Strange but True: Infinity Comes in Different Sizes

[shunyadragon]

Strange but True: Infinity Comes in Different Sizes

Source: http://www.scientificamerican.com/article/strange-but-true-infinity-comes-in-different-sizes/

In the 1995 Pixar film Toy Story, the gung ho space action figure Buzz Lightyear tirelessly incants his catchphrase: "To infinity … and beyond!" The joke, of course, is rooted in the perfectly reasonable assumption that infinity is the unsurpassable absolute—that there is no beyond.

That assumption, however, is not entirely sound. As German mathematician Georg Cantor demonstrated in the late 19th century, there exists a variety of infinities—and some are simply larger than others.

Take, for instance, the so-called natural numbers: 1, 2, 3 and so on. These numbers are unbounded, and so the collection, or set, of all the natural numbers is infinite in size. But just how infinite is it? Cantor used an elegant argument to show that the naturals, although infinitely numerous, are actually less numerous than another common family of numbers, the "reals." (This set comprises all numbers that can be represented as a decimal, even if that decimal representation is infinite in length. Hence, 27 is a real number, as is π, or 3.14159….)
In fact, Cantor showed, there are more real numbers packed in between zero and one than there are numbers in the entire range of naturals. He did this by contradiction, logically: He assumes that these infinite sets are the same size, then follows a series of logical steps to find a flaw that undermines that assumption. He reasons that the naturals and this zero-to-one subset of the reals having equally many members implies that the two sets can be put into a one-to-one correspondence. That is, the two sets can be paired so that every element in each set has one—and only one—"partner" in the other set.

Think of it this way: even in the absence of numerical counting, one-to-one correspondences can be used to measure relative sizes. Imagine two crates of unknown sizes, one of apples and one of oranges. Withdrawing one apple and one orange at a time thus partners the two sets into apple-orange pairs. If the contents of the two crates are emptied simultaneously, they are equally numerous; if one crate is exhausted before the other, the one with remaining fruit is more plentiful.

© Copyright Original Source

[shunyadragon]

Part II

Source: http://www.scientificamerican.com/article/strange-but-true-infinity-comes-in-different-sizes/

Cantor thus assumes that the naturals and the reals from zero to one have been put into such a correspondence. Every natural number n thus has a real partner rn. The reals can then be listed in order of their corresponding naturals: r1, r2, r3, and so on.
Then Cantor's wily side begins to show. He creates a real number, called p, by the following rule: make the digit n places after the decimal point in p something other than the digit in that same decimal place in rn. A simple method would be: choose 3 when the digit in question is 4; otherwise, choose 4.

For demonstration's sake, say the real number pair for the natural number 1 (r1) is Ted Williams's famed .400 batting average from 1941 (0.40570…), the pair for 2 (r2) is George W. Bush's share of the popular vote in 2000 (0.47868…) and that of 3 (r3) is the decimal component of π (0.14159…).

Now create p following Cantor's construction: the digit in the first decimal place should not be equal to that in the first decimal place of r1, which is 4. Therefore, choose 3, and p begins 0.3…. Then choose the digit in the second decimal place of p so that it does not equal that of the second decimal place of r2, which is 7 (choose 4; p = 0.34…). Finally, choose the digit in the third decimal place of p so that it does not equal that of the corresponding decimal place of r3, which is 1 (choose 4 again; p = 0.344…).
Continuing down the list, this mathematical method (called "diagonalization") generates a real number p between zero and one that, by its construction, differs from every real number on the list in at least one decimal place. Ergo, it cannot be on the list.
In other words, p is a real number without a natural number partner—an apple without an orange. Thus, the one-to-one correspondence between the reals and the naturals fails, as there are simply too many reals—they are "uncountably" numerous—making real infinity somehow larger than natural infinity.

"The idea of being 'larger than' was really a breakthrough," says Stanley Burris, professor emeritus of mathematics at the University of Waterloo in Ontario. "You had this basic arithmetic of infinity, but no one had thought of classifying within infinity—it was just kind of a single object before that."

Adds mathematician Joseph Mileti of Dartmouth College: "When I first heard the result and first saw it, it was definitely something that knocked me over. It's one of those results that's short and sweet and really, really surprising."

