Paradoxes: temporal, logical and otherwise.

[Nigh Invulnerable]

To answer the unstoppable force/immovable object "paradox": deflection.

Also, Epicuros' paradox about God only works if you assume God operates according to human standards of morality (good/evil). Some of us say God is God because he sees things from a perspective we can't even comprehend.

 

[The_root_of_all_evil]

I gave you an entire thread full of proofs.

Which I disagree with because you're using your own stated proof as part of the solving equation. That's paradoxical in itself.

But hey, seen as you didn't bother to read my post last time.

See, that's what the problem is. I actually read it - found the problem - and discarded it. You're the one who is being aggressive about YOUR proof, because the onus of proof relies on you.

Here it is again.

Now sum(from k=1 to n) (9 * 1/10^k) is a finite sum - No proof available.

, and so we can calculate that
|1 - sum(from k=1 to n) (9 * 1/10^k)| = |1/10^n| - Assumption - Working not shown.

Try proving that e=0 from first principles.

all you have to do is stop mesuring, and all is solved

But why does the act of measuring corrupt it? Infinite decimals being used with finite operands.

That would mean that Achilles is slowing down until he becomes equal in speed to the tortoise.

If Achilles is running at a constant speed and not slowing down, then f**k you Zeno.

Won't actually matter what speed he's running at, because you're ignoring speed for distance - and marking distance relative to lesser increments of time.

In other words - if Achilles catches the tortoise within 2 seconds, you begin looking further and further into those 2 seconds - upto an infinite point where they cross - and you physically can't count to infinity - you just have to reach a point (given by the infinite decimal series) where you can claim it's equivalent.

If you simply count/measure 1 : 2 : 3 , Achilles zooms past the tortoise.

 

[Valkyrie101]

Also, time travel theories: if we accept the concept of multiple universes, where each action we could possibly take branches off into another universe, then when you travel back in time and say, kill yourself, what you're doing is breaking off the time-line of your universe, and have essentially forcefully created a new universe: one where you are both dead and alive. You can never get back to your universe, you have killed your time-line and all the actions that could have taken.

No, there is no actual conclusion in that.

Define action, though. In cosmic terms, what constitutes an action?

It's generally accepted as the choice between walking left or walking right, talking to someone or ignoring them. Though, theoretically, it could extend hundreds of levels up and down in scale. I'm not too sure.

The problem is that left and right are subjective. They don't actually exist. Besides, there are different degrees of left and right, theoretically infinitely so. Also, one has to consider different velocities.

And then, it all gets very, very messy indeed.

 

[Maze1125]

I gave you an entire thread full of proofs.

Which I disagree with because you're using your own stated proof as part of the solving equation. That's paradoxical in itself.

What? That doesn't make sense in context, at all.

I gave you a thread full of proofs, all of which were correct independently.

So how on Earth does that lead you to conclude that "you're using your own stated proof as part of the solving equation."

But hey, seen as you didn't bother to read my post last time.

See, that's what the problem is. I actually read it - found the problem - and discarded it. You're the one who is being aggressive about YOUR proof, because the onus of proof relies on you.

If you found the problem, why didn't you mention it?

And, as I said, the thread was full of proofs. I'd like to see your problem with each one of then.

Here it is again.

Now sum(from k=1 to n) (9 * 1/10^k) is a finite sum - No proof available.

What? How you do even require a proof of that. a 4 year old could see that was finite.

Anyway, I'll do it for you anyway.

n is finite. There are n terms in the sum, therefore there are a finite number of terms in the sum.

For each k > 0, 0 < 9 * 1/10^k < 1, therefore each term is finite.

Every term is finite and there are only finitely many terms, therefore the sum is finite.

, and so we can calculate that
|1 - sum(from k=1 to n) (9 * 1/10^k)| = |1/10^n| - Assumption - Working not shown.

Okay, they literally teach you how to add and subtract decimals when you're 12. And you expect me to believe that you've done 4 years at university when you can't even subtract a decimal from 1?

Oh well, I'll try and teach you what your school obviously failed to.

For k=1

) 1.0
)-0.9

(I'm using the ")" to maintain the format)

First we subtract the right most terms.
Unfortunately 9 > 0, so what we do is "borrow" a 1 from the next column. Allowing us to do 10 - 9 giving us a 1 in that column. And it turns out that we borrowed everything that was available in the left most column, leaving us with 0 - 0. So, the final result is:

) 1.0
)-0.9
)-------
) 0.1

For k=2 we have:

) 1.00
)-0.99

This time with have to borrow across two columns, which still gives us the 10 - 9 in the right most column, and leaves us with 9 - 9 in the middle column and 0 - 0 in the left most, so the result is:

) 1.00
)-0.99
)--------
) 0.01

Now, for k = n, we have to borrow across n columns, but the principle still works, Giving:

) 1.00...00
)-0.99...99
)-------------
) 0.00...01

(Note in this case the ... stop with another number, indicating the chain is finite.)

