Zeno's Paradox and Flechette Defense

[Ari Rahikkala]

Yep... that's why I said "thresholds of infinity" for specific subjects - such as computing with floating-point numbers (though I don't think the amount of buildings of a type in a base is stored as one ) - are just fine, but you shouldn't confuse stuff like that with just plain normal mathematics.

[smacksim]

That eventually x=x+1 is a convention, like saying 'the sky is the limit' to mean there is no limit. It's completely arbitrary. Remember that it was Bill 'I have 40 Bills' Gates that once said 'Noone is ever going to need more than 512k', which led to some FAT problems, etc... Then there was the problem with processor design and floating point numbers not really being naturally big enough for some of the sciences, hence things like 'double' or 'long'. Computers represent limits to possible numbers, which is opposite the concept of infinity after all. That sufficiently big numbers can be fudged is fine, until you need more than 512k, for instance....

I do wish SMAC could count a bit higher. I think it's somewhere around 300 units of a type before a rollover (that's what I get with clean formers at least). Also, wouldn't it be nifty if SMAC had a reports system as seen in some other strategy games? Even being able to filter the F4 screen by base size/economy/production/drones would be nice. I don't think they anticipated people would have such huge empires, but it would be nice if there is ever a SMAC 2. SMAC is math rich, and would do well with more detailed analytic capabilities. ...Oh, that's what we had Mari One for, never mind

[Natalinasmpf]

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.

Supposedly infinity could have some other threshold not known, but an infinite value can increase, just not increase its effect after crossing said threshold. Suppose I had a box with infinite weight, so after increasing it by infinite kg, it wouldn't take any extra force required to move it (ie. having a force past the threshold can move anything after the threshold, even if the value is 10 times that of the threshold)...

Just wild speculation though. I was just imagining, that sooner or later with a more advanced game, we can reduce that 1/512, 1/1024, 1/2048, 1/4096 chance, etc. to completely zero.

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

Okay, I was just having wild theories...

You'd need an infinate amount of memory just to store the number of ODPs.

I already said IF we had such.

[Ari Rahikkala]

Allright, so nobody here wants to think like a mathematician. Fine then, but at the very least I'll protest this...

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

Surreal numbers are *definitely* a mathematical definition. If you were to say "to be big enough to be considered infinite by the computer", that would be fine, but... I'm a purist and I'm going to keep protesting when you try to bring into mathematics concepts that might work elsewhere but are definitely wrong inside mathematics.

Surreal numbers are a superset of real numbers, BTW, so what you really meant with that quote is that a number isn't real anymore. In essence, you assert that there exists such a number x, which is a real number, that x + 1 is not a real number. Mathematicians don't use bongs .

[Natalinasmpf]

Actually I was just using a crude way to replicate infinity on a primitive little 32-bit/64-bit processing machine, although we might not have to use this in the future should computers be later get a bit more advanced...

And I was just bringing terms off the top of my head....I like the threshold idea for some reason, and all the previous url's you gave me the other time were too much info to take in at once, so we can discuss it string by string to prove me wrong, now. (And then also resolve my crazy ideas of confusion) ).

[smacksim]

Originally posted by Ari Rahikkala
In essence, you assert that there exists such a number x, which is a real number, that x + 1 is not a real number. Mathematicians don't use bongs .

/me tries to remember:

+ + != (bong + idea not equal to cash)

rather: + = (bong + math = dizzy)

[Ari Rahikkala]

Allright, let's get back to the original post, then.

First, you're essentially multiplying infinity by zero (infinite PBs with zero chance to hit). But the answer's obvious, isn't it? Well, I'm sorry - that's not true. Let's look at it through two different, but IMO just as "obvious" operations:

lim x*0 as x approaches infinity: Any real number, no matter how big, times zero is zero, so the result is zero.

lim x*inf as x approaches zero: Any real number, no matter how small, times infinity is infinity, so the result is infinity.

Choose some other terms, and you can give the result of that operation further different values. That's why the value of zero times infinity isn't any of those values, but instead indeterminate. See http://mathforum.org/dr.math/faq/faq.divideby0.html (and note that when you're breaking the rules already, it's trivial to show that 0/0 is the same operation as inf*0).

Oh, and note that the definition I gave is the definition in calculus. In algebra, 0/0 is simply undefined and that's all there is to it.

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

0.[infinite number of zeroes]1 actually *is* zero for real numbers. After all, you never come across the 1, no matter how far you go.

Actually I was just using a crude way to replicate infinity on a primitive little 32-bit/64-bit processing machine, although we might not have to use this in the future should computers be later get a bit more advanced...

What do you mean? As Chaos Theory said, IEEE floating-point numbers already support representing infinities. Of course you could do better and make a surreal number data type, but I'm not sure how many programmers would know what to do with it...

Finally, Zeno's paradox doesn't really have much to do with this. We haven't actually dealt with geometric series, which is what the original paradoxes were represented as. But, just to throw in an example...

There are two bases left in the world, Plex Anthill and Free Drone Central, locked in a bitter war. Plex Anthill is creating one planet buster a turn - that's the reason why there are no other bases left - while Free Drone Central is defending itself by building one orbital defense pod per turn. All satellites were recently wiped out of orbit, and Plex Anthill's production was just sabotaged, so it just happens that both of them are starting with a completely clean slate. The bases are close enough to each other that a planet buster from Plex Anthill can fly to Free Drone Central in one turn. Given an infinite amount of turns, what is the probability of Free Drone Central surviving the buster onslaught for all eternity?

