faced lie? Naturally I should listen to what people on a forum dedicated to a web-comic think about what words mean instead of the dictionary.

[OracleofSilence]

that would be this (http://community.wizards.com/go/thread/view/75882/19868070/LoPs_Omniscificer)

[kharmakazy]

Finite does not mean limited.
.

Weird...



Definition of FINITE
a : having definite or definable limits <a finite number of possibilities>


We should start writing letters.

[Salanmander]

That's so weird. I wonder why the dictionary would tell such a bald-faced lie? Naturally I should listen to what people on a forum dedicated to a web-comic think about what words mean instead of the dictionary.

Because the dictionary is talking about colloquial English, whereas we are talking about mathematical jargon. The OP specifically /made/ it about mathematical jargon, by differentiating between "an arbitrarily large value" and "infinity".

[kharmakazy]

It's not jargon. Those are simple and well understood words with clear definitions that some people choose to ignore.

And the dictionary is not "colloquial English" It is standard English.

[OracleofSilence]

That's so weird. I wonder why the dictionary would tell such a bald-faced lie? Naturally I should listen to what people on a forum dedicated to a web-comic think about what words mean instead of the dictionary

as soon as the dictionary can bring up sources from PhD mathematicians i will listen. cause, don't yah see, that is more or less where i got my definition from. (needless to say, it is a lot simpler, and some of it is just an easy description)

this is more or less taken from a discertation given by Reimann in the 19th century, where he said the following

In the extension of space construction to the infinitely great, we must distinguish between infinite extent and unboundedness; the former belongs to measure relations; the later to extent relations.

this essentially means that infinity is a quantity, while unboundedness/limitlessness is a quality

something that is limitless simply can be grown forever, without end. something that is infinite is already infinitely bigger then that.

does that mkae the difference more clear? and yes, this is the mathematical definition

[squeekenator]

As for turning limitless into infinite via the dominate trick, it doesn't work. Even if you can increase your stats as a free action, the DMG explicitly calls out that there is a limit on the number of free actions you can take in a round. This gets handwaved for the peasant railgun and pony express and things like that, which is kinda okay because a DM could, within RAW, allow any number of free actions on a turn. However, a DM /can't/ actually allow an /infinite/ number of free actions on a turn and stay within RAW.

That's fine. The synchronicity trick doesn't involve free actions. It simply grants you as many standard actions as you want.

[ericgrau]

Well good thing the OP clearly defined each one so that there was no confusion... and then everyone ignores him anyway :smallbiggrin:.

I think if pun-pun can step out of time or otherwise get infinite actions then the issue does indeed become moot, as he's no longer limited by the number of actions he can take.

[Dragonsoul]

You do it and you get-

DIVIDE BY PINEAPPLE ERROR-PLEASE REBOOT UNIVERSE FROM PREVIOUS SAVE

[OracleofSilence]

nice to see a classy quote for once. but i could have sworn that that was an ~ INSERT CHEESE ERROR~~REDO FROM START~

[Urpriest]

It's not jargon. Those are simple and well understood words with clear definitions that some people choose to ignore.

And the dictionary is not "colloquial English" It is standard English.

Which is irrelevant because infinity is a mathematical term and you and I are educated people. You want to know what infinity is, you look in an Analysis textbook.

[Redshirt Army]

I think if pun-pun can step out of time or otherwise get infinite actions then the issue does indeed become moot, as he's no longer limited by the number of actions he can take.

Actually, if Pun Pun was to take an infinite amount of actions in order to get infinitely high stats, wouldn't his turn just never end? :smallconfused:

[kharmakazy]

The only distinction between "countless, innumerable"and "endless, unlimited" or rigidly bounded and loosely bounded infinities in this context is one that you decide to make, and given the desired outcome of declaring this variable there is no rational for choosing the former over the latter...

And the actual definitions of the words still apply.

[OracleofSilence]

that is, until you run up against the previously mentioned build, the Omniscifier, and see what actually infinite stats mean. basically, it doesn't matter until you met the guy that can make you his friend without fail, and then scare you so bad you literally curl up and die, all at fourth level. finite but unbounded cannot beat infinite.

[Dragonsoul]

nice to see a classy quote for once. but i could have sworn that that was an ~ INSERT CHEESE ERROR~~REDO FROM START~

That's for infinite Caster loops.

(In case you were wondering Chain Gating results in PURPLE MONKEY ERROR-INSERT DISC TO CONTINUE)

And remember-

Jesus Saves- Then Quickloads

[MeeposFire]

that is, until you run up against the previously mentioned build, the Omniscifier, and see what actually infinite stats mean. basically, it doesn't matter until you met the guy that can make you his friend without fail, and then scare you so bad you literally curl up and die, all at fourth level. finite but unbounded cannot beat infinite.

Too bad Pun-pun still wins since he gets started at level 1 now.

[OracleofSilence]

Too bad Pun-pun still wins since he gets started at level 1 now.

read the thread (http://web.archive.org/web/20061112212706/boards1.wizards.com/showthread.php?t=546612)

[erikun]

1. Pun-Pun dominates himself and his familiar.
2. He orders himself and his familiar to perform the synchronicity loop and raise everything to infinity in their next turn.
3. Because the dominate effect decides their actions rather than the player, he is forced to continue with the action loop indefinitely, increasing every relevant attribute a truly infinite number of times in a single round.
4. Badass.

Does it work?
I see two problems with this.

First, it order to get infinite1 stats, he would need to spend infinite1 actions doing so. Unless you have some way to grant infinite actions, Pun-Pun will literally be doing this forever.

Second, assuming he dominates himself for an infinite amount of time, he will never be able to break out of it. Infinity is not just an exceptionally high number.

1 I am speaking of actual infinite stats, meaning either continuously increasing or somehow larger than reachable numbers. Whichever definition you use will depend on what might work, but the case presented here still has the same problems.