Now, if you have any more problems, please take them up with your 7th grade maths teacher.

Try proving that e=0 from first principles.

What? You never need to prove e = 0. In fact, e cannot be 0, it is specifically designed to be any positive number that isn't 0.

I have to wonder is you've ever done a limit proof before.

And as for first principles, I proved it using nothing but finite addition and subtraction, limits and the definitions. How much more first principle could I get?

 

[The_root_of_all_evil]

What? That doesn't make sense in context, at all.

Just because you don't understand something doesn't make it wrong.

I gave you a thread full of proofs, all of which were correct independently.

And all rely on the underlying assumption that you're heading towards your own soloution, while ignoring points that don't fit your soloution.

For a start, you're actually proving that e=0 for your solotion to work - given that you're saying that 1-0.9 recurring =e.

So how on Earth does that lead you to conclude that "you're using your own stated proof as part of the solving equation."

Because for each assumption you make without a logical step behind it, you're ignoring potential problems that may come from other assumptions. Pure Mathematicians never "assume" anything, they always provide proof.

If you found the problem, why didn't you mention it?

There's numerous areas of problems. The word "assume" is one of them. You're also using the same equation multiple times as proof of each other, which is a no-no as well.

What? How you do even require a proof of that. a 4 year old could see that was finite.

I doubt they could. And the age range doesn't matter. If you're proving something from first principles, you have to show every step.

You've probably run into trigonometry, irrational numbers, differentials or a number of other areas of maths where you can't just say that N=M, because of the use of finite operands on certain series of numbers. The Fibonacci series for one.

Anyway, I'll do it for you anyway.

n is finite.

But it isn't, is it? Same problem again. n is infinite. It's an infinite series.

From your own line 1: (as n->infinity)

What you're doing, each time, is moving infinite counting into finite counting so that you can ignore that last little minute differential. That's what makes it equivalent to rather than equal than.

How much more first principle could I get?

I could be asking you to define the number line. I think we can both assume that.

And as for the inferred insults against my education, I'd advise you to take more time checking your own working before you pour scorn on mine.

 

[HK_01]

snip

You're still at it? Come on, it's getting old. Also, you're losing the argument, Root of All Evil.

 

[RikSharp]

There is one barber in a town who cuts the hair of everyone who does not cut their own hair. Who cuts the barber's hair?

Himself, you never said that he only cuts everyone else's hair.

That's the paradox. The barber cuts the hair of everyone who does not cut their own hair. So therefor if someone cuts their own hair he does not cut it. So if he cuts his own hair he can't cut his hair... and on and on

or his hair is cut by a guy who is not a barber that cuts his friends hair and vice-versa thus eliminating all three characters from the equation.

 

[RikSharp]

actually, didnt think that through, the above reply was pure nonsense. try this one:

the barber is bald and has no need for hair cuts

 

[The_root_of_all_evil]

snip

You're still at it? Come on, it's getting old. Also, you're losing the argument, Root of All Evil.

It's not even started getting interesting yet.
And the best wars need to be lost before they can be won.

 

[Maze1125]

For a start, you're actually proving that e=0 for your solotion to work - given that you're saying that 1-0.9 recurring =e.

That is completely wrong.

e is an arbitrary number greater than 0. It could be 15, it could be pi, it could be 1/phi[sup]500000000000[/sup]. Anything. So long as it's a positive number greater than 0.

This is the founding principle of limit proofs, and you don't seem to get it at all.
Which, again, makes me question whether you've ever actually done one rigorously before.

Because for each assumption you make without a logical step behind it, you're ignoring potential problems that may come from other assumptions. Pure Mathematicians never "assume" anything, they always provide proof.

I didn't make any assumptions in my proof.
I used definitions and inferences. I occasionally skipped very easy intermediary proofs, but the only assumption there was that you would know enough maths to do the calculation 1 - 0.99999 and the subsequent ones with increasing numbers of 9s.

There's numerous areas of problems. The word "assume" is one of them. You're also using the same equation multiple times as proof of each other, which is a no-no as well.

Please provide an example of this. To quote yourself, just saying it doesn't make it true.

If you're proving something from first principles, you have to show every step.

Only if you're proving it as part of an exam. When you're proving it to another person it's perfectly okay to skip steps that you can reasonably assume they already understand. And if there's a problem, they can ask you to clarify.

You've probably run into trigonometry, irrational numbers, differentials or a number of other areas of maths where you can't just say that N=M, because of the use of finite operands on certain series of numbers. The Fibonacci series for one.

