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Are some infinities bigger than others?

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infinity as snake swallowing tail

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?


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

47 Replies to “Are some infinities bigger than others?

  1. 1
    asauber says:

    a bigger infinity

    Someone seriously wrote this?


  2. 2
    daveS says:

    It turns out that the set of all points on a continuous line is a bigger infinity than the natural numbers;

    There are always philosophical questions lurking in the background, but this is accepted as true by nearly everyone who has examined the issues seriously, regardless of their “worldview”.

  3. 3
    asauber says:

    but this is accepted as true by nearly everyone who has examined the issues seriously

    Wow. A knee-jerk Appeal to Authority

    or Size

    Nearly Everyone is Pretty Big.

    But is it as big as a larger infinity?

    Guess the smart people will tell me.


  4. 4
    daveS says:


    Have you read Cantor’s proof that the real numbers cannot be put into 1-1 correspondence with the natural numbers?

  5. 5
    asauber says:

    Have you read Cantor’s proof that the real numbers cannot be put into 1-1 correspondence with the natural numbers?


    Why would I bother? What in infinity could I would I find meaningful if I read it?


  6. 6
    daveS says:


    Why would I bother? What in infinity could I would I find meaningful if I read it?

    You might discover why people say that some infinities are bigger than others.

  7. 7
    bornagain77 says:

    Gödel, Infinity, and Jesus Christ as the Theory of Everything – video

    Gödel’s incompleteness theorem can be stated simply as such: “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”.

    Even Stephen Hawking himself, who tried to put forth a mathematical theory of everything in his book ‘The Grand Design’, reluctantly admitted in that book that Kurt Gödel, with his incompleteness theorem. ‘halted the achievement of a unifying all-encompassing theory of everything’

    “Note that despite the incontestability of Euclid’s postulates in mathematics, (ref. on cite), Gödel’s incompleteness theorem (1931), proves that there are limits to what can be ascertained by mathematics. Kurt Gödel (ref. on cite), halted the achievement of a unifying all-encompassing theory of everything in his theorem that: “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”. Thus, based on the position that an equation cannot prove itself, the constructs are based on assumptions some of which will be unprovable.”
    Cf., Stephen Hawking & Leonard Miodinow, The Grand Design (2010) @ 15-6

    Gödel’s incompleteness theorems were ultimately a cumulation of work that was originally started by Georg Cantor who lived from 1845 to 1918

    Kurt Gödel
    Excerpt: Gödel made an immense impact upon scientific and philosophical thinking in the 20th century, a time when others such as Bertrand Russell,[3] A. N. Whitehead,[3] and David Hilbert were analyzing the use of logic and set theory to understand the foundations of mathematics pioneered by Georg Cantor.

    Naming and Diagonalization, from Cantor to Gödel to Kleene – 2006
    Excerpt: The first part of the paper is a historical reconstruction of the way Godel probably derived his proof from Cantor’s diagonalization, through the semantic version of Richard. The incompleteness proof-including the fixed point construction-result from a natural line of thought, thereby dispelling the appearance of a “magic trick”. The analysis goes on to show how Kleene’s recursion theorem is obtained along the same lines.

    Cantor was trying to bring a systematic understanding of infinity into mathematics. This endeavor was very much a theological quest for Cantor. Cantor, although he pioneered some very useful tools in mathematics, useful tools which were essential to Gödel in his work on incompleteness, Cantor, none-the less, ultimately failed in his endeavor to bring a systematic understanding of infinity into mathematics. This following video gives us a very interesting glimpse into how Gödel was able to pick up the pieces where Cantor had failed, and bring his incompleteness theorems for mathematics to fruition:

    BBC-Dangerous Knowledge – Part 1 (Cantor, Boltzmann, Gödel and Turing) – video
    Part 2

    And although Cantor was ultimately proven wrong by Gödel in his quest to fully integrate a systematic understanding of infinity into mathematics,,,,

    The God of the Mathematicians – by David P. Goldman – 2010
    The religious beliefs that guided Kurt Gödel’s revolutionary ideas
    Excerpt: That is Cantor’s “continuum hypothesis,” which attempts to identify a first and second transfinite cardinal number. From there, he believed, all the possible orders of infinity could be counted, the same way the integers count groups of one, two, three, and so forth. He not only recognized, but was driven by, the ontological implications of this assertion: If the continuum hypothesis turned out to be true, Spinoza would be vindicated because God’s infinity could be packaged into a neat series of numbers. Cantor spent the last thirty-five years of his life in a vain effort to prove this. He died in 1918 in a mental hospital.
    It was Gödel and, later, Paul Cohen who demonstrated respectively that Cantor’s continuum hypothesis could be neither proved nor disproved within existing set theory. Indeed, Cantor’s hypothesis remains maddeningly undecidable.,,,
    God’s infinitude remains safe in heaven. Mathematicians have proven that an infinite number of transfinite numbers exist but cannot tell what they are or in what order they should be arranged.

