Are some infinities bigger than others?

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February 15, 2018

Cosmology, Intelligent Design, Mathematics, Philosophy

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From theoretical physicist Paul Davies at Cosmos:

It turns out that the set of all points on a continuous line is a bigger infinity than the natural numbers; mathematicians say there is an uncountably infinite number of points on the line (and in three-dimensional space). You simply can’t match up each point on the line with the natural numbers in a one-to-one correspondence.

…

If it is continuous (and some physicists think it may not be) then it will contain an uncountably infinite number of points. But that doesn’t mean it has to go on forever. As Einstein discovered, it may be curved in on itself to form a finite volume. More.

One would have thought that once we get involved with infinities, we are skirting the boundaries of measurement—at best.

See also: God as a necessary, maximally great, endless being vs. the challenge to an actual infinity

Reader: Weirdness of infinity shows that the universe is not infinitely old

Durston and Craig on an infinite temporal past . . .

At Quanta: Is infinity real?

and

Cosmologist: In an infinite multiverse, physics loses its ability to make predictions. And that’s okay.

Comments

@daveS, much appreciated. The discussion of math and physicalism is fascinating, since in my mind math is one of the most concrete pieces of evidence that physicalism is false. Just about everything in math seems to not be physically realizable, while being both true and extremely useful in the physical realm. Yet, it is almost as if these kinds of non-physical abstract concepts are the only things we know. We never know the material world in itself, every description of the material world is only true to a certain level, and fails to capture everything. Yet, if we decided to do away entirely with abstraction due to its illusive nature, and try to live off of direct experience, we'd be no better than the animals around us, and set human progress back by millenia. Why does our very civilization depend on abstraction if it is an illusion? So, with the correspondence mismatch between math (and abstraction in general) with the physical world, yet its extreme usefulness and truthfulness, it is hard to believe such a useful, concrete thing is not real in itself, as opposed to some sort of "ephemeral emergent pattern supervening on a material substrate." Mathematics really does seem to have an independent, non-physical existence which we can grasp with our minds. Which further implies our minds are not themselves purely physical things, but also have an independent, non-physical existence. In which case, engineering is essentially wizarding, where our arcane symbols and algorithmic incantations instantiate otherworldly entities into our physical world, and everyone can possess this magical ability in their mind by understanding math. Doing our taxes is a witching hex to keep away the IRS agents.EricMH_{February 20, 2018
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EricMH, I think some would call this "collection" a proper class. Sets have certain properties (they have elements, they have subsets, you can take the union of a set of sets, etc.), but these properties do not hold for arbitrary classes. I understand the notion of "class" is actually an informal concept. In ZFC, literally the only things that actually exist are sets. I don't really have a good answer for what it means for the number 1 to exist in the world. It certainly doesn't exist physically, or correspond to any physical object which grounds it. But I can talk about 1 and other integers with other people, and treat integers as real, almost concrete objects, so it seems there's "something" there.daveS_{February 20, 2018
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@daveS, thanks for your response. It is interesting there can be a listing of alephs, each of which is a set, but there is no set that can contain them all. Counter intuitive, since in my mind being able to say "them all" entails a collection. But if they cannot be a collection, what is the "all" I am referring to? It is also interesting you say "abstract things exist in the world". I do not understand what you mean. For example, the number 1 does not exist in the world, though our symbolic manipulation methodology can sometimes be embedded in a computer program. Saying the number 1 exists in the world suggests there is a correspondence between "1" and some physical object which grounds 1's existence. In which case, if this physical object is destroyed, then 1 no longer exists. That sounds very counter intuitive to me.EricMH_{February 20, 2018
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EricMH, To answer this question:
To answer my own question, is the set of all the different aleph infinities the biggest infinity?
At least in ZFC, the 'collection' of all cardinal numbers is not a set, so it doesn't have a well-defined cardinality. Furthermore, if S is a set, then the power set of S always has cardinality strictly greater than that of S, so there cannot be a largest cardinal number.
If physically unrealizable entities turn out to be extremely useful for understanding the physical world, does this mean there is more to existence than the physical world?