And? So because something doesn't work for them, it can't work for this? That's an incredible fallacy.

Anyway, I'll do it for you anyway.

n is finite.

But it isn't, is it? Same problem again. n is infinite. It's an infinite series.

From your own line 1: (as n->infinity)

Yes, n tends to infinity. That means it isn't currently infinity.

This is basic limits, the whole point is that you only work with finite numbers and sequences, and from that derive the limit. The definition of a limit never uses anything infinite.

n is purposely finite. It has to be so. If it was already infinite, it couldn't tend to infinity.

The definition of a limit is thus:

For a sequence a[sub]n[/sub].

L = lim[sub]n->infinity[/sub] a[sub]n[/sub] iff for all e > 0 there exists N such that for all n > N |L - a[sub]n[/sub]| < e

There isn't a single piece of infinity in the definition. The infinite limit is defined from finiteness.

What you're doing, each time, is moving infinite counting into finite counting so that you can ignore that last little minute differential. That's what makes it equivalent to rather than equal than.

No I'm not. I'm not touching the infinite counting at all. I'm using limits and finiteness. That is precisely why my proof is rigorous. Because I don't touch infinity at all. I just work from the definitions.

 

[Kraiiit]

What would happen if an unstoppable force hit an immovable object?

Then the Joker says, "You truly are.... incorruptible, aren't you?"

 

[IrradiatedFish]

If I told you "Don't do anything I tell you to.", what do you do?

Doing what I told you to do wouldn't be doing what I told you to and vice-versa

...Wait what?

 

[Maze1125]

If I told you "Don't do anything I tell you to.", what do you do?

You don't do what they tell you to, at that point, by doing something they tell you to in the future.

 

[The_root_of_all_evil]

No I'm not. I'm not touching the infinite counting at all. I'm using limits and finiteness. That is precisely why my proof is rigorous. Because I don't touch infinity at all. I just work from the definitions.

This is why you fail.

You've already accepted a infinite limit rounds to a finite point before you've even started writing the equation.

That's your assumption, and I could prove anything given that I can define the terms beforehand.

There isn't a single piece of infinity in the definition. The infinite limit is defined from finiteness.

See? Please god, tell me that you see.

It's a radix point of numbers that their last trailing decimal is indistinguishable from the next one. It directly ignore the infinite differential.

That's why Zeno's paradox works. Because it constantly alters the limiting point, so you can't use a infinite number series to calculate it.

If you are defining an infinite decimal series to only use finite numbers, you're automatically assuming that there is a finite number beyond which infinity can be reached. That is clearly untrue.

The number e (2.71828182845904523536...) is a prime example. Despite being irrational and transcendental, it's still regarded as being defined by an infinite limit, despite all of it's places not being defined.

lim(n->infinity) of (1+1/n)^n

As it's an assymptote, it can be assumed to be close to the point of infinite trials (like the Bernoulli trials), but it never actually equals them, because each point gives the increasingly small chance of a disruption.

However, if you're trying to reach the point where there will no longer be ANY chance of a trial change, then you can't ignore the possibility. You can just label it statistically insignificant.

 

[Kjakings]

Edit: if anybody mentions the chicken and the egg, I'll be forced to kill you: evolution provides us with the answer: the creature we consider 'chicken' was obviously hatched out of an egg by its evolutionary ancestor divergant enough to NOT be considered 'chicken.' Therefore, the egg was first.

Ah, but that depends on how you define a chicken egg.
Is a chicken egg an egg with a chicken inside, or an egg laid by a chicken?
I say both are requirements for a chicken egg.
Surely the first chicken egg would have to be the first egg to be laid by a chicken and thus contain a chicken. As such, the chicken had to exist in order to lay a chicken egg.

I believe the egg you describe as having come first, is not a chicken egg at all, but rather the egg of a "pre-chicken" which contained a mutated offspring which we now refer to as a chicken.

I did mention this qualifier on your definition of 'chicken egg' at some point, I'm sure of it.

 

[Maze1125]

If you are defining an infinite decimal series to only use finite numbers,

There is no "if" about it.
That is how they are defined.
If you define them some other way then you are not talking about infinite decimals.

It's quite possible that your construct that you've made up in your head doesn't equal 1, but that construct is not 0.999...

But, hey, if you want to disagree with all mathematicians on how to define infinite decimals. Then, please, tell me what your definition is.

 

[ClassicJokester]

All can add is catch-22, which is essentially the same thing.

catch 22 is lame.
i must be insane to want to fly up there but if i'm insane, i cant fly up there. therefore i'm not insane and can fly up there, etc, etc
missing the obvious: you could fly up there and not want to, making you both fly and not insane.

