Paradoxes
Talking about (time) Paradoxes and how could you guys miss the Grandfather Paradox :
-wikipedia
Their is a resolution to it involving parallel (infinite?) universes where you travel back to time in some other stream of time and kill the granddad of your clone(?) and not you.
| the time traveller went back in time to the time when his grandfather had not married yet. At that time, the time traveller kills his grandfather, and therefore, the time traveller is never born when he was meant to be. |
Their is a resolution to it involving parallel (infinite?) universes where you travel back to time in some other stream of time and kill the granddad of your clone(?) and not you.
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Religious folk turn off your computers now.
Given:
(1) "God invented humans."
(2) "Eve is the mother of all humans." (Correct me if I'm wrong)
(3) "Necessity is the mother of all inventions."
(4) "Eve is human."
Deduction:
(*) (1) => Humans are an invention.
(**) (*)&(4) => Eve is an invention.
(***) (*)&(3) => Necessity is the mother of all humans.
(**)&(3) => Necessity is Eve's mother.
(***)&(2) => Necessity and Eve are the same thing.
Conclusion:
Eve is her own mother.
Wut.
Given:
(1) "God invented humans."
(2) "Eve is the mother of all humans." (Correct me if I'm wrong)
(3) "Necessity is the mother of all inventions."
(4) "Eve is human."
Deduction:
(*) (1) => Humans are an invention.
(**) (*)&(4) => Eve is an invention.
(***) (*)&(3) => Necessity is the mother of all humans.
(**)&(3) => Necessity is Eve's mother.
(***)&(2) => Necessity and Eve are the same thing.
Conclusion:
Eve is her own mother.
Wut.
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0. Genealogical trees contain no cycles (graph theory).
#2 contradicts #0. By principle of explosion, anything can be proven.
#2 contradicts #0. By principle of explosion, anything can be proven.
What about compactification? I'm no mathematician, but the idea of approaching infinity and somehow ending up approaching 0 from negative infinity and vice versa sounds interesting.
Jumping from positive to negative without crossing 0 counts as a paradox?
Jumping from positive to negative without crossing 0 counts as a paradox?
Only if you think it's paradoxical that the speed at which the minute hand on a clock gets farther from 12 changes instantly from x to -x as it passes 6.
(Technically clocks move in rather complex ways, but let's ignore that for this analogy.)
A true paradox in mathematics is Russel's paradox in naive set theory. It was so bad, ZFC was developed.
(Technically clocks move in rather complex ways, but let's ignore that for this analogy.)
A true paradox in mathematics is Russel's paradox in naive set theory. It was so bad, ZFC was developed.
Oh right, the whole changing directions... Very interesting readings, Russel's paradox and ZFC.
We have a vase and small cards with numbers writen on them. At one hour before midnight we place ten cards with numbers 1–10 into vase and remove card with number 1. At half a hour before midnight we place cards with numbers 11–20 into vase and remove number 2. At 1/3 hour before midnight we place numbers 21-30 and take number 3. At 1/4 hour before midnight we place numbers 31-40 and remove 4. And so on.
How many cards will be in the vase at midnight?
How many cards will be in the vase at midnight?
That looks like a variation of Zeno's paradox with Achilles and the tortoise?
http://en.wikipedia.org/wiki/Zeno%27s_paradoxes#Achilles_and_the_tortoise
So we tend to remove an infinity amount of cards before reaching midnight. In fact we never reach midnight. At the same time, for the same reason, we add an infinity amount of cards. Right?
http://en.wikipedia.org/wiki/Zeno%27s_paradoxes#Achilles_and_the_tortoise
So we tend to remove an infinity amount of cards before reaching midnight. In fact we never reach midnight. At the same time, for the same reason, we add an infinity amount of cards. Right?