Perhaps so. I operate under the assumption that abstract things actually exist in the world, in any case.
Our daily reasoning and experience is filled with useful physical fictions. All of our decision making revolves around counterfactuals, which by definition do not physically exist. We try to figure out if something is true or false, but false things do not exist. Ideas do not have physical properties. The redness of an apple does not inhere in the physical/chemical sequence of lightwaves hitting rod cells and transmitting signals to the brain. And so on.
Yes, and I've been thinking about some of these "useful" fictions lately, and whether sometimes they create more problems than they solve. We've had some discussions of potentially misleading notations & conventions in calculus, for example.daveS_{February 20, 2018
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@daveS, If physically unrealizable entities turn out to be extremely useful for understanding the physical world, does this mean there is more to existence than the physical world? It seems physics contains many more physically problematic math concepts than just infinitesimals. Numbers, of course, cannot be physical. Zero is one of the original stumpers. Nothing doesn't exist. Negative numbers is another. The discovery of irrational numbers got poor Hippasus killed. Imaginary numbers are another one. Our daily reasoning and experience is filled with useful physical fictions. All of our decision making revolves around counterfactuals, which by definition do not physically exist. We try to figure out if something is true or false, but false things do not exist. Ideas do not have physical properties. The redness of an apple does not inhere in the physical/chemical sequence of lightwaves hitting rod cells and transmitting signals to the brain. And so on. Without the blinding presupposition of physicalism, physicalism is obviously false. Self referentially so, since "physicalism is true" is not a physical object, otherwise I could step on it and physicalism would cease to be.EricMH_{February 19, 2018
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To answer my own question, is the set of all the different aleph infinities the biggest infinity?EricMH_{February 19, 2018
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Yes, something to ponder... Thinking back, I can remember a few times when reasoning about infinitesimals helped illuminate concepts for me, for example infinitesimal transformations in the context of Lie groups and algebras.daveS_{February 17, 2018
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DS, I was taught the old joke about the two drunks on a dark, rainy night. What are you looking for, down on your knees like that? My contact lenses. So down went the second to help. After some time, are you sure you lost them here? I actually lost them over there in the dark but this is where the light is, under this lamp-post. Sometimes, the difference between theory and model is a lot less than we are willing to acknowledge. KFkairosfocus_{February 17, 2018
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KF,
DS,infinitesimals are all over physics.
Yes, at least they appear in arguments and calculations that physicists use. But I don't think there are any physically real infinitesimal entities, which I find somewhat paradoxical.daveS_{February 17, 2018
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DS,infinitesimals are all over physics. So are vectors and matrices, complex numbers etc. Near as I figure, Math and Physics did not get really separated until C19. KFkairosfocus_{February 17, 2018
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KF & johnnyb, One criticism of my own position, in light of the reddit thread that KF linked to: I was saying that I prefer to think of calculus completely in terms of the real numbers (so no infinitesimals and no infinities) because that is more compatible with my physical intuition, including my belief that there are no "infinitesimals" in the physical world. I further suggest that as the "best" approach for elementary calculus education in general. On the other hand, actual physicists often do use infinitesimal-style arguments, and presumably they know what's most productive for them. So maybe my posts above simply reflect the fact that I was taught calculus in a particular way? And perhaps a lack of aptitude for physics... .daveS_{February 17, 2018
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Book: https://www.math.wisc.edu/%7Ekeisler/calc.htmlkairosfocus_{February 16, 2018
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Fun exchanges: https://www.reddit.com/r/math/comments/5tppmy/nonstandard_analysis_in_physics/kairosfocus_{February 16, 2018
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KF,
DS, there is an approach, non-standard analysis.
Of course, I've read about it. But AFAIK, there are no such things as distances/intervals of infinitesimal length in space or time, so for me, sticking with the real numbers works better. YMMV.daveS_{February 16, 2018
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DS, there is an approach, non-standard analysis. It gives fairly obvious meaning to things like dy, dx, dt etc. There is a textbook online. KFkairosfocus_{February 16, 2018
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johnnyb,