[MeeposFire]

read the thread (http://web.archive.org/web/20061112212706/boards1.wizards.com/showthread.php?t=546612)

I did I say to you read the thread to the end. Last post at the very bottom and I quote

"Khan the Destroyer conceeded that Omni could
theoretically know how to beat the level 5 pun-pun build.
It is in the OP of the pun-pun thread.
It doesn't beat the level 1 build, and couldn't beat say a level 3 build."

[OracleofSilence]

that is true, but still, the point remains. for these purposes, infinite is better then unbounded. still, i concede the point.

[kharmakazy]

Math aside, I'm not sure how being forced to do what you tell yourself to do forever is in any way a hindrance. You already do what you tell yourself to do.

[squeekenator]

Unless you have some way to grant infinite actions

He does. Font of power + affinity field + synchronicity. Basically, his familiar can grant both itself and Pun-Pun a standard action as a standard action, thus giving Pun-Pun a potentially infinite number of standard actions per round. The entire process only takes 6 seconds.

[OracleofSilence]

totally true, but the fact remains, this is still an indefinately high number not an INFINITY of standard actions, so he can give his arbitrarily high stats an arbitraritly high boost resulting in another arbitrarily high stat. its like this

Stat x = o

for every standard action stat x is boosted by m

p standard actions are taken, but p is some number defined where p(n) = p1+p2+...p(n-1) by some formula

p however, is still by definition finite (and cannot be infinite because that just doesn't work)

so he now has n=1 / n = n (o+(m(p(n)) (i know i did the formula wrong but still) added to every stat as defined by m and p

here is the trick. m and p are still finite, so his stat can never be infinite

[Frozen_Feet]

And the dictionary is not "colloquial English" It is standard English.

Nevertheless, several words have a specific meaning on some fields that differs from their standard use. In this case, we're talking about what limitless and infinite mean in mathematical sense. That they're synonyms in everyday language is utterly irrelevant.

[erikun]

Font of power + affinity field + synchronicity.
Aha, I see how that is supposed to work now. However:

1.) You're working with a 17th level build, minimum.
2.) Font of Power specifically affects you, not your Psicrystal, so you can give infinite actions to your Psicrystal but not the reverse,
3.) The Pun-Pun loop requires actions from both you and your Psicrystal,
4.) This still isn't infinite (or infinitely increasing) stats, it's just an absurdly large number in a single round, and finally,
4.) You still have no way to break out of the self-dominate loop, if you wanted to use it for some reason, as "Infinite" is not a number you will ever reach.

You also have the problem with old age, unless you're an Elan.

[OracleofSilence]

this is about pun-pun. he is already a n divine ranked god, where n is any number he wants.

[Urpriest]

Dominate is unnecessary. Upon reflection, just the standard loop should be enough, because of the following:

Mathematically speaking (which is the relevant field for this discussion), an infinite number is one such that it is impossible to choose a higher number. The Affinity Field loop takes no more than 6 seconds, and suppose it is used to raise Strength. Then after those six seconds, for any number you could choose, his Strength would be higher, since he had enough actions to raise it higher. Thus you do indeed get infinite ability scores that way.

[ericgrau]

He does. Font of power + affinity field + synchronicity. Basically, his familiar can grant both itself and Pun-Pun a standard action as a standard action, thus giving Pun-Pun a potentially infinite number of standard actions per round. The entire process only takes 6 seconds.

And now limitless is infinite for all practical purposes. /thread?

It's not truly infinite but the 7 comments that ignored the OP are now valid; it doesn't make a difference anymore.

[erikun]

this is about pun-pun. he is already a n divine ranked god, where n is any number he wants.
This is a good point. You could simply have the value of Graham's Number (http://en.wikipedia.org/wiki/Graham%27s_number) in divine ranks, all stats, and all rolls that you wished and call it a day. Graham's Number isn't infinite, but it is "absurdly large".

As in, we can't type it out. We can't even type out how many digits long it is, because that number is larger than the given universe. I can guarantee you that it's larger than anything you've seen in any sourcebook. :smallwink:

[squeekenator]

Aha, I see how that is supposed to work now. However:

1.) You're working with a 17th level build, minimum.
2.) Font of Power specifically affects you, not your Psicrystal, so you can give infinite actions to your Psicrystal but not the reverse,
3.) The Pun-Pun loop requires actions from both you and your Psicrystal,
4.) This still isn't infinite (or infinitely increasing) stats, it's just an absurdly large number in a single round, and finally,
4.) You still have no way to break out of the self-dominate loop, if you wanted to use it for some reason, as "Infinite" is not a number you will ever reach.

You also have the problem with old age, unless you're an Elan.

1. As OracleofSilence said, we already have limitless power. This isn't an exercise in optimisation as such, it's simply an attempt to turn Pun-Pun's impressive and functional arbitrarily large numbers into pointless but cool infinities.
2. I'm not an expert on the trick, I simply read about it in the Pun-Pun thread, but I'm fairly certain that, if there was indeed a hole in a trick used by the best-known CharOp build ever, it would have been pointed out by now. Either way, I'm sure he can find a different infinite action loop, given that he's Pun-Pun.
4. The point of using Dominate is that it forces you to increase your stats an infinite number of times. There simply isn't any way around it, after 6 seconds you must have infinities in every relevant attribute, because the system forces you to.
5. You reach infinity within 6 seconds (well, a single standard action, which is less than 6 seconds, but whatever). The entire process takes a single round, you only age 6 seconds.

[Zale]

Why does he even /need/ infinite stats?

He's god**** Pun-Pun.

He could probably drop kick a mountain into dust.

[NNescio]

Sure... ok.

Originally Posted by The dictionary
Limitless
Definition: never-ending, infinite

That's funny.. what about


Originally Posted by The dictionary
Infinite
Definition: limitless, without end

That's so weird. I wonder why the dictionary would tell such a bald-faced lie? Naturally I should listen to what people on a forum dedicated to a web-comic think about what words mean instead of the dictionary.