It isn't related to Zeno's paradox. (Which have errors in formulation. It is always used as example in schools when we learning limits)
Martin Gardner proves that correct answer is 0: We have infinite number of natural numbers and we made infinite number of operations, so each natural number will be removed. At same time logic tells as that there should be 9 more times cards inside than we removed.
It is not paradox if we will remember that we cannot manipulate and compare infinities like numbers.
And also we might notice that if we would name set of placed cards as X and set of removed cards as Y, then |X| = |Y|
EDIT 2: It is called Tristram Shandy Paradox.
http://en.wikipedia.org/wiki/Paradoxes_of_set_theory#The_diary_of_Tristram_Shandy
I like set theory paradoxes for some reason
EDIT: Another one showing that infinity is not a number:
If I won infinite money in lottery, how much taxes should I pay?
Martin Gardner proves that correct answer is 0: We have infinite number of natural numbers and we made infinite number of operations, so each natural number will be removed. At same time logic tells as that there should be 9 more times cards inside than we removed.
It is not paradox if we will remember that we cannot manipulate and compare infinities like numbers.
And also we might notice that if we would name set of placed cards as X and set of removed cards as Y, then |X| = |Y|
EDIT 2: It is called Tristram Shandy Paradox.
http://en.wikipedia.org/wiki/Paradoxes_of_set_theory#The_diary_of_Tristram_Shandy
I like set theory paradoxes for some reason
EDIT: Another one showing that infinity is not a number:
If I won infinite money in lottery, how much taxes should I pay?
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| We have infinite number of natural numbers and we made infinite number of operations, so each natural number will be removed. |
The answer "0" comes from believing that
(lim 10n) - (lim n) = 0
when in fact it's an ill-formed expression, as it's subtracting infinities.
The answer "∞" comes from believing that (lim f) - (lim g) = lim (f - g) even when neither limit is finite.
| The answer "∞" comes from believing that (lim f) - (lim g) = lim (f - g) even when neither limit is finite. |
Let's say it is number X∈N. But it will be be removed on X-th iteration. So no numbers from N are contained in vase.
Name any number n that has been removed from the vase. After that number was removed, the vase contained 9n other numbers.
You can't reach an answer about the final state of an infinite non-trivial process by citing states at finite times.
As another example: A light switch starts in the ON state and it's flipped every time a card is removed from the vase. What's the state of the switch at midnight?
You won't know by looking at the switch at any time before midnight.
You can't reach an answer about the final state of an infinite non-trivial process by citing states at finite times.
As another example: A light switch starts in the ON state and it's flipped every time a card is removed from the vase. What's the state of the switch at midnight?
You won't know by looking at the switch at any time before midnight.
| What's the state of the switch at midnight? |
Whole message of that paradox is that you cannot apply common logic and arithmetic operation to infinite sets.
http://sguthrie.net/infinity.htm
| The implication here is that since any number added to infinity is still infinity, then the principle that the whole is greater than the parts does not apply here. One component of the equation (X0) is quantitatively equal to the sum of both components (X0 and 1). Russell asserts that given an infinite number of years to write plus the infinite number of days obtained results in an infinite amount of time transpired. Thus, the amount of time to write if obtained would be equal to the amount of time given to write about. Therefore (d = days to write on; y = years to complete; t = time obtained) ... Russell believes that when the presence of infinity is seen all at once, then the concept of infinity is something that can exist as a quantitative property |
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is this a paradox or a fallacy? If pigs fly, then you will understand the Chernoff Bound
so its true because pigs dont fly but if pigs did magically fly then you still wouldnt undersand the chernoff bound
so its true because pigs dont fly but if pigs did magically fly then you still wouldnt undersand the chernoff bound
It should be assumed that the least restrictive variable be the dependent variable - therefore, pigs flying depends on understanding the Chernoff Bound.
| is this a paradox or a fallacy? If pigs fly, then you will understand the Chernoff Bound |
Logically, the fact that the sky is green implies that it is blue, even though causally this makes no sense. It also implies that I'm Superman.