Actually, I find that infinitesimals make it 1000x more intuitive.
Thanks for the offer. I suppose this is largely a matter of taste. I think of the real line, plane, etc. in terms of physical space and time, and as far as I know, infinitesimals don't correspond to real spatial or temporal entities. Two distinct points in space (or time) are always separated by a non-infinitesimal distance (or time interval). At least in a "basic" physics class, they are. On the other hand, it's always good to consider other approaches, and if they are successful at getting students interested in calculus and math in general, that's a positive thing.daveS_{February 16, 2018
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daveS - Actually, I find that infinitesimals make it 1000x more intuitive. I teach homeschool Calculus co-ops every year. Infinitesimals are a lot easier to understand than, say, imaginary numbers. In fact, once you have the machinery to understand real numbers (i.e., the idea that fractions don't give you all the numbers even though they can be super tiny), the extension to infinitesimals is very natural (just like there were numbers in-between the rationals, there are also numbers in-between the reals). You can also represent these algebraically with a "w^n" being the level of infinity, and w^(-n) (or e^n) being the level of infinitesimal. Then they work just as algebraic units, and you can just perform normal algebraic manipulations on them for answers. If you are interested, I can send you the PDF of my calc book as it currently stands. Just email me at jonathan.bartlett@blythinstitute.org Jonjohnnyb_{February 16, 2018
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What is the biggest infinity?EricMH_{February 16, 2018
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johnnyb,
By the way, the application of infinity as a quantity makes doing Calculus and related mathematics a *lot* easier.
:o I agree that the mechanics of calculus can be simplified by using infinitesimals, but would it really be a good idea to introduce them in freshman calculus, for example? Even an introductory analysis class? Those students are really in no position to understand number systems with infinitesimals (granted, even the real number system can be challenging to them). I guess I would be on the side of the purgers. Elementary calculus is a setting in which students build intuition about the topology of the real line, and I think infinitesimals would just muddy the waters.daveS_{February 16, 2018
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By the way, the application of infinity as a quantity makes doing Calculus and related mathematics a *lot* easier. A huge barrier to learning Calculus has been from the attempt to purge infinities and infinitesimals from it. If you've ever had to do limit proofs, you know what I mean. Infinitesimals make Calculus infinitely easier. They even make limits (which were supposed to replace them) infinitely easier.johnnyb_{February 16, 2018
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KF,
Transfinitely many is clear, just what kind is not.
Yes, and I think a skeptic could find this state of affairs problematic.daveS_{February 16, 2018
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DS, that gets into things like W^W and worse, W standing for omega the order type of the natural counting numbers. Transfinitely many is clear, just what kind is not. KFkairosfocus_{February 16, 2018
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KF,
There are transfinitely many transfinites
I would agree with that, or at least I've probably said similar things here myself. For any natural number n, there are more than n different infinite cardinal numbers. But when we say there are "transfinitely many" infinities, that naturally leads to the question of which degree of infinity "transfinitely many" refers to. And that is a question that is impossible or at least difficult to answer.daveS_{February 16, 2018
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DS, it was you who pointed me to the surreals. There are transfinitely many transfinites -- and the infinitesimals (all but zero quantities that appear in say the nonstandard analysis form of Calculus) are the same too. The relocation trick means that a transfinite cloud of infinitesimals surrounds any given specific number, forming an infinitely close neighbourhood, so to speak. KFkairosfocus_{February 16, 2018
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the fact of the matter is that mathematics itself is very much a spiritual exercise.
ba77, I think this is true in the sense that anything could be a spiritual exercise. Everything we do can be a spiritual exercise if we offer it to the Lord. Anyway, my beefs with infinity are perhaps what your comment touches on. It's illogical to appeal to a 1) real infinity if materialism is your religion. There cant be an infinity, because we could never perceive or detect it with physical science. 2) Mathematical infinity is completely imaginary. Its realm is strictly the mind. Its just a way of arranging symbols. Like I said, 3) infinity as theology is a whole 'nother thing, and I get the feeling people want to smear the different notions of infinity together, which doesn't make any sense to me. Andrewasauber_{February 16, 2018