I find it funny that you never named your source.

Then I find it even funnier when I found out that your source turns out to be answers.com's antonym list.


1) http://www.google.com/search?btnG=1&pws=0&q=%22Definition%3A+never-ending%2C+infinite%22
2) http://www.google.com/search?btnG=1&pws=0&q=%22Definition%3A+limitless%2C+without+end%22

They are brief for a reason you know -- they are antonym lists.

Now, we don't need to delve into actual math textbooks. Here's (http://www.thefreedictionary.com/infinite) what a real definition looks like:


in·fi·nite (nf-nt)
adj.
1. Having no boundaries or limits.
2. Immeasurably great or large; boundless: infinite patience; a discovery of infinite importance.
3. Mathematics
a. Existing beyond or being greater than any arbitrarily large value.
b. Unlimited in spatial extent: a line of infinite length.
c. Of or relating to a set capable of being put into one-to-one correspondence with a proper subset of itself.
n.
Something infinite.
[Middle English infinit, from Old French, from Latin nfntus : in-, not; see in-1 + fntus, finite, from past participle of fnre, to limit; see finite.]
infi·nite·ly adv.
infi·nite·ness n.
Synonyms: infinite, boundless, eternal, illimitable, sempiternal
These adjectives mean being without beginning or end: infinite wisdom; boundless ambition; eternal beauty; illimitable space; sempiternal truth. See Also Synonyms at incalculable.

Usage Note: Infinite is sometimes grouped with absolute terms such as unique, absolute, and omnipotent, since in its strict mathematical sense infiniteness is an absolute property; some infinite sets are smaller than others, but they are no less infinite. In nontechnical usage, of course, infinite is often used to refer to an unimaginably large degree or amount, and in these cases it is acceptable to modify or compare the word: Nothing could give me more infinite pleasure than to see you win. Withdrawing the troops would create an even more infinite set of problems for the coalition. · Note that unlike other incomparable adjectives, infinite when used in its strict literal sense cannot be modified by words like nearly, since quantities do not approach infinity by degrees. This constraint, too, can be ignored when the word is used simply to refer to a very large number: You need a nearly infinite amount of patience to do the job. See Usage Notes at absolute, unique.

And if you want the Oxford definition: (http://oxforddictionaries.com/definition/infinite?region=us)


infinite(in·fi·nite)

Pronunciation:/ˈinfənit, /
adjective

1 -limitless or endless in space, extent, or size; impossible to measure or calculate:the infinite mercy of Godthe infinite number of stars in the universe
-very great in amount or degree:he bathed the wound with infinite care
-Mathematics greater than any assignable quantity or countable number.
-Mathematics (of a series) able to be continued indefinitely.
2 Grammar another term for nonfinite

Note the different definitions? By your logic I can say that "kill" and "turn off" are the same, just because they share certain definitions.

Practically almost every single online dictionary of note take care to distinguish the slight difference in meaning of "infinite" (from the colloquial, which is implicit) when it's used in 'mathematics'.

[erikun]

2. I'm not an expert on the trick, I simply read about it in the Pun-Pun thread, but I'm fairly certain that, if there was indeed a hole in a trick used by the best-known CharOp build ever, it would have been pointed out by now. Either way, I'm sure he can find a different infinite action loop, given that he's Pun-Pun.
I don't automatically put much faith in a specific board just because it's well known for it's optimization. I seem to recall another well-known optimization board posting a build granting access to every single domain in the game, but it did so by taking five different variants of 1st level Cleric - something clearly against the rules, even with the most liberal interpretations.

Other than that, I guess we're using slightly different definitions of the term. Or perhaps looking at it from different angles. I'm saying you can't reach infinity through numerical addition, and thus Pun-Pun would be stuck in his loop (or remain locked in his domination loop if forced out of his 6-second round somehow) because there is no way to reach an infinite number through addition. You're saying that he will come out with infinite stats, because his 6-second turn must end sometime and it can only end when his stats are infinite.

I suppose there might be some interesting philosophical discussion there, but probably not relevant to the point of the thread.

[squeekenator]

Why does he even /need/ infinite stats?

He's god**** Pun-Pun.

He could probably drop kick a mountain into dust.

Because it's cooler?

This is not an exercise in practical optimisation. Anything that is designed to improve upon an already omnipotent character does not have any particular purpose. It's merely designed to make builds with access to infinite actions per round even more impressively broken.

(That's not aimed specifically at you, by the way.)

[kharmakazy]

If you order yourself to up your stats an unlimited number of times, your turn never ends does it?

[Urpriest]

If you order yourself to up your stats an unlimited number of times, your turn never ends does it?

It does though, it ends in precisely 6 seconds, same as every other turn.

[kharmakazy]

It does though, it ends in precisely 6 seconds, same as every other turn.

A round lasts that long in game time, yes. So if he ever ended his turn 6 seconds would have elapsed, but if he never stops upping his stats, then he never stops taking actions, and his turn never ends. The universe is frozen on his turn.

[squeekenator]

A round lasts that long in game time, yes. So if he ever ended his turn 6 seconds would have elapsed, but if he never stops upping his stats, then he never stops taking actions, and his turn never ends. The universe is frozen on his turn.

By the same logic, Floppsy the Destroyer, a simple rabbit, can end life as we know it by simply refusing to take an action in his turn, thus causing the entire multiverse to freeze permanently. Such interpretations of the rules are somewhat silly. Though if we do intepret the rules like that, I think I just created the most powerful character ever. Even Pun-Pun cannot stand up to Floppsy's might.

[kharmakazy]

By the same logic, Floppsy the Destroyer, a simple rabbit, can end life as we know it by simply refusing to take an action in his turn, thus causing the entire multiverse to freeze permanently. Such interpretations of the rules are somewhat silly. Though if we do intepret the rules like that, I think I just created the most powerful character ever. Even Pun-Pun cannot stand up to Floppsy's might.