© Copyright Original Source

[JonF]

Cantor's diagonal proof that the infinity of reals is greater than the infinity of natural numbers is a favored target of math loons. As I bet you know. Mark Chu-Carroll

[klaus54]

What does the set theory abstraction of uncountable infinities have to do with the real world of matter and energy?

[klaus54]

If one were to apply a mathematical notion of infinity to multiple universes, it would be in the sense of "no upper bound". {1, 2, 3, ...}

And if one were to assign a transfinite cardinal to this concept, it would be aleph-null, although I don't recall ever seeing the "no-upper-bound" concept mixed with transfinite cardinals.

[robrecht]

What does the set theory abstraction of uncountable infinities have to do with the real world of matter and energy?

Abstract algebra is evil and of the devil. Calculus and diffyq are good and reflect the real three-dimensional world of matter and energy. You need to add in some non-Euclidean geometry or theoretical physics to incorporate time into the the mix, 'though philosophy or theology might do in a pinch.

[oxmixmudd]

If one were to apply a mathematical notion of infinity to multiple universes, it would be in the sense of "no upper bound". {1, 2, 3, ...}

And if one were to assign a transfinite cardinal to this concept, it would be aleph-null, although I don't recall ever seeing the "no-upper-bound" concept mixed with transfinite cardinals.

But as I proposed in the other thread, what about the set of potential initial conditions for any universe about to be born? What if the set of potential initial states can only be represented by the power set of aleph null, i.e. aleph one? And suppose the probability distribution for initial states is even, with the probability of any specific initial state then equal to zero? Then the probability any newly born universe will match any of its predecessors would also be zero.

Jim

[shunyadragon]

But as I proposed in the other thread, what about the set of potential initial conditions for any universe about to be born? What if the set of potential initial states can only be represented by the power set of aleph null, i.e. aleph one? And suppose the probability distribution for initial states is even, with the probability of any specific initial state then equal to zero? Then the probability any newly born universe will match any of its predecessors would also be zero.

Jim

Yes, this a possibility, but very unlikely. IF this extreme view is true we would be the only universe. There is an assumption in science of uniformity that it is unlikely that there is 'only one' of anything, which is unique. If science discovers something, there has always been more than one. Based on our present knowledge of Quantum Mechanics there is the possibility that the initial conditions of singularities may result from variations in the physics constants that result in the formation of our universe, and the formation of the universe fails, or one forms that will not result in a universe like ours that supports life. At present this possible range of constants is unknown, and they may be limited to the point that all possible universes will form when the same initial conditions of the singularity that forms a universe.

[oxmixmudd]

Yes, this a possibility, but very unlikely. IF this extreme view is true we would be the only universe. There is an assumption in science of uniformity that it is unlikely that there is 'only one' of anything, which is unique. If science discovers something, there has always been more than one. Based on our present knowledge of Quantum Mechanics there is the possibility that the initial conditions of singularities may result from variations in the physics constants that result in the formation of our universe, and the formation of the universe fails, or one forms that will not result in a universe like ours that supports life. At present this possible range of constants is unknown, and they may be limited to the point that all possible universes will form when the same initial conditions of the singularity that forms a universe.

I take exception to the term 'extreme' view. It is simply a possibility. What do we know about whatever it is that provides the 'substrate' for multiple universes, assuming there is >1? And without knowing much of anything about it, why should we necessarily extend observations about our universe to it? It by definition must be in some ways different in that somehow it contains or facilitates all the universes - yes?

In our universe, as far as I/we know, continua does not actually exist except as a mathematical abstraction. Everything from matter to energy to space itself appear defined as some sort of discrete quanta of energy or time or both. Yet something in the early universe defined what those quanta would be. Are these values fixed across all possible universes, or are then themselves a variable, something that can take on a continua of possible values. And if so, what do differing quanta (different energy, space, or time 'packet sizes' do to the constraints of the resulting universe. Is quantization itself a variable - are there possible universes without quanta?