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I can think of one objection that a skeptic might raise. If we claim that there are multiple "degrees of infinity", then one might expect us to state precisely how many there are. Perhaps two? Aleph-null? Aleph-seventeen? But no, a mathematician (working in ZFC, say) will mutter something about proper classes having no well-defined cardinality, and slam the door in your face. Some people will find that response unsatisfactory.daveS_{February 16, 2018
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This comment,,,
"you may be persuaded that this has any meaning spiritual or physical"
,,, Reminded me of this comment from Berlinski
An Interview with David Berlinski - Jonathan Witt Berlinski: There is no argument against religion that is not also an argument against mathematics. Mathematicians are capable of grasping a world of objects that lies beyond space and time…. Interviewer:… Come again(?) … Berlinski: No need to come again: I got to where I was going the first time. The number four, after all, did not come into existence at a particular time, and it is not going to go out of existence at another time. It is neither here nor there. Nonetheless we are in some sense able to grasp the number by a faculty of our minds. Mathematical intuition is utterly mysterious. So for that matter is the fact that mathematical objects such as a Lie Group or a differentiable manifold have the power to interact with elementary particles or accelerating forces. But these are precisely the claims that theologians have always made as well – that human beings are capable by an exercise of their devotional abilities to come to some understanding of the deity; and the deity, although beyond space and time, is capable of interacting with material objects. http://tofspot.blogspot.com/2013/10/found-upon-web-and-reprinted-here.html
And where Darwinian evolution is based on a materialistic view of reality which denies that anything beyond nature exists, on the other hand, Mathematics exists in a transcendent, beyond space and time, realm which is not reducible any possible material explanation. This transcendent mathematical realm has been referred to as a Platonic mathematical world.
Platonic mathematical world - image https://image.slidesharecdn.com/quantuminformation2-120301000431-phpapp01/95/quantum-information-14-728.jpg?cb=1330561190
As the following article points out, 'Mathematical platonism enjoys widespread support and is frequently considered the default metaphysical position with respect to mathematics.,,, and that arguments for mathematical platonism typically assert,,, that mathematical entities are not constituents of the spatio-temporal realm.'
Mathematical Platonism Excerpt: Mathematical platonism enjoys widespread support and is frequently considered the default metaphysical position with respect to mathematics. This is unsurprising given its extremely natural interpretation of mathematical practice. In particular, mathematical platonism takes at face-value such well known truths as that "there exist" an infinite number of prime numbers, and it provides straightforward explanations of mathematical objectivity and of the differences between mathematical and spatio-temporal entities. Thus arguments for mathematical platonism typically assert that in order for mathematical theories to be true their logical structure must refer to some mathematical entities, that many mathematical theories are indeed objectively true, and that mathematical entities are not constituents of the spatio-temporal realm. The most common challenge to mathematical platonism argues that mathematical platonism requires an impenetrable metaphysical gap between mathematical entities and human beings. Yet an impenetrable metaphysical gap would make our ability to refer to, have knowledge of, or have justified beliefs concerning mathematical entities completely mysterious. Frege, Quine, and "full-blooded platonism" offer the three most promising responses to this challenge.,,, http://www.iep.utm.edu/mathplat/
Simply put, Mathematics itself, contrary to the materialistic presuppositions of Darwinists, does not need the physical world in order to exist. And yet Darwinists, although they deny that anything beyond nature exists, need this transcendent world of mathematics in order for their theory to be considered scientific in the first place. The predicament that Darwinists find themselves in regards to denying the reality of this transcendent world of mathematics, and yet needing validation from this transcendent world of mathematics in order to be considered scientific, should be the very definition of self-refuting. And although some mathematicians who are of the atheistic persuasion may be tempted to believe that this transcendent Platonic mathematical world can exist without God, yet, due to Gödel’s incompleteness theorem, they would be wrong to believe that.