That's hyperbolic at best. If you never stop taking actions your turn never ends. Turn 2 never starts if turn 1 never ends. If flopsy refuses to take an action that is his business, but it does not prevent the next turn from starting. You aren't not taking your action, you are taking a ludicrously large amount of actions. If you didn't ever stop taking actions on your last turn, how does your next turn conceivably start?

[RaggedAngel]

By the same logic, Floppsy the Destroyer, a simple rabbit, can end life as we know it by simply refusing to take an action in his turn, thus causing the entire multiverse to freeze permanently. Such interpretations of the rules are somewhat silly. Though if we do intepret the rules like that, I think I just created the most powerful character ever. Even Pun-Pun cannot stand up to Floppsy's might.

THIS.

This needs to be a thing. Pun-Pun is cool and awesome and all, but still.

All hail Floppsy the Destroyer! All hail his Apathatic Might!

[kharmakazy]

If floppsy can somehow gain an amount of extra actions that does not end, and you are operating under the rules of initiative...

So long as floppsy goes on first in the initiative anyone who goes after floppsy will, in fact, not get a turn until floppsy decides to stop taking actions on his turn.

[MeeposFire]

If floppsy can somehow gain an amount of extra actions that does not end, and you are operating under the rules of initiative...

So long as floppsy goes on first in the initiative anyone who goes after floppsy will, in fact, not get a turn until floppsy decides to stop taking actions on his turn.

But Pun Pun could do the same exact thing but can do it better since his initiative is better.

[kharmakazy]

I guess... I don't think it was intended to be a combat tactic so much as it was a drawback to taking an unspecified number of standard actions in one round. But I guess so long as one was so inclined he could prevent everything but immediate actions nigh indefinitely.

I still don't see the mechanical difference between saying "I take infinite actions to boost STR" and dominating yourself and ordering yourself "Take infinite actions to boost STR."

[Divide by Zero]

The point is, no matter how many actions Pun-Pun decides to take, he will have to eventually stop taking actions, and the number he has taken will never be infinite. It can be arbitrarily large, certainly, but never infinite.

[Salanmander]

This is a good point. You could simply have the value of Graham's Number (http://en.wikipedia.org/wiki/Graham%27s_number) in divine ranks, all stats, and all rolls that you wished and call it a day. Graham's Number isn't infinite, but it is "absurdly large".

As in, we can't type it out. We can't even type out how many digits long it is, because that number is larger than the given universe. I can guarantee you that it's larger than anything you've seen in any sourcebook. :smallwink:

Ahh, good ol' Graham's Number. I'm a firm believer in the factually false statement that Graham's Number is closer to infinity than it is to zero.

You know, I once tried to calculate G0. I failed. All I managed to calculate was the order of magnitude of the time it would take me to calculate a number one step removed from G0.

Anyway, the problem is that even the pun-pun with a Con of G64 will be killed in one hit by the 1d2 crusader. That's why we want our Con score to be infinite, not just...Big.

[MeeposFire]

Ahh, good ol' Graham's Number. I'm a firm believer in the factually false statement that Graham's Number is closer to infinity than it is to zero.

You know, I once tried to calculate G0. I failed. All I managed to calculate was the order of magnitude of the time it would take me to calculate a number one step removed from G0.

Anyway, the problem is that even the pun-pun with a Con of G64 will be killed in one hit by the 1d2 crusader. That's why we want our Con score to be infinite, not just...Big.

Actually it won't be a problem since Pun Pun is immune to damage.

[Salanmander]

Actually it won't be a problem since Pun Pun is immune to damage.

Oh...well, bugger that then. Graham's number is big enough for me =P.

[NoldorForce]

No infinite action loop is required. Instead, you want traps to instantly reset. The gist is to set up a scenario whereupon Polymorph (to change type into an animal) and Awaken (to grant +2 HD) traps are invoked in an infinite loop to grant infinite HD. Have them both set with proximity triggers, invoked by a third trap also with an insta-reset proximity trigger that summons Vivacious (incorporeal for infinite stacking) monsters. (Note that the third trap keeps invoking itself in a feedback loop.)

Anyway, once Pun-Pun has infinite HD, he also gets infinite feats. Doesn't matter what they are, because he should next use Psychic Reformation in conjunction with Supernatural Transformation to custom-fit all (infinite) feats and (infinite) skill points. The result? Pun-Pun has infinite HD, infinite ability scores (via the epic feats to boost each, infinity's weird that way), infinite bonuses to all skills (partly through ability scores, partly through ranks), and various other infinite stats. All with a finite (two digits at most) quantity of actions.

[Cicciograna]

Just to satisfy the nerd in me...
Guys, keep yourselves up to date! Graham number pales in comparison to TREE(3) (http://en.wikipedia.org/wiki/TREE%283%29) :smallsmile:

[erikun]

Just to satisfy the nerd in me...
Guys, keep yourselves up to date! Graham number pales in comparison to TREE(3) (http://en.wikipedia.org/wiki/TREE%283%29) :smallsmile:
Good lord, I remember back in high school when mathematics made sense. I will fully admit that I do not understand a single word of that article, although I can easily see the vast size difference between TREE(3) and Graham's Number.

[Moriato]

Anyway, once Pun-Pun has infinite HD, he also gets infinite feats.

But... he'll never get to infinite HD. Not matter how many HD you add, it will still be finite. You can't "count up to infinity", because infinity is not a number. Any number, no matter how high, is by definition not infinite.

[MeeposFire]

But... he'll never get to infinite HD. Not matter how many HD you add, it will still be finite. You can't "count up to infinity", because infinity is not a number. Any number, no matter how high, is by definition not infinite.

It would not surprise me if there was a loop for infinite HD and if there is then pun pun has it.

