#as aleph null is the cardinality of the naturals

[simply-disconnected]

To add to all of the above, there is another way to construct countable many different cardinality, and it's with power sets! For any given set A, the power set which I will denote P(A) is the set of all subsets of A. For example, P({1,2})={{},{1},{2},{1,2}}. With our friend cantor's diagonalization argument, it can be shown the cardinality of P(A) is strictly greater than that of A, even if A was infinite. That means that we can construct infinitely many cardinalities simply by taking power sets of power sets. If we denote the cardinality of A to be |A|, then |A|<|P(A)|<|P(P(A))|<|P(P(P(A)))|<... and so we have countable many infinite infinities.

You can even complicate things! What if you take the union of all of A,P(A),P(P(A))... ? You get a set that's even larger than all the previous ones! let's denote it A_2. What then of P(A_2)? and P(P(A_2))? We can take the infinite union of all of those and get an even larger set, which we will call A_3. We can then recursively define A_4, A_5 etc. And if we take the infinite union of all the A_n? We get a new larger set! If we denote it B, and then repeat this process taking the infinite union of B, P(B), P(P(B))... we get B_2 and similarly B_3 and so on. We can in fact repeat this process infinitely many times, getting C, then D, and continuing even after we run out of letters! And THEN we can take the union of this infinite process, and denote that Alpha, and take P(Alpha) and P(P(Alpha)) and so on and so forth, and the more you look at this, the more similar it starts to seem to cardinals.

Tell me everything about infinity.

Oh, a very loaded question! All right. Let's start with the sizes of infinity!

Roughly speaking, there are two sizes of infinity; or, in proper terminology, "cardinalities." (There is some debate, as I recall, over whether there are more sizes of infinity. But we know there are two.) The first cardinality is the same size as the integers, which are the positive and negative whole numbers; essentially tick marks going in a line forever. 1, 2, 3, and so on; and in the opposite direction, -1, -2, -3, and so on.

The second cardinality is the same size as the real numbers, which are all the numbers that most people use on a daily basis; think, instead of tick marks, a line, and every place on that line is a number. No matter how close two places on that line are, there's always another number in between them. So you have 2.5 and 2.6, but between them you have 2.55 and 2.5932, and infinitely many more.

The concept of infinity, of course, gets a bit weird once you move into more than one dimension; it's easy enough to point in the same direction as a line and say, "that goes to infinity," but once you have multiple dimensions, is it meaningful to talk about a negative or positive infinity? Oh, and adding and subtracting get weird once you start adding infinitely many things together. Addition loses commutativity (e.g. 5+3 = 3+5), which still blows my mind, even though I've seen the proof for it. It's the kind of thing that keeps me up at night.

Generally mathematicians get around the multiple-dimensions problem by using the modulus of a number, which is the distance from the number to zero, measured using our dear old friend the Pythagorean Theorem: so, if your point is at 4 in one dimension and at 3 in another, you use the Pythagorean theorem to get 5, and then you consider that point to be 5 away from zero. (This would be easier to explain if I had a chalkboard!) Then, infinity is sort of a circle that surrounds the whole plane; or, if you think of your 2d plane like a flat circle, if you folded it up into a ball, infinity would be all the points at the very top of the ball, and zero would be the point at the bottom. (Obviously this gets weirder if we have three dimensions, but you get the idea.) Okay, so that's a quick introduction to infinity from a mathematical perspective. I think there were also some physics questions re: the expanding universe and spacetime? I'm happy to write a bit about that, too, but I think it belongs in a separate post! So if you have questions about that, please let me know and I'll try to share what I do know! Disclaimer: while I DO know more math than the average person, I have essentially a bachelor's-degree level of knowledge in math. I think everything I've typed out is correct, but I may very well have missed some details! Dear readers, please feel free to correct me if you have greater knowledge than I.

Edit: also, I should have mentioned that the pythagorean theorem only works in a euclidean space. But I feel like going into non-Euclidean stuff is a bit past the scope of (this) tumblr post.

[janmisali]

Number Tournament: NOT A NUMBER vs ALEPH-NULL

NaN (Not a Number)

seed: 26 (20 nominations)

class: ERROR

description: ERROR

aleph-null (countable infinity)

seed: 39 (19 nominations)

class: infinite cardinal

description: "infinity" in the sense of "the number of natural numbers"

[xyymath]

Exploring the Infinity Paradox: What Does Infinity Really Mean?