    ,,, none-the-less, I believe that both Cantor, and Gödel, (since they were both Christians), would be very pleased, since it is very comforting to overall Christian concerns, to see how infinity is now used, mathematically, in Quantum Mechanics and in modern day Quantum Electrodynamics,

    Double Slit, Quantum-Electrodynamics, and Christian Theism – video

    From the preceding video, it is also very interesting to note that Quantum-Electrodynamics was the first attempt at unifying relativity and quantum mechanics. Specifically, in Quantum-Electrodynamics special relativity was merged with electromagnetism to produce what is held to be the ‘most precise theory of natural phenomena ever developed’:

    Theories of the Universe: Quantum Mechanics vs. General Relativity
    Excerpt: The first attempt at unifying relativity and quantum mechanics took place when special relativity was merged with electromagnetism. This created the theory of quantum electrodynamics, or QED. It is an example of what has come to be known as relativistic quantum field theory, or just quantum field theory. QED is considered by most physicists to be the most precise theory of natural phenomena ever developed.

    Richard Feynman was only able to unify special relativity and quantum mechanics into Quantum Electrodynamics by quote unquote “brushing infinity under the rug” with a technique called Renormalization

    THE INFINITY PUZZLE: Quantum Field Theory and the Hunt for an Orderly Universe
    Excerpt: In quantum electrodynamics, which applies quantum mechanics to the electromagnetic field and its interactions with matter, the equations led to infinite results for the self-energy or mass of the electron. After nearly two decades of effort, this problem was solved after World War II by a procedure called renormalization, in which the infinities are rolled up into the electron’s observed mass and charge, and are thereafter conveniently ignored. Richard Feynman, who shared the 1965 Nobel Prize with Julian Schwinger and Sin-Itiro Tomonaga for this breakthrough, referred to this sleight of hand as “brushing infinity under the rug.”

  8. 8
    bornagain77 says:

    In the following video, Richard Feynman rightly expresses his unease with “brushing infinity under the rug.” in Quantum-Electrodynamics:

    “It always bothers me that in spite of all this local business, what goes on in a tiny, no matter how tiny, region of space, and no matter how tiny a region of time, according to laws as we understand them today, it takes a computing machine an infinite number of logical operations to figure out. Now how can all that be going on in that tiny space? Why should it take an infinite amount of logic to figure out what one stinky tiny bit of space-time is going to do?”
    – Richard Feynman – one of the founding fathers of QED (Quantum Electrodynamics)
    Quote taken from the 6:45 minute mark of the following video:
    Feynman: Mathematicians versus Physicists – video

    I don’t know about Richard Feynman, but as for myself, as a Christian Theist, I find it rather comforting to know that it takes an ‘infinite amount of logic to figure out what one stinky tiny bit of space-time is going to do’:

    “Why should it take an infinite amount of logic to figure out what one stinky tiny bit of space-time is going to do?”
    – Richard Feynman

    “In the beginning was the Word, and the Word was with God, and the Word was God.”

    of note: ‘the Word’ in John1:1 is translated from ‘Logos’ in Greek. Logos is also the root word from which we derive our modern word logic

    ,,, And whereas special relativity, by ‘brushing infinity under the rug’, has been successfully unified with quantum theory to produce Quantum Electrodynamics, no such mathematical ‘sleight of hand’ exists for unifying general relativity with quantum mechanics.

    General relativity simply refuses to be mathematically unified with quantum mechanics in any acceptable way. In technical terms, Gravity has yet to be successfully included into a theory of everything since the infinities that crop up in that attempt are not renormalizable as they were in Quantum-Electrodynamics.

    Unified field theory
    Excerpt: Gravity has yet to be successfully included in a theory of everything.
    Simply trying to combine the graviton with the strong and electroweak interactions runs into fundamental difficulties since the resulting theory is not renormalizable. Theoretical physicists have not yet formulated a widely accepted, consistent theory that combines general relativity and quantum mechanics. The incompatibility of the two theories remains an outstanding problem in the field of physics.
    Some theoretical physicists currently believe that a quantum theory of general relativity may require frameworks other than field theory itself, such as string theory or loop quantum gravity.

