I find it quite interesting to discuss on how one to the power of infinity is assumed to be undefined. We all know that if we take one to any number, it will always be one. But with the limit, we can also assume that 1 ^ infinity is equal e (with the lim of (1+1/x)^x as x gets increased without bound. All we have to do is subtitute the x with infinity [as it is allowed in finding the infinity limit in normal circumstance] and you will end up getting 1^infinity as its final result). Is this enough to justify that 1^infinity is equal undetermined?
One to the power of infinity
- Thread starter embargiel
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When you have something like "infinity," you have to realize that it's
not a number. Usually what you mean is some kind of limiting process.
So if you have "1^infinity" what you really have is some kind of limit:
the base isn't really 1, but is getting closer and closer to 1 perhaps
while the exponent is getting bigger and bigger, like maybe (x+1)^(1/x)
as x->0+.
The question is, which is happening faster, the base getting close to
1 or the exponent getting big? To find out, let's call:
L = lim x->0 of (x+1)^(1/x)
Then:
ln L = lim x->0 of (1/x) ln (x+1) = lim x->0 of ln(x+1) / x
So what's that? As x->0 it's of 0/0 form, so take the derivative of the
top and bottom. Then we get lim x->0 of 1/(x+1) / 1, which = 1.
So ln L = 1, and L = e. Cool!
Is it really true? Try plugging in a big value of x. Or recognize this
limit as a variation of the definition of e. Either way, it's true. The
limit is of the 1^infinity form, but in this case it's e, not 1. Try
repeating the work with (2/x) in the exponent, or with (1/x^2), or with
1/(sqrt(x)), and see how that changes the answer.
That's why we call it indeterminate - all those different versions of
the limit approach 1^infinity, but the final answer could be any
number, such as 1, or infinity, or undefined. You need to do more work
to determine the answer, so 1^infinity by itself is not determined yet.
In other words, 1 is just one of the answers of 1^infinity.
Does that help any?
not a number. Usually what you mean is some kind of limiting process.
So if you have "1^infinity" what you really have is some kind of limit:
the base isn't really 1, but is getting closer and closer to 1 perhaps
while the exponent is getting bigger and bigger, like maybe (x+1)^(1/x)
as x->0+.
The question is, which is happening faster, the base getting close to
1 or the exponent getting big? To find out, let's call:
L = lim x->0 of (x+1)^(1/x)
Then:
ln L = lim x->0 of (1/x) ln (x+1) = lim x->0 of ln(x+1) / x
So what's that? As x->0 it's of 0/0 form, so take the derivative of the
top and bottom. Then we get lim x->0 of 1/(x+1) / 1, which = 1.
So ln L = 1, and L = e. Cool!
Is it really true? Try plugging in a big value of x. Or recognize this
limit as a variation of the definition of e. Either way, it's true. The
limit is of the 1^infinity form, but in this case it's e, not 1. Try
repeating the work with (2/x) in the exponent, or with (1/x^2), or with
1/(sqrt(x)), and see how that changes the answer.
That's why we call it indeterminate - all those different versions of
the limit approach 1^infinity, but the final answer could be any
number, such as 1, or infinity, or undefined. You need to do more work
to determine the answer, so 1^infinity by itself is not determined yet.
In other words, 1 is just one of the answers of 1^infinity.
Does that help any?
Also remember that infinity is not a single value...there is "infinite" and "bigger than infinite", which validates the schoolyard "infinity plus ONE!" arguments..
Basically, a brilliant mathematician named Cantor found out about this (and other stuff) and went crazy dying in a sanitarium, but the basic theory is around the idea of lining up numbers one after another, and analyzing that there is a way of lining up a diagonal in a certain way to prove that (insert interesting infinity here) > (normal infinity).
Basically, a brilliant mathematician named Cantor found out about this (and other stuff) and went crazy dying in a sanitarium, but the basic theory is around the idea of lining up numbers one after another, and analyzing that there is a way of lining up a diagonal in a certain way to prove that (insert interesting infinity here) > (normal infinity).
Can someone tell me how infinity relates to time?
I had a crazy thought once and proved to myself an infinite 'event' does not need to take an infinite amount of time to complete.
Or something like that, I cant remember - I was smoking something hallucinogenic at the time.
I had a crazy thought once and proved to myself an infinite 'event' does not need to take an infinite amount of time to complete.
Or something like that, I cant remember - I was smoking something hallucinogenic at the time.
How are you defining an event? You mean like a computation?
It is possible for an infinite amount of computation to occur in a finite amount of proper time as long as energy increases towards infinity. This was proved by Frank Tipler.
You can then surmise a "matrix" like computation where this infinite computing device is simulating reality. In which case, all events that ever occured, or ever will occur, or ever COULD occur, could be simulated in finite time.
Tipler used this to show how "God" could exist via the laws of physics.
IF the universe is closed
and IF the collapse can be manipulated by sentient life
and IF they use it to extract an infinite amount of gravitational energy
and IF this is used to build an "infinite" Omega-point computer
THEN in the final seconds of the Big Crunch
the machine could
Resurrect and simulate any human being who ever existed, or indeed, anything in the entire universe.
It could even simulate all things which could possibly exist
It could do so for an infinity of subjective time (immortality of simulants), provide them a garden of eden, bliss, yadda yadda.
In other words, once you actually have a real physical infinity at you disposal, you can achieve omnipotentence according to current physics.
The only problem is, perhaps real infinities don't exist, and quantum mechanics prohibits an actual infinite amount of energy from existing during collapse.
It is possible for an infinite amount of computation to occur in a finite amount of proper time as long as energy increases towards infinity. This was proved by Frank Tipler.
You can then surmise a "matrix" like computation where this infinite computing device is simulating reality. In which case, all events that ever occured, or ever will occur, or ever COULD occur, could be simulated in finite time.