The essence of science, of exploration is to be capable of thinking outside the box of currently accepted norms when the evidence shows there is more than was thought to be. When you start talking about a multiverse, you have left the norm. So why not ask questions of what might be beyond this universe, why not at least explore the possibility some of the things we take for granted as fixed properties are in fact not? This is what Einstein did - space and time are not absolutes. Yet the prevailing view at the time was that they were. That was a big part of his genius - being able to see the possibilities. He saw the evidence and explored a possibility for reconciliation of the evidence that others did not. I'm not Einstein by any stretch, but I do embrace the idea of exploring what might be that is different from what is the accepted norm.

Jim

[shunyadragon]

I take exception to the term 'extreme' view. It is simply a possibility. What do we know about whatever it is that provides the 'substrate' for multiple universes, assuming there is >1? And without knowing much of anything about it, why should we necessarily extend observations about our universe to it? It by definition must be in some ways different in that somehow it contains or facilitates all the universes - yes?

In our universe, as far as I/we know, continua does not actually exist except as a mathematical abstraction. Everything from matter to energy to space itself appear defined as some sort of discrete quanta of energy or time or both. Yet something in the early universe defined what those quanta would be. Are these values fixed across all possible universes, or are then themselves a variable, something that can take on a continua of possible values. And if so, what do differing quanta (different energy, space, or time 'packet sizes' do to the constraints of the resulting universe. Is quantization itself a variable - are there possible universes without quanta?

The essence of science, of exploration is to be capable of thinking outside the box of currently accepted norms when the evidence shows there is more than was thought to be. When you start talking about a multiverse, you have left the norm. So why not ask questions of what might be beyond this universe, why not at least explore the possibility some of the things we take for granted as fixed properties are in fact not? This is what Einstein did - space and time are not absolutes. Yet the prevailing view at the time was that they were. That was a big part of his genius - being able to see the possibilities. He saw the evidence and explored a possibility for reconciliation of the evidence that others did not. I'm not Einstein by any stretch, but I do embrace the idea of exploring what might be that is different from what is the accepted norm.

Jim

The belief that there would be only one universe, ours, is an extreme view. Since the more general view of science is that more universes within a multiverse matrix is possible, and more likely. The substrate of the multiverse would be the timeless Quantum zero-point energy existence at T=0.

Source: http://www.calphysics.org/zpe.html

Quantum mechanics predicts the existence of what are usually called ''zero-point'' energies for the strong, the weak and the electromagnetic interactions, where ''zero-point'' refers to the energy of the system at temperature T=0, or the lowest quantized energy level of a quantum mechanical system. Although the term ''zero-point energy'' applies to all three of these interactions in nature, customarily (and hereafter in this article) it is used in reference only to the electromagnetic case.

In conventional quantum physics, the origin of zero-point energy is the Heisenberg uncertainty principle, which states that, for a moving particle such as an electron, the more precisely one measures the position, the less exact the best possible measurement of its momentum (mass times velocity), and vice versa. The least possible uncertainty of position times momentum is specified by Planck's constant, h. A parallel uncertainty exists between measurements involving time and energy (and other so-called conjugate variables in quantum mechanics). This minimum uncertainty is not due to any correctable flaws in measurement, but rather reflects an intrinsic quantum fuzziness in the very nature of energy and matter springing from the wave nature of the various quantum fields. This leads to the concept of zero-point energy.

Zero-point energy is the energy that remains when all other energy is removed from a system. This behaviour is demonstrated by, for example, liquid helium. As the temperature is lowered to absolute zero, helium remains a liquid, rather than freezing to a solid, owing to the irremovable zero-point energy of its atomic motions. (Increasing the pressure to 25 atmospheres will cause helium to freeze.)
. . .
From this line of reasoning, quantum physics predicts that all of space must be filled with electromagnetic zero-point fluctuations (also called the zero-point field) creating a universal sea of zero-point energy. The density of this energy depends critically on where in frequency the zero-point fluctuations cease. Since space itself is thought to break up into a kind of quantum foam at a tiny distance scale called the Planck scale (10-33 cm), it is argued that the zero point fluctuations must cease at a corresponding Planck frequency (1043 Hz). If that is the case, the zero-point energy density would be 110 orders of magnitude greater than the radiant energy at the center of the Sun.