    Does quantum mechanics contradict the theory of relativity?
    Sanjay Sood, Microchip Design Engineer, Theoretical and Applied Physicist – Feb 14, 2016
    Excerpt: quantum mechanics was first integrated with special theory of relativity by Dirac in 1928 just 3 years after quantum mechanics was discovered. Dirac produced an equation that describes the behavior of a quantum particle (electron). In this equation the space and time enter on the same footing – equation is first order in all 4 coordinates. One startling by product of this equation was the prediction of anti matter. It also gave the correct explanation for the electron’s spin. Dirac’s equation treats an electron as a particle with only a finite degrees of freedom.
    In 1940s Dirac’s equation was incorporated into the relativistic quantum field theory that’s knowns as quantum electrodynamics (QED) independently by Feynman, Schwinger and Tomonaga. This is the theory that describes the behavior of electrons and photons and their interactions with each other in terms of relativistic quantum fields that have infinite degrees of freedom. QED allowed extremely precise calculation of anomalous magnetic dipole moment of an electron. This calculated value matches the experimentally measured value to an astonishing precision of 12 decimal places!
    The integration of Einstein’s general theory of relativity and quantum mechanics has proved to be far more difficult. Such an integration would give a quantum theory of gravity. Even after a sustained effort lasting more than half a century, no renormalized quantum field theory of gravity has ever been produced. Renormalization means a theory that’s free of infinities at zero distance or infinite energy because 2 point particles can interact with each other at zero distance. A non renormalizable theory has no predictive value because it contains an infinite number of singular coefficients.

    Some theoretical physicists have remarked that this failure to mathematically unify Quantum Mechanics and General Relativity into a mathematical “Theory of Everything” is ‘the collapse of physics as we know it’
    At the 4:30 minute mark of the following video, Michio Kaku, after going through some calculations with the equation of General Relativity showing how it breaks down at black holes, states that, ‘It means the collapse of everything we know about the physical universe.’

    “Here is the problem (with black holes), right there, when ‘r’ (radius) is equal to zero, The point at which physics itself breaks down. So 1 over ‘r’ equals 1 over 0 equals infinity. To a mathematician infinity is simply a number without limit. To a physicist it is a monstrosity. It means first of all that gravity is infinite at the center of a black hole. That time stops. And what does that mean? Space makes no sense. It means the collapse of everything we know about the physical universe. In the real world there is no such thing as infinity. Therefore there is a fundamental flaw in the formulation of Einstein’s theory.”
    Quantum Mechanics & Relativity – Michio Kaku – The Collapse Of Physics As We Know It ? – video

    Then in the same video at the 7:08 minute mark, Michio Kaku,, after going through some calculations trying to integrate Quantum Mechanics and Gravity at a black hole, goes on to state “And when you do this integral, you get something which makes no sense whatsoever. An infinity. Total nonsense. In fact, you get an infinite sequence of infinities. Infinitely worse than the divergences of Einstein’s original theory.”

    “And when you do this integral, you get something which makes no sense whatsoever. An infinity. Total nonsense. In fact, you get an infinite sequence of infinities. Infinitely worse than the divergences of Einstein’s original theory.”
    Quantum Mechanics & Relativity – Michio Kaku – The Collapse Of Physics As We Know It ? – video

    As the preceding video clearly illustrated, the main conflict of reconciling General Relativity and Quantum Mechanics appears to arise from the inability of either theory to successfully deal with the Zero/Infinity conflict that crops up between each theory:

    Excerpt: The biggest challenge to today’s physicists is how to reconcile general relativity and quantum mechanics. However, these two pillars of modern science were bound to be incompatible. “The universe of general relativity is a smooth rubber sheet. It is continuous and flowing, never sharp, never pointy. Quantum mechanics, on the other hand, describes a jerky and discontinuous universe. What the two theories have in common – and what they clash over – is zero.”,, “The infinite zero of a black hole — mass crammed into zero space, curving space infinitely — punches a hole in the smooth rubber sheet. The equations of general relativity cannot deal with the sharpness of zero. In a black hole, space and time are meaningless.”,, “Quantum mechanics has a similar problem, a problem related to the zero-point energy. The laws of quantum mechanics treat particles such as the electron as points; that is, they take up no space at all. The electron is a zero-dimensional object,,, According to the rules of quantum mechanics, the zero-dimensional electron has infinite mass and infinite charge.