Infinity is a concept that challenges our intuitions. It isn’t just a number; it’s an idea that can stretch across mathematical fields, appearing both simple and mind-boggling.

1. Zeno’s Paradox: Halfway There… Forever

Zeno’s paradoxes, particularly the famous Achilles and the Tortoise, deal with infinity in the context of motion and time. In the paradox, Achilles runs ten times faster than the tortoise, but the tortoise gets a head start. Zeno argues that Achilles can never overtake the tortoise because, by the time he reaches the point where the tortoise was, the tortoise will have moved further ahead.

While the reasoning seems absurd, it introduces the idea of infinite divisions of time or space. In reality, calculus helps us solve this paradox—by summing the infinite series of steps, we conclude that Achilles will indeed surpass the tortoise in a finite amount of time.

2. Cantor’s Set Theory: The Sizes of Infinity

The concept of infinity is explored profoundly in Cantor's set theory, which shows that not all infinities are equal. Cantor introduced the idea of countable and uncountable infinities. A countable infinity is an infinity where you can list all elements, like the set of natural numbers {1,2,3,4,… }\{1, 2, 3, 4, \dots\}. This type of infinity is represented by the cardinal number ℵ0\aleph_0 (aleph-null).

However, uncountable infinity is much more interesting. Consider the set of real numbers between 0 and 1. No matter how much you try, you can’t list them all in a sequence because for any number you list, there’s always another real number that isn’t on the list. This is a larger infinity—the continuum. Cantor showed that the real numbers form an uncountable set, and the cardinality of this infinity is represented by 2ℵ02^{\aleph_0}, often referred to as the cardinality of the continuum.

3. Hilbert’s Hotel: A Hotel with an Infinite Number of Rooms

Hilbert's Hotel is a famous thought experiment that demonstrates the paradoxes of infinity in a concrete way. Imagine a hotel with infinitely many rooms, all occupied. The hotel manager is faced with a new guest arriving, yet there is no available room. But because the hotel has infinitely many rooms, the manager asks each guest to move from room nn to room n+1n+1. This frees up room 1 for the new guest. The paradox shows that even with all rooms occupied, infinity can accommodate more guests.

The real kicker? If an infinite number of new guests arrive, the manager can still accommodate them by shifting guests in a similar pattern. Infinity has no "end," and this seemingly impossible scenario exposes the counterintuitive nature of infinite sets.

4. The Paradox of Infinite Sets: The Size of the Continuum

Cantor’s discovery wasn’t just theoretical; it had deep implications for how we think about the continuum of real numbers. The real number line is uncountably infinite—there are more real numbers between 0 and 1 than there are natural numbers, even though both sets are infinite. This raises the issue of infinite sets with different cardinalities. While ℵ0\aleph_0 is the size of the set of natural numbers, the set of real numbers between 0 and 1 has a larger size, denoted 2ℵ02^{\aleph_0}. This revelation left mathematicians questioning the true nature of infinity, as it seems there’s no upper bound on the size of sets.

5. The Infinite Hotel of Cantor: A Deeper Dive

Cantor’s paradoxes are further developed by thinking of sets of different "sizes" of infinity. Consider Hilbert's Hotel again, but now with infinitely many floors, and each floor is also infinite, forming a 2D array. This two-dimensional infinity shows that even within the same "size" of infinity, there are many different levels of infinity. As Gödel and Cohen later proved, infinity is not fully understood: questions like the Continuum Hypothesis—whether there is a set whose size is strictly between the size of the natural numbers and the real numbers—remain unresolved.

6. The Unknowable Future: Limits of Infinity

Infinity isn’t just something that mathematicians wrestle with—it’s tied to the very limits of knowledge and computation. Turing’s halting problem and the limits of computability further illustrate the practical consequences of infinity. No matter how powerful a computer becomes, it will never be able to calculate all the possible outcomes of an infinitely large problem.

Infinity is also central to modern physics—from the infinite expanse of space to the singularity at the center of black holes. As much as we strive for understanding, the paradoxes of infinity challenge even our fundamental laws of the universe.

[stuart-boehmer]

TWO DISCOVERIES IN MATHEMATICAL PHILOSOPHY

1  HYPERNUMBERS

            First, allow me to interject that the idea that I am about to describe seems more extensive than the hyper-reals of non-standard analysis and is not to be confused with them.