The probability of a planet buster hitting on any specific turn n is P(n) = 1/(2^n) (since each ODP halves the possibility of a buster getting through and we're assuming that the Free Drone player is future-conscious and never sacrifices pods). That's a simple converging geometric series. Use it as the probability function of a geometric distribution. You'll note that geometric distributions are normalised, meaning that we can just ignore the rest of this calculation and state that the probability of the Hive getting a buster through is exactly 1, so the Drones have no chance of surviving. In fact, I'm pretty sure there's a way to prove this that's even more direct. The fact that 1/(2^n) defines a converging series (its sum is 2) doesn't affect this, contrary to my original expectations (I honestly spent quite a while figuring that out - I'm almost ashamed for it).

[Natalinasmpf]

Ari hit what I wanted to say (for the example), as well as clearing up some mathematical misconceptions of mine.

*applause*

Which brings me to another thought, though: do you get clean minerals limit decreased for every PB you launch or attempts to detonate, or every PB that actually is successful? Because if the Hive detonated an infinite amount of PB's (if clean minerals decreased for every PB launched/detonated successfully or unsucessfully), this would mean an infinite amount of ecological damage wouldn't it?

After all, you never come across the 1, no matter how far you go.

But it is an infinitesimal, which is quite different from zero.

Although if you use my weird threshold crazy idea, theoretically, one would come across the zero after crossing a threshold, but pushing it even farther back would have no further effect.

[Ari Rahikkala]

I'm still not convinced that I'm right, though... two things about that solution vex me:

1) The probability function, P(n) = 1/(2^n). I haven't proved that that's the right function. However, doing so should be a simple matter...

2) More importantly... I still can't "accept" that conclusion. I think I got confused by the normalisation thing. I mean, I'm asserting here that it doesn't matter how quickly the probability of an event occurring drops toward zero (and how close it started), if you repeat it infinite times the probability of... erm... *something happening* becomes 1.

I'll check item one first... and as for item two, well, I can think of different approaches... perhaps one of them is more helpful than this.

But it is an infinitesimal, which is quite different from zero.

Ayup, but I was talking about real numbers, and real numbers don't include infinitesimals.

[Natalinasmpf]

and real numbers don't include infinitesimals.

For some reason, I have a strong conviction/gut feeling infinitesimials and infinites are real numbers (as in, certainly not in the same category as the square root of minus two!).

I have no idea why I think that. I think because I was told something on that subject before, but I forgot...

[Ari Rahikkala]

OK then. Let's take a statement ("infinitesimals are real numbers") and disprove it, then, that's how you do things in mathematics . I'll use a counterexample.

In real numbers, when you substract a nonzero number from any number x, the resulting number is not equal to x. I'll assume you agree. Now let's substract an infinitesimally small number, 0.[an infinite number of zeroes]1 from 1. The result is, intuitively, 0.999.... (*)

Let's give that value to a variable y, and explore the value of that variable a bit.

y = 0.999...
10y = 9.999... (since the string of nines is infinite, it hasn't gotten any shorter on the "other" end - the other end does not even exist!)

10y - y = 9y
Or, with numbers: 9.999... - 0.999... = 9
9y = 9
y = 1

(This is, by the way, a very common operation for converting repeating decimals into (much nicer) fractions. If the period of repetition is longer, you choose a larger power of ten to multiply with. Something like 5.125125125... you'd multiply with 1000, for instance.)

So, the result is that y is both 1 and 0.999... here at the same time. Now you can either disagree with me on that substracting numbers from numbers changes them (i.e. what I assumed you'd agree to in the beginning), disagree with me on some other point, or concede that the infinitesimal 0.[infinite number of zeroes] is actually zero here. It's your choice, but try to be sane about it .

(*) No ellipses here . The first three dots mean that the nines just go on and on forever, and the last dot ends the sentence.

[livid imp]

Nerds!

[Chaos Theory]

Originally posted by smacksim
Also, wouldn't it be nifty if SMAC had a reports system as seen in some other strategy games? Even being able to filter the F4 screen by base size/economy/production/drones would be nice.

You can - go to the F4 screen and look at the buttons on the lower left. Play with them. Enjoy!

[Ari Rahikkala]

Right. Here's part 1:

Each ODP has a 50% chance of failing. Each test for whether an ODP catches a PB is a Bernoulli trial. We want to test if *all* n trials fail, and for that, we will use a binomial distribution.

P_p (n|N) = (N over n) * p ^ n * q ^ (N - n)

p=0,5
n=N

P_0,5 (n|n) = 1 * p ^ n * q ^ 0
= p ^ n
= (1/2) ^ n
= (1^n) / (2^n)
= 1 / (2^n)

... which is where we wanted to get. I know I'm being a bit more verbose than I really need to be, but I think it's important that natalinasmpf and everyone here should be able to follow easily...

[Ari Rahikkala]

Those alternate methods don't seem to be helping, and I am afraid I was right initially. That's all I can say here. That's how daunting mathematics is.

(of course, the other possibility I failed to mention here is that I was still just confused... and I think I'm still confused at the time of this edit...)