    Yet, the unification, into a ‘theory of everything’, between what is in essence the ‘infinite Theistic world of Quantum Mechanics’ and the ‘finite Materialistic world of the space-time of General Relativity’ seems to be directly related to what Jesus apparently joined together with His resurrection, That is to say, it looks to be directly related to the unification of infinite God with finite man.

    Dr. William Dembski in this following comment, although he was not addressing the Zero/Infinity conflict in General Relativity and Quantum Mechanics, offers this insight into what the ‘unification’ of infinite God and the finite man might look like mathematically:

    The End Of Christianity – Finding a Good God in an Evil World – Pg.31
    William Dembski PhDs. Mathematics and Theology
    Excerpt: “In mathematics there are two ways to go to infinity. One is to grow large without measure. The other is to form a fraction in which the denominator goes to zero. The Cross is a path of humility in which the infinite God becomes finite and then contracts to zero, only to resurrect and thereby unite a finite humanity within a newfound infinity.”

    Philippians 2:8-9
    And being found in appearance as a man, He humbled Himself and became obedient to the point of death, even the death of the cross. Therefore God also has highly exalted Him and given Him the name which is above every name,

  9. 9
    bornagain77 says:

    Of personal note: I hold it to be fairly obvious that ‘growing large without measure’ is a lesser quality infinity than a fraction in which the denominator goes to zero.

    Can A “Beginning-less Universe” Exist? – William Lane Craig – video
    ,,”the impossiblity of forming an actually infinite number of things by adding one member after another.,,,
    1. A collection formed by adding one member to another cannot be actually infinite,,,”

    Simply put, the reason why ‘growing large without measure’ is a lesser quality infinity than ‘a fraction in which the denominator goes to zero’ is because anything that begins to grow large without measure must necessarily have some sort of beginning in time and must also necessarily have some sort of preexistent space to grow into. Whereas, on the other hand, to form a fraction in which the denominator goes to zero is to force a finite object into a type of infinity that can have no discernible beginning in space or time. That is to say, since zero represents the absence of everything including the absence of space and time, then to form a fraction in which the denominator goes to zero is to force a finite object into a true infinity that is beyond space and time.

    In more technical terms, I view growing large without measure to be a potential infinite and I view a fraction in which the denominator goes to zero to be an actual infinite.

    Potential Infinity vs. Actual Infinity – June 7, 2012 by Ryan
    Excerpt: In a potential infinity, one can keep adding or subdividing without end, but one never actually reaches infinity. In a sense, a potential infinity is an endless process that at any point along the way is finite. By contrast, in an actual infinity, the infinite is viewed as a completed totality.

    Moreover, if we rightly let the Agent causality of God ‘back’ into the picture of modern physics, as the Christian founders of modern science originally envisioned, (Newton, Maxwell, Faraday, and Planck, to name a few), , then an empirically backed reconciliation between Quantum Mechanics and General Relativity, i.e. the ‘Theory of Everything’, readily pops out for us in Christ’s resurrection from the dead.
    Specifically, the fact that Jesus Christ dealt with both general relativity and quantum mechanics in His resurrection from the dead is made evident by the Shroud of Turin.

    Particle Radiation from the Body – July 2012 – M. Antonacci, A. C. Lind
    Excerpt: The Shroud’s frontal and dorsal body images are encoded with the same amount of intensity, independent of any pressure or weight from the body. The bottom part of the cloth (containing the dorsal image) would have born all the weight of the man’s supine body, yet the dorsal image is not encoded with a greater amount of intensity than the frontal image. Radiation coming from the body would not only explain this feature, but also the left/right and light/dark reversals found on the cloth’s frontal and dorsal body images.

    The absorbed energy in the Shroud body image formation appears as contributed by discrete (quantum) values – Giovanni Fazio, Giuseppe Mandaglio – 2008
    Excerpt: This result means that the optical density distribution,, can not be attributed at the absorbed energy described in the framework of the classical physics model. It is, in fact, necessary to hypothesize a absorption by discrete values of the energy where the ‘quantum’ is equal to the one necessary to yellow one fibril.