[danzibr]

No matter how I see it, you can't actually get infinity as a stat, for the mere reason that no matter how large a number you're talking about, it's still infinitely far from infinity. I mean, I really don't see any way to get infinitely many actions in a finite amount of time.

[Moriato]

It would not surprise me if there was a loop for infinite HD and if there is then pun pun has it.

Well, as mentioned earlier, he could just give himself infinite stats using his Manipulate Form ability. The point is that you can never really get to "infinite" by adding up numbers, because a number can never be infinite.

[MeeposFire]

Well, as mentioned earlier, he could just give himself infinite stats using his Manipulate Form ability. The point is that you can never really get to "infinite" by adding up numbers, because a number can never be infinite.

I was talking about an infinte loop for HD.

[Cicciograna]

Good lord, I remember back in high school when mathematics made sense. I will fully admit that I do not understand a single word of that article, although I can easily see the vast size difference between TREE(3) and Graham's Number.

Believe me, I have a degree in Physics, and Math still does not make sense :smallbiggrin:

[danzibr]

Believe me, I have a degree in Physics, and Math still does not make sense :smallbiggrin:
I just got a Master's in Math and most of it still does not make sense :smallbiggrin:

[NoldorForce]

But... he'll never get to infinite HD. Not matter how many HD you add, it will still be finite. You can't "count up to infinity", because infinity is not a number. Any number, no matter how high, is by definition not infinite.See my previous post. If you set things up properly, a no-decision feedback loop can be set up to cast Awaken an infinite number of times.

How is this infinite? Consider:
-Awaken is successfully cast each time the feedback loop is triggered. (This is performed by Polymorphing into an animal for the type.)
-The feedback loop, in the very act of firing, triggers itself. (The trap that triggers the Polymorph and Awaken traps also triggers itself with its own summoned monster.) Thus, for each time the loop fires, it must fire an additional time.
-Setting off the first instance of the loop is trivial.

To determine whether an infinite number of applications of Awaken have been cast, let us consider the related set of applications thereof and its cardinality. A set is said to be countably infinite (http://en.wikipedia.org/wiki/Aleph_number#Aleph-naught) iff it can be put into a one-to-one correspondence with the natural numbers. Can this be done?

Let the set of Awaken applications be A, with each element in A being Ai for some distinct i. We let A0 be the first Awaken application, and let all others be defined such that Aj is the Awaken application caused by the feedback loop calling itself when it also calls Aj-1. (Essentially, the elements of A are ordered based on when they are called.) The set of naturals is of course N.

To perform a bijection between the two, we may first match A0 (in A) with 0 (in N). Now, suppose that we have matched Ai (in A) with i (in N). Then because the feedback loop triggers itself by its very construction, there exists Ai+1 (in A), which we may match with i+1 (in N). By mathematical induction (http://en.wikipedia.org/wiki/Mathematical_induction), then, all members of N may be matched with distinct members of A (and vice versa; matching is symmetric) in a one-to-one correspondence. Thus, A is in fact countably infinite. (Ie, Awaken has been applied a countably infinite number of times.)

If you have issues/questions with the mathematics above, let me know. Infinity's a weird concept and has some paradoxical dealings, but they're mostly beyond the scope of this discussion. You may have some other concerns with this, such as:

-Actually performing an infinite sequence of anything is impossible in D&D because each whatever takes a nonzero amount of time to process. True, but each instance of the loop triggers itself in the same round unless noted otherwise. Consider the timestamps of loop instances to be real numbers. Even if they're all distinct, the total distance in time spanned between them is not prohibited from being finite; after all, there's an infinite set of reals between 0 and 1.
-Actually performing an infinite sequence of anything is impossible in D&D because there's no stopping point. Well, what happens if you try to do so in a completely decision-free manner? Does the universe go Zonk or something? And also consider the argument above; just because there's no final element to the sequence, doesn't mean the sequence can't span a finite length.
-You can't stack an infinite set of monsters in one spot. This is actually a valid point. I'd thought incorporeals could stack with each other, but they can't. (They can stack with corporeals.) Instead, just set up enough entropic creatures in range to insta-kill whatever gets summoned by the feedback loop. As a side bonus, "killed" summoned monsters leave no corpses. You'll want to set things up as to not have the entropic stuff setting off the traps, however.

[danzibr]

True, but each instance of the loop triggers itself in the same round unless noted otherwise.
Just to point out this one thing... I read your other post and I'm not sure what traps you're talking about. But anyways, what kind of action is a trap setting off? If it's at least a free action this won't work because only a reasonable amount of free actions occur per turn. If it doesn't count as an action or something... then yeah, this seems like the way to go. The real problem is getting infinitely many actions in a finite amount of time, but if this trap thing does it, then problem solved.

[erikun]

To determine whether an infinite number of applications of Awaken have been cast, let us consider the related set of applications thereof and its cardinality. A set is said to be countably infinite (http://en.wikipedia.org/wiki/Aleph_number#Aleph-naught) iff it can be put into a one-to-one correspondence with the natural numbers. Can this be done?

[...]

Thus, A is in fact countably infinite. (Ie, Awaken has been applied a countably infinite number of times.)

If you have issues/questions with the mathematics above, let me know. Infinity's a weird concept and has some paradoxical dealings, but they're mostly beyond the scope of this discussion. You may have some other concerns with this, such as:
From the looks of the definition, it seems that "countably infinite" means a set of real numbers that can be counted from one to another. That is, like 1, 2, 3, 4, 5, 6, and so on for the set whole numbers. The opposite ("non-countably infinite"?) would then be a set where you could not count from one number to another, perhaps because for any two numbers you pick, there would be another number between them. For example, in the set of real numbers, you could not count from 1 to 2 because 1.1 is between them. You could not count from 1 to 1.1 because 1.01 is between them. You could not count from 1 to 1.0000001 because 1.00000000001 is between them.

Am I understanding that correctly?