Actually, that's not quite what Catch-22 is.
Catch-22 is like this: I cannot fly any more missions if I'm insane, but to be considered insane, you have to say that you are, and ask to be grounded. The act of asking proves you to be sane (you would HAVE to be insane to fly more), so I have to fly more missions.

 

[The_root_of_all_evil]

If you are defining an infinite decimal series to only use finite numbers,

There is no "if" about it.
That is how they are defined.
If you define them some other way then you are not talking about infinite decimals.

.9 recurring is an infinite sum that cannot be used in finite operands. It is an unmeasurable point, and therefore has to be taken to the next measurable point, which is its closest integer. You are placing finite limits on an infinite sum in order to work on it.

And let's just have a look at that in action.

But, hey, if you want to disagree with all mathematicians

As usual, you're assuming, rather than proving.

Simply put, .9 recurring < 1 because an infinite series can't equal finite. It can be equivalent to, by the use of an infinite decimal series, but that's just rounding off a statistically insignificant point.

Infinity by definition is a concept that refers to a quantity without bound or end.

And if I take a bite out of an apple, I still have an apple left. Maths is only as accurate as it allows itself to be.

 

[Lombax302]

Traveling backwards in time is my favorite.

Asuming that time is linear (and that time travel is possible), traveling back in time is a bit of a mind fuck. Because if you go back in time to do something, there won't be a reason for you to go back in time when you did because there is you already did what you did. So whatever you have done will not happen because it happened. Kinda. Confusing stuff.

after changing what you were going to change you go forward and tell yourself to go back and change it. if you do that then it makes perfect sense.

True. Moving on to the classic "kill your grandparrent" example, no shit would make sense. If you would kill say your grandpa before your parrents where born, you would have never been born to go back in time to prevent yourself from being born. Can't think of anything to make that make any sense at all (still assuming linear time and the possibility of time travel).

You can't go back in time and kill your grandparents prior to the conception of your parent, you will be thwarted in any attempt assuming time is linear and time travel is possible then any actions in the past conducted by a future interloper have already occurred. Your grandparent didn't die prior to the conception of your parent therefore you will never go back and kill them.

In that sense the future is written aswell as the past - in which case, isn't everything inevitable - the future is written on the deeds of the past. I know some people will throw in "chaos theory" here but isn't that just a poor excuse of lack of knowledge in a Newtonian model, yes a butterfly flapping it's wings in one hemisphere could cause a hurricane in another but if we had all the relevant data up to that point we would be able to accurately predict it. A person will react in a given situation based upon the knowledge they've collected in their lifetime - whether that event happened ten years ago or ten minutes in the future the outcome is set. What has happened, happened and what will happen, will happen - just try to look surprised.

Assuming time is linear that is.

to further the killing of a grandparents discussion, u wouldn't even get a chance to kill them because if hypothetically u did kill them, u wouldn't be around to contemplate killing them in the first place. so if u do end up contemplating killing ur grandparents, that means that u never actually do kill them because ur still there. so basically if you are in the position to change something, u won't because then it would have happened already. (i love talking about this all day) so going back in time isn't actually changing something in the first place, but rather fulfilling what has already happened. and if u do change something u would cease to be in your original time line and be in another time line. yes, there are infinite time lines and this is due to the existence of choice. if i hesitate typing the letter A to the letter B for one second more than i really am, that is creating a completely different time line with it diverting within that second. now consider that there is an infinite amount of options of when i type A from B, so whether it's hesitating for a second or 57/1000000ths a second, an infinite amount of timelines diverged in that one second. now wrap ur head around putting that model onto all of history.

 

[Maze1125]

I'm beginning to suspect you're just trolling me.

After all, you make statements like this:

.9 recurring is an infinite sum that cannot be used in finite operands.

And then go on to criticise me for "assuming, rather than proving."
So, either you're a troll, or a hypocrite of the highest degree.

Why exactly can't an infinite sum be used in finite operands?
After all, the definition of an infinite sum is one of limits, and the definition of a limit is nothing but finite.

The sum

sum (k = 0 to infinity) a[sub]k[/sub] is nothing more than lim(n->infinity)sum(k = 0 to n) a[sub]k[/sub]. Which can easily be dealt with using limits and finite operations.

The only reason they tell you in courses such as physics or, for example, computer science, that you cannot use the two together is because doing so requires a level or rigour and precision that you will never be taught in that course.

I mean, just look at you.
You were asked to prove that the numbers were different and your response was this:

Easily. 0.99999 recurring < 1.

Not a single bit of precision or rigour.

I ask you to give your definition of what an infinite decimal is and you just dodge the question, and don't even seem to be aware how infinite sums or limits are defined.

Your mathematical rigour is non-existent, yet you constantly criticise me for my supposed lack of the same.