            The hyper-reals, in my understanding (and I haven’t yet thoroughly studied non-standard analysis, so this may be a mis-perception), are just an alternative description of the real number system with the same cardinality of the reals—the power set of the naturals, c := 2^(aleph-null).  according to a book I have on non-standard analysis by Mark Davis published in 1977 & 2005, to the date of publication, only one theorem of real analysis has been proven by non-standard methods—not too powerful a method?

            Now to describe what I mean by “hyper-numbers:” H1 is the first class of hypernumber, of cardinality of the power set of the reals, c1 := 2^c.  The “integers” of this number system are reals, with the density of the reals—no “first,” “second,” &c., integers & the “digits” of a hyper-number of first class are reals, with the density of the reals—no “first,” “second,” &c., digits.

            From this, we can likewise define hyper-numbers of the second class, H2 = power set of those of the first class, whose “integers” and “digits” are H1 numbers, with the density of H1 numbers.  This set has cardinality c2 := 2^c1.

            And, more generally, we can continue in a sequence of HN with cardinality cN := 2^c(N-1), where N is any finite natural number. Consider N = 100, N= 1,000,000, or N = 2^100!

            Mind blowing!

            I’d like to study this theory in detail someday, maybe establishing a formal theory by proving a few seminal theorems.  Right now, I’m still busy completing and writing up my theory of black holes for publication in a peer-reviewed journal (as you may know if you’ve been reading my prior missives)—pretty bland stuff by comparison, no?

2  THE INEFFABLES. IDEALISM AND REALITY.

            In quantum theory there is the well-known Wigner Probability Density Distribution [WQPD], which is not necessarily globally > or = 0 (and what, pray tell, does it mean that your chances of, say, getting lung cancer under certain circumstances is -5%?).  In quantum theory, however, it remains the case that the probability density of a single observable is always globally > or = 0.   This seems to have something to do with the hypothesis that any discrete or single (continuous) observable has a presumed exact value that can be determined exactly in a single measurement (while, for example, q, position, and p, momentum, cannot both be determined exactly simultaneously—the uncertainty principle +, really, only one thing can be measured at a time; perhaps this has something to do with why the WQPD may sometimes be < 0).

            But what quantity can be measured to arbitrary precision with any real measuring instrument (I mean out to aleph-null decimal places), and in what conceivable machine or human memory device could all these digits be recorded or written down for even just one measurement, let alone scads of them?

            Thus, I conjecture that not even the probability of a single (continuous) observable or variable need be globally > or = 0—and, indeed, why not dispense as well with the stipulation that it be normalized to unity (i.e., that the “total probability” = 100%)?  What if sometimes the attempt to measure a quantity didn’t result in anything that could be represented by a number, or that, though it was in principle a number, it couldn’t be measured or calculated (like Chaitin’s Omega—the “probability” that a computer program loaded with a specific input would ever stop and not get caught in an infinite loop [the Turing “halting problem”])?

            This brings me to my concept of “ineffables”—things that cannot, even in principle, be expressed by measurable or calculable “numbers” (isn’t the number concept itself an ineffable itself—WHAT IS a “number” [or, similarly, WHAT DOES it “mean” to “mean”?].  Mind you, I’m not looking for definitions in terms of other words—the attempt to describe an infinite reality with a paltry 100,000 or 300,000 words [the English language, the language with the most extensive vocabulary by far on earth] is really pathetic, and, ultimately, a vicious circle [all the words in a dictionary must be defined by other words—or by “pointing” to something: when you point to something, a dog will assume you’re trying to draw attention to your finger, not the object being “pointed” to—and, how do you point to “love” or “growth”?]).

            Ineffables that we can mention, besides Chaitin’s Omega, are Cantor’s “Absolute” (infinity), which he associated with God, and consciousness, not to mention “color” (I mean the raw, conscious experience of a color, not its wavelength or frequency—I very much doubt that telling a congenitally blind person that the wavelength of blue light is 4000 Angstroms would in any way convey what a sighted person experiences as the sky).

            What of the phenomenon of synesthesia, where a person might associate colors with music or numbers?  And who says that what I experience as blue, if I could experience it through your sensory system, I might not experience as teal or red—or the note C-flat (synesthesia again!)?  Science can describe only the “average,” “typical” case—these machines that can determine what a person is thinking about fairly accurately describe only what is common in the experience—what can be described in the public sphere.  To what is atypical or unique in an experience—the private, subjective, world—science will forever be blind.  Study your Nietzsche!—these things have been known a long time!