If that is the case, then the set of numbers possible to achieve by Awaken (which is basically the set of possible Intelligence value numbers) is basically the set of positive whole numbers, and so would share the same properities (being countably infinite, for one).

The question, however, is if one can achieve infinite (run through the Awaken set) within one round (the set of real numbers between 0 seconds and 6 seconds). It seems to me that we would need to compart the Awaken set with this Single-Round set to see if they are "equal", meaning that they would be considered to have the same number of terms, or could relate to one another on a 1:1 correspondence. If they can, then it would be reasonable to conclude that the Awaken set had reached infinity during the run of the Single-Round set. If not, then that means either we didn't have enough time during the Single-Round set to finish the Awaken set, or that the Awaken set completed and we still have an infinite amount of values remaining in the Single-Round set.

I'll leave it up to others to determine things from there, though.

[NoldorForce]

Just to point out this one thing... I read your other post and I'm not sure what traps you're talking about. But anyways, what kind of action is a trap setting off? If it's at least a free action this won't work because only a reasonable amount of free actions occur per turn. If it doesn't count as an action or something... then yeah, this seems like the way to go. The real problem is getting infinitely many actions in a finite amount of time, but if this trap thing does it, then problem solved.Take a look here (http://www.d20srd.org/srd/traps.htm) for details on traps, but so far as I can see traps perform no actions of their own; they just react to things around them. The setup I was referring to had a Summon Monster trap, a (Reach Spell) Awaken trap, and a (Reach Spell) Polymorph trap. All three have proximity triggers and reset automatically. Thus the Summon Monster trap can keep triggering itself and the others. (The Awaken and Polymorph traps aren't set to cast on the summoned stuff, but rather on whatever's in another predefined space.)
From the looks of the definition, it seems that "countably infinite" means a set of real numbers that can be counted from one to another. That is, like 1, 2, 3, 4, 5, 6, and so on for the set whole numbers. The opposite ("non-countably infinite"?) would then be a set where you could not count from one number to another, perhaps because for any two numbers you pick, there would be another number between them. For example, in the set of real numbers, you could not count from 1 to 2 because 1.1 is between them. You could not count from 1 to 1.1 because 1.01 is between them. You could not count from 1 to 1.0000001 because 1.00000000001 is between them.

Am I understanding that correctly?

If that is the case, then the set of numbers possible to achieve by Awaken (which is basically the set of possible Intelligence value numbers) is basically the set of positive whole numbers, and so would share the same properities (being countably infinite, for one).

The question, however, is if one can achieve infinite (run through the Awaken set) within one round (the set of real numbers between 0 seconds and 6 seconds). It seems to me that we would need to compart the Awaken set with this Single-Round set to see if they are "equal", meaning that they would be considered to have the same number of terms, or could relate to one another on a 1:1 correspondence. If they can, then it would be reasonable to conclude that the Awaken set had reached infinity during the run of the Single-Round set. If not, then that means either we didn't have enough time during the Single-Round set to finish the Awaken set, or that the Awaken set completed and we still have an infinite amount of values remaining in the Single-Round set.

I'll leave it up to others to determine things from there, though."Countable" doesn't quite mean "able to be counted" in mathematics, but rather it refers to something being *a subset (proper or not) of the naturals (0 plus the positive integers). (Edit: *in a one-to-one correspondence with such a subset.) The naturals themselves are countable by definition (a set is always a subset of itself), but as it turns out the rational numbers (any that can be expressed as a/b where a and b are integers with b nonzero) are also countable (http://en.wikipedia.org/wiki/Countable). This is despite the fact that the rationals are also dense (http://en.wikipedia.org/wiki/Densely_ordered), that there are infinite rationals between any two distinct arbitrary rationals. (The naturals, despite having a bijection with the rationals, are not dense - infinity's weird like that.) The reals are not countable (http://en.wikipedia.org/wiki/Cantor%27s_first_uncountability_proof) precisely because they're not a subset of the naturals.

But whether you regard the timestamps of the Awaken set as rationals or reals between 0 and 6, both sets are dense and thus contain an infinite number of members within that space. Thus, they can contain the Awaken set (an injection can map that to the reals or the rationals).

[erikun]

"Countable" doesn't quite mean "able to be counted" in mathematics, but rather it refers to something being a subset (proper or not) of the naturals (0 plus the positive integers). The naturals themselves are countable by definition (a set is always a subset of itself), but as it turns out the rational numbers (any that can be expressed as a/b where a and b are integers with b nonzero) are also countable (http://en.wikipedia.org/wiki/Countable). This is despite the fact that the rationals are also dense (http://en.wikipedia.org/wiki/Densely_ordered), that there are infinite rationals between any two distinct arbitrary rationals. (The naturals, despite having a bijection with the rationals, are not dense - infinity's weird like that.) The reals are not countable (http://en.wikipedia.org/wiki/Cantor%27s_first_uncountability_proof) precisely because they're not a subset of the naturals.
Aha, I see where I was getting mixed up. "Countable" means the same cardinality (number of items) as a set of natural numbers, not a subset of natural numbers. That would be why rational numbers can be countable despite being different from the set of natural numbers.

I'm not sure how the difference between rational numbers and real numbers makes one countable and one not, but I'll assume the proofs are valid.


But whether you regard the timestamps of the Awaken set as rationals or reals between 0 and 6, both sets are dense and thus contain an infinite number of members within that space. Thus, they can contain the Awaken set (an injection can map that to the reals or the rationals).
I'm forced to go with your conclusion, trusting that you have already accounted for any objections I might raise due to my limited understanding. And so, I'll save us all the time.

[danzibr]

Just to be picky...

"Countable" doesn't quite mean "able to be counted" in mathematics, but rather it refers to something being a subset (proper or not) of the naturals
I think you stated this earlier, but we need bijections, not set theoretic containment.

"Countable" means the same cardinality (number of items) as a set of natural numbers, not a subset of natural numbers. That would be why rational numbers can be countable despite being different from the set of natural numbers.

