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Tomdstone
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Are you going to show us where the error is or not?
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AnimalSpirits
Guest
I do.
“Countable” just means “of carnality equal to some subset of the integers.” That is, a finite set or one that is infinite but not “bigger” than the integers. So yes, “uncountable therefore bigger than the integers” would be trivial- but luckily that’s not what the proof does.
Sets A and B are said to be of the same size if you can “map” each element of A to and element of B such that each element of A goes to exactly one element of B, each element of B gets “hit” by exactly one element of A, and every element of B gets hit. Cantor shows that this isn’t possible from the natural numbers to the real numbers.
You’ll have to explain here. What do you mean by “compare”, and what makes you think that special knowledge is required to do so?
Furthermore, why on Earth would you suppose that the idea of infinity is incoherent? “infinite” is simply “not finite” and a “finite” set is one with some natural number (0 included) amount of elements.
This seems like a fatal misunderstanding. Sets are not ordered, sets are just lists of unique elements. If your set is {1,2} then there is nothing “between” them, as the concept of “between” doesn’t apply to this set any more than it does to the set of {apple, orange}.Even if you have {apple, orange, dog} it doesn’t follow that “orange” is between apple and dog- you could alter the order in which you write them and have the same set.
The appropriate statement is: the finite interval [1,2] on the real numbers contains infinitely many rational numbers. The set of all rational numbers contained within this interval is infinite. This isn’t a contradiction by any means- a finite interval simply has finite end points, but the number of elements could be finite or infinite depending on your domain ([1,2] has infinite elements on the reals, finite elements on the integers).
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Tomdstone
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See post 779 and tell us where Cantor’s error is.
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Aloysium
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LoL, you know what he meant. “Fatal misunderstanding”, LoL. This isn’t a math site, LoL.
Just for my own edification,
I would say that:
whereas we can imagine things as being infinite, this is not the case when it comes to physical reality.
Although, I can suppose space or time may be divided into an infinite number of parts, in reality, this does not happen:
Planck Time, the shortest physically meaningful interval of time ≈ 5.4×10^-44 s
Planck Length, the smallest length that makes any sense = 1.62×10^−35 m
Is there a fatal error in saying this?
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Simca
Guest
Is this thread still going ? think we have coved all bases by now…/QUOTE]
I’ve only just found this thread.
So maybe this has been said before…
Could the Universe have created itself?
Would it answer if we asked it this question?
Wouldn’t we get more of a response by asking the Creator?
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AnimalSpirits
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Fatal error as in a mistake that was leading to a false conclusion, although I see how it could have been taken in a more demeaning way. Confusing “finite interval” and “finite set” was leading to an apparent contradiction, where no exists. “Finite interval” isn’t actually a mathematical term (too my knowledge)- the more appropriate one is “bounded.”
And this is more physics, which has always been fuzzier to me. Planck time is the shortest amount of time a “meaningful event” can take place during. But, to me, that doesn’t mean that no shorter units of time exist. The definition doesn’t seem to preclude the following, where time=t, measured in Planck time.
t=0: World’s shortest “process” begins.
t=.5: An identical process begins.
t=1: First process finishes.
t=1.5 Second process finishes.
See what I mean? For all I know this isn’t actually possible- I’m not sure.
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Peter_Plato
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Excellent!
That much is clear.
I am not the one who attempted to “compare” the two. That would have been Tomdstone, here:
That seems incoherent to me, at least if the word “larger” can be used in any meaningful way when it is used to refer to and compare two sets that both have no determinable (infinite) quantity.
The term “not finite” means essentially the same as “not bounded,” “not complete” or “without a determinable bound.” If that is the case, then “infinite” may have no actual possible referent since it can only be used as a negation with reference to the word finite.