How could such an enormous energy not be wildly evident? There is one major difference between zero-point electromagnetic radiation and ordinary electromagnetic radiation. Turning again to the Heisenberg uncertainty principle one finds that the lifetime of a given zero-point photon, viewed as a wave, corresponds to an average distance traveled of only a fraction of its wavelength. Such a wave ''fragment'' is somewhat different than an ordinary plane wave and it is difficult to know how to interpret this.

© Copyright Original Source

[37818]

Looking at an infinity being at the decimal point.

...000.000... or ...999.999... being the same infinity.

G64 not infinity ...987.000...
Gn where n is infinity. ...987.000... having no left starting digit. Is a smaller infinity than ...999.999... Infinity.

[shunyadragon]

Looking at an infinity being at the decimal point.

...000.000... or ...999.999... being the same infinity.

G64 not infinity ...987.000...
Gn where n is infinity. ...987.000... having no left starting digit. Is a smaller infinity than ...999.999... Infinity.

Explain!?!?!?! This is not a very meaningful infinity when considering the large scale of infinities.

[oxmixmudd]

The belief that there would be only one universe, ours, is an extreme view. Since the more general view of science is that more universes within a multiverse matrix is possible, and more likely. The substrate of the multiverse would be the timeless Quantum zero-point energy existence at T=0.

Source: http://www.calphysics.org/zpe.html

Quantum mechanics predicts the existence of what are usually called ''zero-point'' energies for the strong, the weak and the electromagnetic interactions, where ''zero-point'' refers to the energy of the system at temperature T=0, or the lowest quantized energy level of a quantum mechanical system. Although the term ''zero-point energy'' applies to all three of these interactions in nature, customarily (and hereafter in this article) it is used in reference only to the electromagnetic case.

In conventional quantum physics, the origin of zero-point energy is the Heisenberg uncertainty principle, which states that, for a moving particle such as an electron, the more precisely one measures the position, the less exact the best possible measurement of its momentum (mass times velocity), and vice versa. The least possible uncertainty of position times momentum is specified by Planck's constant, h. A parallel uncertainty exists between measurements involving time and energy (and other so-called conjugate variables in quantum mechanics). This minimum uncertainty is not due to any correctable flaws in measurement, but rather reflects an intrinsic quantum fuzziness in the very nature of energy and matter springing from the wave nature of the various quantum fields. This leads to the concept of zero-point energy.

Zero-point energy is the energy that remains when all other energy is removed from a system. This behaviour is demonstrated by, for example, liquid helium. As the temperature is lowered to absolute zero, helium remains a liquid, rather than freezing to a solid, owing to the irremovable zero-point energy of its atomic motions. (Increasing the pressure to 25 atmospheres will cause helium to freeze.)
. . .
From this line of reasoning, quantum physics predicts that all of space must be filled with electromagnetic zero-point fluctuations (also called the zero-point field) creating a universal sea of zero-point energy. The density of this energy depends critically on where in frequency the zero-point fluctuations cease. Since space itself is thought to break up into a kind of quantum foam at a tiny distance scale called the Planck scale (10-33 cm), it is argued that the zero point fluctuations must cease at a corresponding Planck frequency (1043 Hz). If that is the case, the zero-point energy density would be 110 orders of magnitude greater than the radiant energy at the center of the Sun.

How could such an enormous energy not be wildly evident? There is one major difference between zero-point electromagnetic radiation and ordinary electromagnetic radiation. Turning again to the Heisenberg uncertainty principle one finds that the lifetime of a given zero-point photon, viewed as a wave, corresponds to an average distance traveled of only a fraction of its wavelength. Such a wave ''fragment'' is somewhat different than an ordinary plane wave and it is difficult to know how to interpret this.

© Copyright Original Source

Frank - I think we are talking past each other? Do you have a preconcieved notion of what I must believe about the universe/multiverse because of my Christian faith?


Jim

[klaus54]

But as I proposed in the other thread, what about the set of potential initial conditions for any universe about to be born? What if the set of potential initial states can only be represented by the power set of aleph null, i.e. aleph one? And suppose the probability distribution for initial states is even, with the probability of any specific initial state then equal to zero? Then the probability any newly born universe will match any of its predecessors would also be zero.

