Zeno's Paradox and Flechette Defense

[Natalinasmpf]

http://en.wikipedia.org/wiki/Zeno's_paradoxes

The first two of Zeno's paradoxes goes along like this: to go some distance, or to conceive some goal, I would first have to go halfway, than the half of that half, and that half of that half of that half, and this goes on infinitely....

Thus the goal will never be reached. We know, by say, walking from A to B, that this isn't true....even though its quite true we go through infinite halves of terms everytime we walk.

This was resolved and the paradox solved through the mathematical conclusion later on, that an infinite amount of terms can make a finite amount.

Thus, suppose I have an infinite amount of planet busters (or just so much, it has crossed the threshold), a huge amount of RAM and such, and a critical city, lets say Free Drone Central, since with planned and eudaimonic, they can halve satellite costs, and with space elevator, get it done in 25% of the time, say one turn. Thats not really important though. Now, lets say there was no turn limit, and suppose (anyway there's 1 million+ mission years in one of the interludes ) the base now has an infinite amount of Flechette Defenses, or enough to cross the threshold of infinity into the surreal number category (where adding +1 no longer matters, same effect).

Now, it seems without infinite amount of satellites, University Base will never be immune to my planet buster attacks, since I can just rain all my infinite (or enough to cross the limit) PB's....but if the flechette defenses are infinite, then theoretically, it will no longer be 0.01+e(infinity), or 0.000000000000000000000000000001, and so on, etc.

It will be zero, completely immune. Adding another planet buster will always be 0%, and not just say 0.000000000000000000000000(and add an infinite amount of zero's and a 1) % of suceeding. Thus, I bring you the conclusion:

Also, say, you have zero orbital defense pods....zilch.

The mathematical resolution of Zeno's paradox, its proves its possible for a base to be completely immune (to missile or PB attacks) using Flechette Defense alone.

Or even a more radical conclusion: if I have enough ODP's (or enough to cross the threshold), it will completely be immune anyway, for all bases...since after crossing the threshold, the extra PB will have no effect on the probability being 0%.

Or if an armour value was added with the value of infinity, (or just *)....no amount of troops, even if I put an infinite amount of units (or say 10000000 to the power of 100000) of scout patrols, or even string disrupting infantry.

[livid imp]

Natalinasmpf, put your hands up and step away from the bong.

[Natalinasmpf]

Well, its just my thoughts on some radical ideas...trying to examine mathematics and such, then decided to put it in an Alien Crossfire style, (after all it is applicable)...

And no, I don't take drugs.

[livid imp]

Originally posted by Natalinasmpf
And no, I don't take drugs.

You just do a little SMAC

[Natalinasmpf]

SMAC isn't a narcotic, its a miracle, beneficial, eye-opening drug.

[Chaos Theory]

You can't have an unlimited number of Flechette Defenses - you can only have 1 per base, and they only affect their own base, and bases 2 tiles away (according to the game). Thus, with maximum packing, you could only benefit from 9 at a time, yielding (supposedly) a 1/512 chance that a PB would slip through.

What I haven't tested is whether the Flechettes fire before the ODPs, or vice versa. Clearly the unlimited-use Flechettes going first would be more desirable.

[livid imp]

While we are gushing about SMAC (the game, not heroin) How cool is a game that has quotes from Nietzsche, Kant, and Einstein.
Anyway, sorry I have nothing to say about theoretical infinite Fletchette systems.

[Natalinasmpf]

You can't have an unlimited number of Flechette Defenses - you can only have 1 per base, and they only affect their own base, and bases 2 tiles away (according to the game). Thus, with maximum packing, you could only benefit from 9 at a time, yielding (supposedly) a 1/512 chance that a PB would slip through.

Ack, misread the datalinks, it gave me the impression you could build two.

Still, infinite ODP's are possible...

[smacksim]

Definately not! How would a computer count up to infinity? You'd need an infinate amount of memory just to store the number of ODPs. Besides which, SMAC has trouble with even small numbers. I am constantly rolling over the number of formers I have active, for instance. One could make a reasonable guess that the number of ODPs maxes out at 511 or 999, or some other low number.

But still, it is interesting to imagine a perfectly defended base, Zeno or no.

[Whoha]

It depends on the coding. If every fletchette defense system and ODP has its chance independently to stop each missile, it is within the realm of freakish mathematical probabilities that a PB will get through every one.