A BIBLICAL VIEW OF MATHEMATICS Vern Poythress - Doctorate in theology, PhD in Mathematics (Harvard) 15. Implications of Gödel’s proof B. Metaphysical problems of anti-theistic mathematics: unity and plurality Excerpt: Because of the above difficulties, anti-theistic philosophy of mathematics is condemned to oscillate, much as we have done in our argument, between the poles of a priori knowledge and a posteriori knowledge. Why? It will not acknowledge the true God, wise Creator of both the human mind with its mathematical intuition and the external world with its mathematical properties. In sections 22-23 we shall see how the Biblical view furnishes us with a real solution to the problem of “knowing” that 2 + 2 = 4 and knowing that S is true. http://www.frame-poythress.org/a-biblical-view-of-mathematics/ Taking God Out of the Equation - Biblical Worldview - by Ron Tagliapietra - January 1, 2012 Excerpt: Kurt Gödel (1906–1978) proved that no logical systems (if they include the counting numbers) can have all three of the following properties. 1. Validity ... all conclusions are reached by valid reasoning. 2. Consistency ... no conclusions contradict any other conclusions. 3. Completeness ... all statements made in the system are either true or false. The details filled a book, but the basic concept was simple and elegant. He (Godel) summed it up this way: “Anything you can draw a circle around cannot explain itself without referring to something outside the circle—something you have to assume but cannot prove.” For this reason, his proof is also called the Incompleteness Theorem. Kurt Gödel had dropped a bomb on the foundations of mathematics. Math could not play the role of God as infinite and autonomous. It was shocking, though, that logic could prove that mathematics could not be its own ultimate foundation. Christians should not have been surprised. The first two conditions are true about math: it is valid and consistent. But only God fulfills the third condition. Only He is complete and therefore self-dependent (autonomous). God alone is “all in all” (1 Corinthians 15:28), “the beginning and the end” (Revelation 22:13). God is the ultimate authority (Hebrews 6:13), and in Christ are hidden all the treasures of wisdom and knowledge (Colossians 2:3). http://www.answersingenesis.org/articles/am/v7/n1/equation
,,, Kurt Gödel himself stated,,,,
“In materialism all elements behave the same. It is mysterious to think of them as spread out and automatically united. For something to be a whole, it has to have an additional object, say, a soul or a mind.”,,, Kurt Gödel – Hao Wang’s supplemental biography of Gödel, A Logical Journey, MIT Press, 1996. [9.4.12]
The following article puts the implications of Gödel's incompleteness theorem for mathematics as such,
THE GOD OF THE MATHEMATICIANS - DAVID P. GOLDMAN - August 2010 Excerpt: we cannot construct an ontology that makes God dispensable. Secularists can dismiss this as a mere exercise within predefined rules of the game of mathematical logic, but that is sour grapes, for it was the secular side that hoped to substitute logic for God in the first place. Gödel's critique of the continuum hypothesis has the same implication as his incompleteness theorems: Mathematics never will create the sort of closed system that sorts reality into neat boxes. http://www.firstthings.com/article/2010/08/the-god-of-the-mathematicians
Thus, although some people may be presupposed to think that the mathematical realm and the spiritual realm are not even in the same ballpark, the fact of the matter is that mathematics itself is very much a spiritual exercise. In fact, when mathematicians are shown equations such as Euler's identity or the Pythagorean identity the same area of the brain used to appreciate fine art or music lights up:
Mathematics: Why the brain sees maths as beauty – Feb. 12, 2014 Excerpt: Mathematicians were shown "ugly" and "beautiful" equations while in a brain scanner at University College London. The same emotional brain centres used to appreciate art were being activated by "beautiful" maths.,,, One of the researchers, Prof Semir Zeki, told the BBC: "A large number of areas of the brain are involved when viewing equations, but when one looks at a formula rated as beautiful it activates the emotional brain - the medial orbito-frontal cortex - like looking at a great painting or listening to a piece of music." http://www.bbc.com/news/science-environment-26151062
Moreover, besides mathematics being incomplete, and as Dr. Bruce Gordon points out in the following article, the evidence from cosmology and quantum mechanics now also proves that the physical universe is itself also causally incomplete.
BRUCE GORDON: Hawking’s irrational arguments – October 2010 Excerpt: ,,,The physical universe is causally incomplete and therefore neither self-originating nor self-sustaining. The world of space, time, matter and energy is dependent on a reality that transcends space, time, matter and energy. This transcendent reality cannot merely be a Platonic realm of mathematical descriptions, for such things are causally inert abstract entities that do not affect the material world,,, Rather, the transcendent reality on which our universe depends must be something that can exhibit agency – a mind that can choose among the infinite variety of mathematical descriptions and bring into existence a reality that corresponds to a consistent subset of them. This is what “breathes fire into the equations and makes a universe for them to describe.” Anything else invokes random miracles as an explanatory principle and spells the end of scientific rationality. http://www.washingtontimes.com/news/2010/oct/1/hawking-irrational-arguments/