Jim

Again, I've never seen ANY application of transfinite cardinals to physical reality. And according to your example of initial conditions, a countable infinity will also produce every possible arrangement of states according the (countable) infinity of universes. But, I don't understand the no-repeats thingy. Why couldn't one have repeats, or in fact repeats an unlimited number of times?

[robrecht]

The belief that there would be only one universe, ours, is an extreme view. Since the more general view of science is that more universes within a multiverse matrix is possible, and more likely. The substrate of the multiverse would be the timeless Quantum zero-point energy existence at T=0.

Source: http://www.calphysics.org/zpe.html

Quantum mechanics predicts the existence of what are usually called ''zero-point'' energies for the strong, the weak and the electromagnetic interactions, where ''zero-point'' refers to the energy of the system at temperature T=0, or the lowest quantized energy level of a quantum mechanical system. Although the term ''zero-point energy'' applies to all three of these interactions in nature, customarily (and hereafter in this article) it is used in reference only to the electromagnetic case.

In conventional quantum physics, the origin of zero-point energy is the Heisenberg uncertainty principle, which states that, for a moving particle such as an electron, the more precisely one measures the position, the less exact the best possible measurement of its momentum (mass times velocity), and vice versa. The least possible uncertainty of position times momentum is specified by Planck's constant, h. A parallel uncertainty exists between measurements involving time and energy (and other so-called conjugate variables in quantum mechanics). This minimum uncertainty is not due to any correctable flaws in measurement, but rather reflects an intrinsic quantum fuzziness in the very nature of energy and matter springing from the wave nature of the various quantum fields. This leads to the concept of zero-point energy.

Zero-point energy is the energy that remains when all other energy is removed from a system. This behaviour is demonstrated by, for example, liquid helium. As the temperature is lowered to absolute zero, helium remains a liquid, rather than freezing to a solid, owing to the irremovable zero-point energy of its atomic motions. (Increasing the pressure to 25 atmospheres will cause helium to freeze.)
. . .
From this line of reasoning, quantum physics predicts that all of space must be filled with electromagnetic zero-point fluctuations (also called the zero-point field) creating a universal sea of zero-point energy. The density of this energy depends critically on where in frequency the zero-point fluctuations cease. Since space itself is thought to break up into a kind of quantum foam at a tiny distance scale called the Planck scale (10-33 cm), it is argued that the zero point fluctuations must cease at a corresponding Planck frequency (1043 Hz). If that is the case, the zero-point energy density would be 110 orders of magnitude greater than the radiant energy at the center of the Sun.

How could such an enormous energy not be wildly evident? There is one major difference between zero-point electromagnetic radiation and ordinary electromagnetic radiation. Turning again to the Heisenberg uncertainty principle one finds that the lifetime of a given zero-point photon, viewed as a wave, corresponds to an average distance traveled of only a fraction of its wavelength. Such a wave ''fragment'' is somewhat different than an ordinary plane wave and it is difficult to know how to interpret this.

© Copyright Original Source

Are you sure this is the article you meant to cite here? It does not seem to say that a multiverse is more likely than a universe.

But it does make some very interesting points. Though not definitively so, this point seems to run counter to your view: "The Casimir force is widely cited as evidence that underlying the universe there must be a sea of real zero-point energy. This argument follows from Casimir's analysis and prediction. It is not necessarily true, however. It is perfectly possible to explain the Casimir effect by taking into account the quantum-induced motions of atoms in each plate and examining the retarded potential interactions of atoms in one plate with those in the other.

This point may be interesting to some: "Zero-point energy behaves differently. For ordinary radiation, the ratio of pressure to energy density is w=1/3c2, which is customarily expressed in units whereby c=1, and thus the ratio is expressed as w=+1/3. But for zero-point energy the ratio is w=-1. This is owing to the circumstance that the zero-point energy density is assumed to be constant: no matter how much the universe expands it does not become diluted, but instead more zero-point energy is assumed to be created out of nothing."

In general, the idea that dark energy is in fact zero-point energy is very interesting. So thanks for that. A patent for tapping zero-point energy as a usable source of energy, what an incredible idea!