Just as incomplete means “not complete” and has no coherency apart from whatever the object is that the word “complete” refers to, “not finite” may implicitly require a finite referent, if only conceptually. We know what an incomplete decks of cards is because we know what a complete deck is. We know what a finite set of numbers is but the question is whether “infinite” does have any sense without reference to an actual set of numbers that lacks members even if we can (or can’t) identify the missing members. It would seem as if an open ended (infinite) set is possible by continuing to add members, but the question is whether infinite itself can logically refer to any actual set. If it can’t, the idea is incoherent as a quantifier because it doesn’t mean anything except in contrast to finite, just as incomplete means nothing without reference to some complete reality. Infinite, on its own, would seem to mean something like a “not finite finite” which appears self-contradictory.
Perhaps that wasn’t the best example. Consider instead a length such as the distance between any Point A and any Point B. It appears to be a finite distance, but if the distance can be infinitely divided into smaller and smaller lengths then the finite length can be made up of an infinite number of infinitely small lengths. No need to invoke apples, oranges or dogs.
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Tomdstone
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Yes there is. The error has to do with the associated theory of discrete quantum gravity based on Planck lengths and areas which has serious issues which is not the case with GR.
If you assume that spacetime has a discrete or foamy structure at the Planck length scale, then because of quantum effects, the Planck length would be the shortest measurable length. However, there are a few issues here.
First of all, although the Planck length is the shortest measurable distance, there is some philosophical question as to whether or not this would necessarily imply that there is not shorter distance in reality.
Secondly, the foam based loop quantum gravity theory is not tenable for several reasons. Perhaps the most serious is that the spin foam version of loop quantum gravity breaks unitarity. Another problem is Lorentz invariance. And of course, loop quantum gravity would require us to use nonseparable Hilbert spaces involving an uncountable set of superselection sectors. There are a whole lot of other problematic issues which arise if you assume a foam based, discrete, noncontinuous, quantum gravity theory.
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Tomdstone
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Some concepts such as length apply on a macroscopic scale, but are redefined to apply to infinitesimal quantities. Some sets of infinitesimal quantites are shown to be non-measurable, in the sense that they will not have a length in the macroscopic sense.
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AnimalSpirits
Guest
“Larger” is a colloquial term. The “mathematical” statement would be that the reals are of greater cardinality than the naturals. A lot of the conclusions involving infinities are far from intuitive- for example, I’ll never quite forgive the universe for the fact that a “dense” set (infinitely many elements on any given interval) can have the same cardinality as a non dense set. Intuitively, one would think that there should be “more” rationals than there are integers- but this is not the case. I understand the proof, I’ve reproduced it, but I don’t like it.
In mathematics, terms get defined as easily as possible. Yes, infinite is just a negation of finite- but you can define it in other terms. You could write up a definition that didn’t invoke the concept of finiteness, but why bother- it would be wordier and the two would be equivalent (unless you felt like redefining infinity).
This is allowed because all sets are either finite or infinite, and none are both. Same as even and odd- you could say that all even integers are divisible by 2, and all odd integers are not even.
But if you like- “An infinite set is at least as big as the integers.” Or, more formally, “A set “A” is infinite if and only if there exists an onto function f A–>N, where N is the set of all integers.” Onto meaning “hits every element.” This is exactly equivalent to the negation of finite, so there’s no issue. This doesn’t always work- for example, an “open” set can also be “closed” and a set can be neither open nor closed- so we can’t definite one as the negation of the other.
As far as the second paragraph goes, I’m not sure I follow. It’s fairly easy to show that the set of all integers isn’t finite- wouldn’t that answer the “Can we construct an infinite set” question. This works by both of the definitions I gave, trivially in the case of the second one (surprise! The integers are as big as the integers!)
If this “length” is in the real world, we’re getting into physics- less my area. If this “length” is in a R3 (Euclidean space of real numbers), then yes, any given length is infinitely divisible. Adding these up is the domain of real analysis/calculus.
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Peter_Plato
Guest
That would make a great signature line
The question might be asked if you have even considered whether the universe will forgive you for holding such a thought if it turns out NOT to be true
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inocente
Guest
Yes.