And I would also argue that the mathematical infinities that are encountered at the Big Bang, Blackholes, and the quantum wave function, are some of the primary reasons, if not THE primary reason, why we know the physical universe is itself causally incomplete, i.e. neither self-originating nor self-sustaining. Therefore, since infinity crops up in places that are of great interest in science, I would argue that a fairly accurate understanding of infinity would be of great benefit. Of supplemental note is this quote from Newton:
The Supreme God is a Being eternal, infinite, absolutely perfect;,,, from his true dominion it follows that the true God is a living, intelligent, and powerful Being; and, from his other perfections, that he is supreme, or most perfect. He is eternal and infinite, omnipotent and omniscient; that is, his duration reaches from eternity to eternity; his presence from infinity to infinity; he governs all things, and knows all things that are or can be done. He is not eternity or infinity, but eternal and infinite; he is not duration or space, but he endures and is present. He endures for ever, and is every where present”: - Sir Isaac Newton - Quoted from what many consider the most influential science book of all time, his book "Principia"
bornagain77_{February 16, 2018
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Here's two articles I've written on infinity: How to Count to Infinity (basically what I've written here) Seeing Infinity (the usage of infinity for practical applications, integrating scientific and theological outlooks)johnnyb_{February 15, 2018
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asauber - There are multiple ways to compare sizes of infinities. There is a difference between "ordinality" and "cardinality" of infinite numbers, but in each you have different sizes (ordinals can be ordered, but they aren't necessarily "bigger" in an absolute sense). The way that you can compare infinities is this: 1) If every member of set A can be matched to a unique member of set B, then B is at least as large as A 2) If every member of set B can be matched to a unique member of set A, then A is at least as large as B 3) If 1 & 2 can be shown, then A is the same size as B Now, if (1) holds, but it can be proven that (2) does not hold (usually shown by demonstrating that (2) leads to a contradiction), then B is *larger* than A. This demonstration has been done to show that the reals are a larger infinity than the integers. If you want the details, google Cantor and "diagonalization". I can post it myself, but probably not until later.johnnyb_{February 15, 2018
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Of related note:
Mathematicians Measure Infinities and Find They’re Equal - 2017 In a breakthrough that disproves decades of conventional wisdom, two mathematicians have shown that two different variants of infinity are actually the same size. The advance touches on one of the most famous and intractable problems in mathematics: whether there exist infinities between the infinite size of the natural numbers and the larger infinite size of the real numbers. The problem was first identified over a century ago. At the time, mathematicians knew that “the real numbers are bigger than the natural numbers, but not how much bigger. Is it the next biggest size, or is there a size in between?” said Maryanthe Malliaris of the University of Chicago, co-author of the new work along with Saharon Shelah of the Hebrew University of Jerusalem and Rutgers University. In their new work, Malliaris and Shelah resolve a related 70-year-old question about whether one infinity (call it p) is smaller than another infinity (call it t). They proved the two are in fact equal, much to the surprise of mathematicians. “It was certainly my opinion, and the general opinion, that p should be less than t,” Shelah said. Consider the real numbers, which are all the points on the number line. The real numbers are sometimes referred to as the “continuum,” reflecting their continuous nature: There’s no space between one real number and the next. Cantor was able to show that the real numbers can’t be put into a one-to-one correspondence with the natural numbers: Even after you create an infinite list pairing natural numbers with real numbers, it’s always possible to come up with another real number that’s not on your list. Because of this, he concluded that the set of real numbers is larger than the set of natural numbers. Thus, a second kind of infinity was born: the uncountably infinite. What Cantor couldn’t figure out was whether there exists an intermediate size of infinity — something between the size of the countable natural numbers and the uncountable real numbers. He guessed not, a conjecture now known as the continuum hypothesis.,,, The details of the two sizes don’t much matter. What’s more important is that mathematicians quickly figured out two things about the sizes of p and t. First, both sets are larger than the natural numbers. Second, p is always less than or equal to t. Therefore, if p is less than t, then p would be an intermediate infinity — something between the size of the natural numbers and the size of the real numbers. The continuum hypothesis would be false.,,, Malliaris and Shelah proved that p and t are equal by cutting a path between model theory and set theory https://www.quantamagazine.org/mathematicians-measure-infinities-find-theyre-equal-20170912/
bornagain77_{February 15, 2018
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