    Astonishing discovery at Christ’s tomb supports Turin Shroud – NOV 26TH 2016
    Excerpt: The first attempts made to reproduce the face on the Shroud by radiation, used a CO2 laser which produced an image on a linen fabric that is similar at a macroscopic level. However, microscopic analysis showed a coloring that is too deep and many charred linen threads, features that are incompatible with the Shroud image. Instead, the results of ENEA “show that a short and intense burst of VUV directional radiation can color a linen cloth so as to reproduce many of the peculiar characteristics of the body image on the Shroud of Turin, including shades of color, the surface color of the fibrils of the outer linen fabric, and the absence of fluorescence”.
    ‘However, Enea scientists warn, “it should be noted that the total power of VUV radiations required to instantly color the surface of linen that corresponds to a human of average height, body surface area equal to = 2000 MW/cm2 17000 cm2 = 34 thousand billion watts makes it impractical today to reproduce the entire Shroud image using a single laser excimer, since this power cannot be produced by any VUV light source built to date (the most powerful available on the market come only to several billion watts)”.
    The ENEA study of the Holy Shroud of Turin concluded that it would take 34 Thousand Billion (trillion) Watts of VUV radiation to make the image on the shroud. This output of electromagnetic energy remains beyond human technology.


    Matthew 28:18
    And Jesus came to them and spake unto them, saying, All authority hath been given unto me in heaven and on earth.

    Colossians 1:15-20
    The Son is the image of the invisible God, the firstborn over all creation. For in him all things were created: things in heaven and on earth, visible and invisible, whether thrones or powers or rulers or authorities; all things have been created through him and for him. He is before all things, and in him all things hold together. And he is the head of the body, the church; he is the beginning and the firstborn from among the dead, so that in everything he might have the supremacy. For God was pleased to have all his fullness dwell in him, and through him to reconcile to himself all things, whether things on earth or things in heaven, by making peace through his blood, shed on the cross.

    This following video also touches on the ‘higher dimensional’ math behind Quantum Mechanics, Special Relativity and General Relativity

    Quantum Mechanics, Special Relativity, General Relativity and Christianity

  10. 10
    BPS from AZ says:

    Albeit in a somewhat different context, William James, riffing on Bergson, once said this about the nature of mathematical mind:

    ‘When the mathematician,’ Bergson writes, ‘calculates the state of a system at the end of a time t, nothing need prevent him from supposing that betweenwhiles the universe vanishes, in order suddenly to appear again at the due moment in the new configuration. It is only the t-th moment that counts—that which flows throughout the intervals, namely real time, plays no part in his calculation…. In short, the world on which the mathematician operates is a world which dies and is born anew at every instant, like the world which Descartes thought of when he spoke of a continued creation.’ To know adequately what really happens we ought, Bergson insists, to see into the intervals, but the mathematician sees only their extremities. He fixes only a few results, he dots a curve and then interpolates, he substitutes a tracing for a reality.

    Multiple infinities? Could all those countless “betweenwhiles” perhaps just be so many figments of the limited, dividing mind? I’d prefer to trust the perennial sages who say “One being, without a second”.

  11. 11
    asauber says:

    You might discover why people say that some infinities are bigger than others.

    You’re digging the hole deeper.


  12. 12
    asauber says:


    To be straightforward in letting you know where I’m coming from, someone is going to have to produce evidence that Infinity is more than just a entertaining mental exercise. I have only so much time and mental space to use, so I try to make all my exercising worth the while, whether it be physical excercising, mental or spiritual.


  13. 13
    ET says:

    Infinity is infinity. The alleged difference is how many elements each numerical infinity contain.

  14. 14
    JDH says:

    asauber@12 That’s an interesting question. As many have noted in the field of mathematics intuition from FINITE sets CANNOT answer questions about INFINITE sets. Thus when one studies infinities one comes up with things which are finite minds have no intuition for.

    Thus it is easy to prove that two countable infinite sets have the EXACT same number of elements. Mathematicians do this by proving there is a 1-1 mapping from every element of one set to an element in the other set. So for example consider the natural numbers (counting numbers) starting from 1. ( 1, 2, 3, 4, 5..) And the set of counting numbers starting from 2. ( 2, 3, 4, 5…)

    Using the intuition from finite sets one could reason that the first set has every element that the second set does AND the number 1 (one). Since the first set has every number the second set does + 1 more, finite reasoning (i.e. reasoning about finite sets) would conclude the first set has more elements. This finite reasoning would be dead wrong. Infinity as a concept does not become bigger by adding a finite number to it. There is a simple mapping ( n2 = n1 + 1) for each n1 in the first set that makes a 1-1 correspondence between the two sets. This is what is meant that they have equal cardinality or in layman’s terms are the same size infinity.