If the universe couldn’t create itself then God must have done it.
But then we would know that God exists.
But then we wouldn’t need faith.
But then Jesus would not have said we need faith.
Therefore, by God’s own Word, the universe could have created itself.
QED
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yppop
Guest
To those of you that are skeptical about the nature of infinite sets (transfinite numbers), let me remind you that there are plenty of persons with brilliant minds far greater than any that I have run into in this forum, including my own, that understand and accept Cantor’s ideas. Here are my thoughts on the subject.
Yppop
- Cantor, using a very elegant proof called the "Cantor’s Diagonalization Proof of the Denumerability of the Rational Numbers", proved that **rational **numbers are an infinite set, that he designated aleph (0), the first transfinite number. Denumerability means putting numbers in a one to one correspondence with the natural numbers. Natural numbers are infinite a priori, therefore, numbers like the rational and algebraic numbers that can be lined up with one to one with the natural numbers are also an infinity of the same transfinite number, aleph (0).
- Using a similar diagonalization method, Cantor proved that the real numbers are not denumerable and are infinitely greater than the rational numbers. He called the infinite set of real numbers,* aleph(1)*, the second transfinite number, and using what is called a power set, showed that: aleph (1) = 2^aleph (0).
- Cantor also demonstrated that using a unique numbering system he could write a number on the real number line to represent the two numbers (x and y) for every point in a unit square or the three numbers (x, y, and z) in the unit volume, thus proving the number of real numbers on the unit line is the same as the number of points in 2 or 3 or any higher dimension space.
- Because of denumerability, it can be shown that there as many points on any segment of the real number line as there are on the entire line. An infinite set has the unusual property that parts can be equal to the whole. Hard to imagine, but completely explicable in a logically irrefutable way.
Yppop
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Linusthe2nd
Guest
When the cat is gone, the mice will playYes.
If the universe couldn’t create itself then God must have done it.
But then we would know that God exists.
But then we wouldn’t need faith.
But then Jesus would not have said we need faith.
Therefore, by God’s own Word, the universe could have created itself.
QED
.For all us ol’ foggies, please explain your abbreviations as you go along.
Linus2nd
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inocente
Guest
Your surrender is accepted.When the cat is gone, the mice will play
If you mean QED, 'tis quod erat demonstrandum, meaning “I dun proved it, so there”.
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alevanpa
Guest
Fantastic, Black. The problem is, that any, and all, infinities that can be proven are all a part of a single universe. Their value outside of the constraints of this universe is merely conjectural. The question is still, “How can a given “infinity” prove anything at all about what is outside the universe in which it exists?” Furthermore, since any “infinity” you can propose is constrained by the universe in which it exists, is it actually an “infinity?” Maybe “infinity” is only a philosophical, argument, that does not exist in reality. Certainly, the concept is useful, and valid, as far as it goes; but there can be no difference between them–unless you can prove that our concept of “infinity” will be valid in any universe. In fact, can anyone prove that our concept of Infinity is valid at the farthest reaches of our own universe–where the farthest galaxies have shifted to light speed and whatever is past that?
This, debate is actually a lot of fun.
This, debate is actually a lot of fun.
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Peter_Plato
Guest
Shortened from the original, “I dun proved it to myself, so there.”
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Aloysium
Guest
Thanks for the information.
I understand that the size of the universe to the size of a dot is roughly equal to the size of that dot to a plank length. It would take as many dots to fill the universe as plank lengths fill the dot.
From what you stated above and other things I have read, there are events/objects/whatever that are smaller than a plank length.
In order to describe the events that occur within the microcosm, models and concepts that you address and which sound rather complex to me, must be employed.
At the other end of the scale of relative magnitude we have clusters of Galaxies.
I am in awe of God’s creation.
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inocente
Guest
This is a cool animation of objects at different scales. Use the scroll bar to zoom in or out.
Scale Of The Universe 2 - htwins.net/scale2/
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alevanpa
Guest
You’ve got that right.
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