If an ODP tries and fails to stop one(a PB), can it(the odp) be sacraficed for a 100% stop rate?

Fletchettes fire first if the manual/datalinks stuff I've read is correct btw.


as for the infinite armor stuff, psi and probe team combat rules.

[molly bloom]

Originally posted by livid imp
While we are gushing about SMAC (the game, not heroin) How cool is a game that has quotes from Nietzsche, Kant, and Einstein.
Anyway, sorry I have nothing to say about theoretical infinite Fletchette systems.

Very cool, in a non-hippy 'cool' way.


I love the fact that both SMAC and SMAX have had so much thought, care, love and imagination put into them, even when I'm only 5 years into the game, stuck between Yang in the Monsoon Jungle, Miriam with New Jerusalem in the centre of the Ruins, Santiago in the crater with a monolith, and I'm playing Lal surrounded almost entriely by fungus, with not a single mineral/arid lozenge in sight.




Yes, even then I love SMAX.

[Chaos Theory]

A computer can represent infinity very well, at least as a floating point number. Of course, infinity - 1 = infinity.

If a PB gets through your defenses, you can sacrifice an ODP to be guaranteed to stop it. Interestingly, in my meager experience with PBs vs ODPs, each PB seems to go against the normal ODP defenses *twice* and with a much higher than 50% chance to penetrate each one. The result it that it often takes deploying ~10 ODPs to stop a PB. If I don't have enough left, I have to sacrifice one, not that it's much of a cost.

I do not know whether you must decide to sacrifice an ODP before Flechette defenses kick in.

[smacksim]

Originally posted by Chaos Theory
A computer can represent infinity very well, at least as a floating point number. Of course, infinity - 1 = infinity.

Faking it is fine, and would work, but actually counting to infinity is a proposition that even quantum computer models dare not tread upon....

I suppose SMAC could/should say 'They have over 1000 ODPs, so I'll give them infinite ODPs from now on', in which case, a PB would never get through, even if it dodged every single one. This is because each check takes some non-zero time, thus, with infinite checks (if it were to keep dodging the bullet), yields infintite time. A stalemate on the PB...

[Ari Rahikkala]

or enough to cross the threshold of infinity into the surreal number category (where adding +1 no longer matters, same effect).

Show me where this threshold is. Heck, as smacksim said, infinity - 1 = infinity. If you add 1 to a number googolplex times, then take 1 from it googolplex times, you end up with 0 - not infinity. A googolplex is much larger a number than you'll ever have any sort of a need for - you couldn't even seriously use it anyway since there are more zeroes in it than there are elementary particles in the Universe... still, it's just a number. A real number (actually, a natural number, too). It's not infinity, not even close.

You're welcome to, say, define a mathematical method for determining how big a number should be "infinite" for the purposes of a specific subject. After all, it *is* true what you said about being overweight (well, except that a person whose weight is infinite by any sane consideration would be very much dead), but only when you happen to consider that specific subject.

14:23 < natalinasmpf> well, to me, so far, I'm guessing infinity is a state of a number....after one crosses a threshold, lets say, x...x+1 has the same effect...although not the same value
14:23 < natalinasmpf> so if you have infinite weight, you would be very obese, and it would be life threatening, but if you gained an infinite pounds more, it wouldn't affect your health any further
14:23 < natalinasmpf> the threshold has been crossed!

However... please do understand that mathematics itself has no such thresholds nor need for them.

(I'm putting this line here because I noticed I'd already used three different kinds of emphasisation in this message. God, what snobbery :/. )

[Chaos Theory]

In computers, if you're working with floating point numbers, eventually x = x+1. At this point, x ~= 2^52 for IEEE double-precision numbers. You could consider x to be infinite if only adding and subtracting 1 were allowed, though I'm not sure if this x also has the property that x = x-1. Whether or not that holds should be in the standard, but I don't care to check.

You also have the real floating-point infinity that you get if you add a sufficiently large number to the largest non-infinite number, or continue multiplying a number by a large enough factor. At that point, x - y = x, if y != infinity, and x - infinity = NaN. Any math ops on NaN yield NaN, except a handful of comparisons (NaN stands for Not a Number, the result of ill-defined operations like 0/0 or infinity - infinity or a representation of missing data).