Tipler used this to show how "God" could exist via the laws of physics.
IF the universe is closed
and IF the collapse can be manipulated by sentient life
and IF they use it to extract an infinite amount of gravitational energy
and IF this is used to build an "infinite" Omega-point computer
THEN in the final seconds of the Big Crunch
the machine could
Resurrect and simulate any human being who ever existed, or indeed, anything in the entire universe.
It could even simulate all things which could possibly exist
It could do so for an infinity of subjective time (immortality of simulants), provide them a garden of eden, bliss, yadda yadda.
In other words, once you actually have a real physical infinity at you disposal, you can achieve omnipotentence according to current physics.
The only problem is, perhaps real infinities don't exist, and quantum mechanics prohibits an actual infinite amount of energy from existing during collapse.
Thanks for the input Democoder. I was thinking on similar lines but necessarily a computation but rather a physical event for example blackholes as they represent infinities (or so I thought).
I was thinking about how a blackhole or wormhole can come into being and collapse again in an infinitesimal small time.
I really can't remember my exact thoughts any longer but it was crystal clear to me in the thought experiment I did (and it was most likely completely flawed).
I think I agree with you a physical infinity requiring infinite energy cannot exist or our concept of infinity is actually a bit limited.
I was thinking about how a blackhole or wormhole can come into being and collapse again in an infinitesimal small time.
I really can't remember my exact thoughts any longer but it was crystal clear to me in the thought experiment I did (and it was most likely completely flawed).
I think I agree with you a physical infinity requiring infinite energy cannot exist or our concept of infinity is actually a bit limited.
I recommend reading a book called Zero. It dives into infinity a fair bit, after all how fair would it be to pay attention to one but not it's partner in crime.
Sabastian you're touching upon the duality between zero and infinity.
If you're calculating all the possible paths the universe could take, then considering the universe is governed by rules (physics) and assuming these rules only allow for certain things to happen. Then, would it be possible to calculate everything necessary without an infinite amount of energy, since the number of possible steps are finite? Mind you this is only one step to the next and following even two steps is a combinatoral explosion of doom!
Sabastian you're touching upon the duality between zero and infinity.
If you're calculating all the possible paths the universe could take, then considering the universe is governed by rules (physics) and assuming these rules only allow for certain things to happen. Then, would it be possible to calculate everything necessary without an infinite amount of energy, since the number of possible steps are finite? Mind you this is only one step to the next and following even two steps is a combinatoral explosion of doom!
I think it was Euler, who felt that infinity was some sort of limit between the negative numbers and the positive numbers. Zero and that number (+/- infinity) would then be identifications much like the north and south pole.
The reason for this was because there is a natural compactification that takes the complex numbers into the 2 sphere.
The reason for this was because there is a natural compactification that takes the complex numbers into the 2 sphere.
I believe it was Kurt Godel.Simon F said:
I don't see why Cantor's diagonal argument can't be used with integers over the interval [0, infinity]. It seems to me that the fundamental problem is precision. Cantor's argument for real numbers only works with infinite precision. However, you can get infinite precision with integers by using the interval [0, infinity]. This means that the set of integers in the interval [0, infinity] should have the same cardinality as the set of real numbers in the interval [0, 1]. Since it can be proved that the set of real numbers in the interval [0, 1] has the same cardinality as real numbers in [0, infinity], integers and reals must both be uncountably large.
Now, obviously I don't think that I'm a better mathematician than Cantor. So where is my argument going wrong? Could it be that an infinite set of numbers does not require that the numbers have infinite precision.
Stating it a different way:
As the value of an integer tends to infinity, the precision converges to a finite value.
I can see why he went crazy...
I belive that zero "0" is much more interesting than infinity.
2X + 6 = X + 3
Solve this equation for X and you get
X = -3
But use the rules of algebra and factor out 2 from 2X + 6 the result is:
2(X + 3) = X + 3
Divide both sides by X + 3 and the result is:
2 = 1
I love this shit. So now define zero.
2X + 6 = X + 3
Solve this equation for X and you get
X = -3
But use the rules of algebra and factor out 2 from 2X + 6 the result is:
2(X + 3) = X + 3
Divide both sides by X + 3 and the result is:
2 = 1
I love this shit. So now define zero.
Nathan:
The difference is that each real number in the interval [0 1] has an infinite amount of decimals (although some of the numbers just has an infinite amount of zeros at the end).
But each integer in the interval [0 infinity] is finite.
So pick a real number => it still has an infinite amount of "information" in it.
Pick an integer => finite amount of "information".
The difference is that each real number in the interval [0 1] has an infinite amount of decimals (although some of the numbers just has an infinite amount of zeros at the end
). But each integer in the interval [0 infinity] is finite.
So pick a real number => it still has an infinite amount of "information" in it.
Pick an integer => finite amount of "information".
And now a question for those who likes infinities fighting each other.
Say that you have a random number generator that generate uniformly distributed real numbers on the interval [0 1]. (Note: real numers, not just rational/float.)
Then you code the real numbers with binals (think decimals, but binary), and chop the binary representation into bytes.
Now, what's the probability that the full design and documentation of R500 is present somewhere in a sample from the random number from said source?
(Since the full design and doc of R500 likely isn't done yet, we'll have to wait for the verification of course.)
Say that you have a random number generator that generate uniformly distributed real numbers on the interval [0 1]. (Note: real numers, not just rational/float.)
Then you code the real numbers with binals (think decimals, but binary), and chop the binary representation into bytes.
Now, what's the probability that the full design and documentation of R500 is present somewhere in a sample from the random number from said source?
(Since the full design and doc of R500 likely isn't done yet, we'll have to wait for the verification of course.
)