    Cantor showed that there is no way to build a map between the counting numbers and the set of real numbers ( points on the number line ). So it is true that in mathematical reasoning the set of real numbers is somehow bigger.

    The interesting question you raise is as to whether this is any more than a mathematical exercise.

    I will leave the question of whether you may be persuaded that this has any meaning spiritual or physical to the ample list of links shared by BA77

  15. 15
    asauber says:

    you may be persuaded that this has any meaning spiritual or physical


    Infinity as Theology is an entirely different matter. But then you have to throw the numbers (as we can use them) away, if you go there. I don’t think mathematics apply. I think mathematics are a tool given to us finite creatures by God. You get mysteries like 3 and 1 at the same time when you think about God in a numerical context. No calculator has that function.


  16. 16
    kairosfocus says:


    Davies is demonstrably right, C exceeds aleph-null, as can be shown. For that matter, the reciprocal of every counting number from 1 up lies in the interval (0,1], as a dust, these are not connected points. The vast bulk, in the close neighbourhood of zero — not a proof but an illustration. So a dual to the set of naturals is found in the continuum between 0 and 1. Astonishing, if you ponder it.


    The numbers apply to different aspects. In orthodox Christian theology, God is unified as to being, diverse as to person; thus, triune. Infinity as applied to the Godhead has to do with being the greatest, beyond finite bound. But at the same time God cannot make a square circle, that is impossible of being. Likewise God will not violate his core character so will be truthful, etc.


  17. 17
    kairosfocus says:

    H’mm, I just thought of a way to drive the matter home conceptually. Not a proof, an illustration. Let any given reciprocal be 1/n, from 1, 2, 3 . . . then do the set for the interval (1,2] by adding 1 +1/n in succession, i.e. 1 +1, 1 + 1/2, 1 + 1/3 etc. The same dust reappears in (1,2]. Then, do it for 2 + 1/n, 3 + 1/n etc. Copies of the dust appear in the intervals between all counting numbers. And in no case does it begin to exhaust the interval (n, n+1]. If that does not astonish you, it should. KF

    PS: try the von Neumann construction of the naturals:

    {} –> 0
    {0} –> 1
    {0,1} –> 2
    etc, endlessly.

    Now recognise that we have a new class of quantity, the first transfinite, of cardinality aleph null. Then go on from there. There are transfinitely many transfinites as the surreals tell us.

  18. 18
    bornagain77 says:

    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

  19. 19
    johnnyb says:

    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.

  20. 20
    johnnyb says:

    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)

  21. 21
    bornagain77 says:

    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.

    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

    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.,,,

    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.

    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.

    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).

    ,,, 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,

    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.

    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.”

    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.

    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”

  22. 22
    daveS says:

    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.

  23. 23
    asauber says:

    the fact of the matter is that mathematics itself is very much a spiritual exercise.


    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.


  24. 24
    kairosfocus says:

    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. KF

  25. 25
    daveS says:


    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.

  26. 26
    kairosfocus says:

    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. KF

  27. 27
    daveS says:


    Transfinitely many is clear, just what kind is not.

    Yes, and I think a skeptic could find this state of affairs problematic.

  28. 28
    johnnyb says:

    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.

  29. 29
    daveS says:


    By the way, the application of infinity as a quantity makes doing Calculus and related mathematics a *lot* easier.


    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.

  30. 30
    EricMH says:

    What is the biggest infinity?

  31. 31
    johnnyb says:

    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


  32. 32
    daveS says:


    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.

  33. 33
    kairosfocus says:

    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. KF

  34. 34
    daveS says:


    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.

  35. 35
  36. 36
  37. 37
    daveS says:

    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… .

  38. 38
    kairosfocus says:

    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. KF

  39. 39
    daveS says:


    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.

  40. 40
    kairosfocus says:

    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. KF

  41. 41
    daveS says:

    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.

  42. 42
    EricMH says:

    To answer my own question, is the set of all the different aleph infinities the biggest infinity?

  43. 43
    EricMH says:

    @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.

  44. 44
    daveS says:


    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.

  45. 45
    EricMH says:

    @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.

  46. 46
    daveS says:


    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.

  47. 47
    EricMH says:

    @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.

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