I'm not sure how the difference between rational numbers and real numbers makes one countable and one not, but I'll assume the proofs are valid.
To be picky again... the naturals and rationals don't have the same number of elements (infinity isn't a number). Countably infinite and uncountably infinite are both infinite. It comes down to bijections.

[AppleChips]

Can't Pun pun just do the omniscificer trick? Temporarily set aside powers that prevent damage, after putting giant amounts of protective spells around to prevent interference while weak.

[Cicciograna]

Aha, I see where I was getting mixed up. "Countable" means the same cardinality (number of items) as a set of natural numbers, not a subset of natural numbers. That would be why rational numbers can be countable despite being different from the set of natural numbers.

I'm not sure how the difference between rational numbers and real numbers makes one countable and one not, but I'll assume the proofs are valid

The proof that Rationals are countable (which is to say, can be put in a 1:1 relationship with Naturals) is not so difficult, and there's a cute visual aid to understand this feature (http://www.homeschoolmath.net/teaching/rational-numbers-countable.php).

The uncountability of Reals, which characterizes them as an aleph-1 set, is not so easy, but again there's a visual aid for it too (http://en.wikipedia.org/wiki/Cantor%27s_diagonal_argument).

[NoldorForce]

I think you stated this earlier, but we need bijections, not set theoretic containment.Sorry, got confused - I'll amend my statement.

[RS14]

I take issue with the notion that a non-terminating loop "produces" values that are infinite.

The rules of D&D describe a system, the state of which is changed by various events. An event can occur which increases stats as many times as desired, and upon terminating, the state is well-defined and gameplay continues. There is, however, no provision for evaluating an infinite series of events. Asking "what happens when this infinite loop finishes" is completely meaningless--as much so as asking "what number will I have reached by the time I am at the end of the hallway, if halfway down the hallway I stop and count forever." No. The loop continues. You can go home and come back next week and keep evaluating that loop. You can grow old and die waiting for that loop to terminate. Any supposed game which takes place after the completion of such a loop is non-RAW, because the RAW consider only finite numbers of events.

The rules are defined only for integers. Not for extended integers (Z \cup \{\infty,-\infty\}). A precise analogy would be the infinite sum

\sum_{k=0}^{\infty} 1

which is not an integer. There exists no such limit in the integers.


Well, what happens if you try to do so in a completely decision-free manner? Does the universe go Zonk or something? And also consider the argument above; just because there's no final element to the sequence, doesn't mean the sequence can't span a finite length.
If playing by RAW, your DM says "Awaken is cast, so Awaken is cast, so Awaken is cast, so Awaken is cast, so Awaken is cast, so Awaken is cast, so Awaken is cast, ... so Awaken is cast, and come back next week for an exciting session of Infinite Loops, the Continuing!"

[Pie Guy]

This looks like a job for limits! (Maybe, my dad's the mathematician, not me.)

Anyway, isn't this just another version of Zeno's paradox?

[RS14]

This looks like a job for limits! (Maybe, my dad's the mathematician, not me.)

Anyway, isn't this just another version of Zeno's paradox?

It's not that you can't define away this problem, but rather that doing so is not RAW, and tends to produce its own set of issues, such as arithmetic not being fully defined on the extended integers.

[MeeposFire]

Can't Pun pun just do the omniscificer trick? Temporarily set aside powers that prevent damage, after putting giant amounts of protective spells around to prevent interference while weak.

Yes due to the fact that Pun-Pun get started at level1 means he goes before the omniscificer. Originally the O "beat" Pun-Pun because it could get going earlier but once Pun-Pun could get going at level 1 then the O just can't compete anymore.

[erikun]

To be picky again... the naturals and rationals don't have the same number of elements (infinity isn't a number). Countably infinite and uncountably infinite are both infinite. It comes down to bijections.
Yes, well, I see where I went wrong. I was assuming that sets behaved like limits, and drew logic from there; clearly my conclusion came out incorrect when my assumptions were as well. Thanks for the correction.


The proof that Rationals are countable (which is to say, can be put in a 1:1 relationship with Naturals) is not so difficult, and there's a cute visual aid to understand this feature (http://www.homeschoolmath.net/teaching/rational-numbers-countable.php).

The uncountability of Reals, which characterizes them as an aleph-1 set, is not so easy, but again there's a visual aid for it too (http://en.wikipedia.org/wiki/Cantor%27s_diagonal_argument).
Interesting. I suppose that does clear things up a bit.

However, I have to wonder; cannot the Cantor's diagonal argument (the second link) be used against the natural number line (in binary form) to produce a s0-like binary number that, by definition, is not part of the natural number line set? Or is this trickery somehow avoided through interesting application of infinity rules?

To clarify what I mean, assume we produce the same setup in Cantor's diagonal argument, but with natural numbers (in binary, read right-to-left, taken to infinite length). We would then have the numbers:

1 = 100000...
2 = 010000...
3 = 110000...
4 = 001000...
5 = 101000...
6 = 011000...

An so on. We could also produce a number, our "s0", where the nth digit is the opposite of the nth digit of number n. We would thus get the number:

n0 = 001111...

However, by the logic in Cantor's diagonal argument, n0 cannot be a natural number. But I see no reason why n0 would not be a natural number; by the rules of binary notation, it should be.

Thus, I am left to conclude that either n0 is somehow not a number, or that Cantor's diagonal argument indicates that the set of natural numbers in uncountable (which causes all kinds of problems).


Anyway, isn't this just another version of Zeno's paradox?
Any talk of infinite division would probably be related to Zeno's paradox, yes. The difference is that we're using tools to come up with an answer. :smalltongue:

[danzibr]

I take issue with the notion that a non-terminating loop "produces" values that are infinite.

The rules of D&D describe a system, the state of which is changed by various events. An event can occur which increases stats as many times as desired, and upon terminating, the state is well-defined and gameplay continues. There is, however, no provision for evaluating an infinite series of events. Asking "what happens when this infinite loop finishes" is completely meaningless--as much so as asking "what number will I have reached by the time I am at the end of the hallway, if halfway down the hallway I stop and count forever." No. The loop continues. You can go home and come back next week and keep evaluating that loop. You can grow old and die waiting for that loop to terminate. Any supposed game which takes place after the completion of such a loop is non-RAW, because the RAW consider only finite numbers of events.

The rules are defined only for integers. Not for extended integers (Z \cup \{\infty,-\infty\}). A precise analogy would be the infinite sum

\sum_{k=0}^{\infty} 1

which is not an integer. There exists no such limit in the integers.
Yeah, writing in that sideways 8 in a stat box isn't really reasonable... but something interesting is happening. Through rule abuse you can boost your HD countably-infinitely many times in a single round. Assuming you terminate the loop on the round after, the loop does not continue. This sounds like a job for... the extended naturals! \mathbb{N}_\infty. For perhaps a terrible analogy, it'd be like integrating 1/(x-2)^2 from 0 to 4. There's some sort of non-integrable singularity going on. You can either say it doesn't exist or define it to be infinite. I personally would define it as plus infinity. In my thesis I work with quasi-metrics which take values in the non-negative extended real axis. You can totally treat infinity as a number and keep consistency.

Sidenote, do you use LaTeX?

[BobVosh]

If allowing for constantly increasing stats to be considered infinite, wouldn't it be easy for him to just give himself :
I win even harder [EX]: Each round +2 to each stat.

Or if he wants to meta game, as I'm sure that would be hard for anyone attempting pun-pun and do stuff to just annoy people
Schrodinger Stat [EX]: Each time my character sheet is observed, each stat goes up +2.

Or contingency spell component pouch [EX]: He sets his stats equal to the cubic feet of bat guano he can pull out of a spell component pouch as a free action.
As several truly infinite things do exist in the game and the fact he can literally make up any ability (as far as I understand, maybe people exaggerate his powers) if he can link the two somehow he is fine. Such as the size of the 9 levels of hell, spell component pouches, or the number of levels of the abyss.

[Bakkan]

0[/SUB]-like binary number that, by definition, is not part of the natural number line set? Or is this trickery somehow avoided through interesting application of infinity rules?

To clarify what I mean, assume we produce the same setup in Cantor's diagonal argument, but with natural numbers (in binary, read right-to-left, taken to infinite length). We would then have the numbers:

1 = 100000...
2 = 010000...
3 = 110000...
4 = 001000...
5 = 101000...
6 = 011000...

An so on. We could also produce a number, our "s0", where the nth digit is the opposite of the nth digit of number n. We would thus get the number:

n0 = 001111...

However, by the logic in Cantor's diagonal argument, n0 cannot be a natural number. But I see no reason why n0 would not be a natural number; by the rules of binary notation, it should be.

Thus, I am left to conclude that either n0 is somehow not a number, or that Cantor's diagonal argument indicates that the set of natural numbers in uncountable (which causes all kinds of problems).


Ah, but n0 is not a natural number, because a natural number must have a terminating binary numeral. To take your example, your n0 = 001111... equals 0x20+0x21+1x22+1x23+1x24+1x25+...., which does not converge to a natural number unless the sequence of digits is after some point all zeros, which is not the case in your example. However, suppose that we reorder the natural numbers in the original argument so that we do eventually have all zeros , i.e. so that n0 is a natural number. Let n0=k0k1k2k3.... Let N be the position of the first zero in our final run of zeros. That is, for all n >= N, suppose kn=0. This means that in the diagonal argument, the Nth digit of the Nth number is 1, the N+1th digit of the N+1th number is 1, and so on ad infinitum. Note that there are 2N-1-1 numbers with a binary numeral N-1 digits or fewer in length. If N>0, then 2N-1-1 > N-1, which means that at some point at or after the Nth number in the diagonal argument, an N-1 or fewer digit number must appear. But such a number will have a 0 in all places after the N-1st, including the place corresponding to that number's position in the argument. Therefore n0 will have a 1 in that position, which is a contradiction. If N=0, then every digit of n0 is 0, which means that n0=0, which is not a natural number, so again we have a contradiction.

Therefore n0 cannot be a natural number.

[Cicciograna]

To clarify what I mean, assume we produce the same setup in Cantor's diagonal argument, but with natural numbers (in binary, read right-to-left, taken to infinite length). We would then have the numbers:

1 = 100000...
2 = 010000...
3 = 110000...
4 = 001000...
5 = 101000...
6 = 011000...

An so on. We could also produce a number, our "s0", where the nth digit is the opposite of the nth digit of number n. We would thus get the number:

n0 = 001111...

However, by the logic in Cantor's diagonal argument, n0 cannot be a natural number. But I see no reason why n0 would not be a natural number; by the rules of binary notation, it should be.

Thus, I am left to conclude that either n0 is somehow not a number, or that Cantor's diagonal argument indicates that the set of natural numbers in uncountable (which causes all kinds of problems).


I'd say that n0 is not a natural number. This should stem from the iterative nature of the construction of n0: at any given moment, the set of sn,i does not contain, by construction, the numer n0; even by projecting the process to infinity, the number n0 will never be part of the set, because if that happened then, as Cantor said, one would find n0,n = sn,n a particular, which goes against hypotesis. Given that the set of the sn is constructed by bijection from the Natural, I'd conclude that there does not exist any number who can be converted to n0.
Note, probabily there's a deeper reason that at the moment escapes me, and besides I'm no expert.

EDIT: swordsaged! actually, Bakkan gave a more correct, more precise proof of the fact. Forget what I said.

[erikun]

Ah, but n0 is not a natural number, because a natural number must have a terminating binary numeral.

I'd say that n0 is not a natural number.
Aha, I see! Well thanks for taking the time to clarify it for me, then.