            Mind blowing!

            This is another thing that makes my continued interest in black hole theory look bland by comparison, no?

            The notion that “empirical facts” can always be reduced quantitatively to numbers is hereby completely exploded.  And the Hawkings of the world are not concerned with matters of “mere” philosophy!  They’re missing half the fun!  Why DO we perform our experiments if not precisely to interpret them and adjust our pardi pris to what they are trying so desperately to tell us about how the world really is—do we continue to stick to our dogmas in defiance of proper understanding of the evidence?  Are we just to be content with the literal results of our piddlin’ experiments conducted in the sterile, controlled environment of the laboratory and not venture a guess as to what might be happening in the “wild?”

            My feelings towards religion are mixed—part Hobbesian materialistic, quantitative atheist/tea kettle agnostic (Russell’s phrase), part Berkeleyian fundamentalist idealist-theist.  There are NO certainties in life—but how much education is required to understand this!  I would venture that the entire purpose of education (as opposed to the vocational training that the modern university system has devolved into) is precisely to depart some sense of just how small the collective knowledge of Man is (Feynman once famously calculated that every document ever produced—most of it the dross of phone books, suspense novels, restaurant receipts, X-rated movies, &c; the true knowledge is a very small percentage of even this tiny amount—could be stored on an ideal memory device of approximately one cubic centimeter—one seventeenth of a cubic inch!), let alone what is known to a single individual, be he a sports news anchor or the highest of artistic geniuses.

            Dogma is never pretty nor correct. How arrogant—let alone unimaginative—is a person to think that he is in possession of even a single atom of the “Truth!”

            As a banner in a church I attend states, “God is too big to fit into one religion.”

            COEXIST!!

[brickwall2]

TWO DISCOVERIES IN MATHEMATICAL PHILOSOPHY

1  HYPERNUMBERS

            First, allow me to interject that the idea that I am about to describe seems more extensive than the hyper-reals of non-standard analysis and is not to be confused with them.

            The hyper-reals, in my understanding (and I haven’t yet thoroughly studied non-standard analysis, so this may be a mis-perception), are just an alternative description of the real number system with the same cardinality of the reals—the power set of the naturals, c := 2^(aleph-null).  according to a book I have on non-standard analysis by Mark Davis published in 1977 & 2005, to the date of publication, only one theorem of real analysis has been proven by non-standard methods—not too powerful a method?

            Now to describe what I mean by “hyper-numbers:” H1 is the first class of hypernumber, of cardinality of the power set of the reals, c1 := 2^c.  The “integers” of this number system are reals, with the density of the reals—no “first,” “second,” &c., integers & the “digits” of a hyper-number of first class are reals, with the density of the reals—no “first,” “second,” &c., digits.

            From this, we can likewise define hyper-numbers of the second class, H2 = power set of those of the first class, whose “integers” and “digits” are H1 numbers, with the density of H1 numbers.  This set has cardinality c2 := 2^c1.

            And, more generally, we can continue in a sequence of HN with cardinality cN := 2^c(N-1), where N is any finite natural number. Consider N = 100, N= 1,000,000, or N = 2^100!

            Mind blowing!

            I’d like to study this theory in detail someday, maybe establishing a formal theory by proving a few seminal theorems.  Right now, I’m still busy completing and writing up my theory of black holes for publication in a peer-reviewed journal (as you may know if you’ve been reading my prior missives)—pretty bland stuff by comparison, no?

2  THE INEFFABLES. IDEALISM AND REALITY.

            In quantum theory there is the well-known Wigner Probability Density Distribution [WQPD], which is not necessarily globally > or = 0 (and what, pray tell, does it mean that your chances of, say, getting lung cancer under certain circumstances is -5%?).  In quantum theory, however, it remains the case that the probability density of a single observable is always globally > or = 0.   This seems to have something to do with the hypothesis that any discrete or single (continuous) observable has a presumed exact value that can be determined exactly in a single measurement (while, for example, q, position, and p, momentum, cannot both be determined exactly simultaneously—the uncertainty principle +, really, only one thing can be measured at a time; perhaps this has something to do with why the WQPD may sometimes be < 0).

            But what quantity can be measured to arbitrary precision with any real measuring instrument (I mean out to aleph-null decimal places), and in what conceivable machine or human memory device could all these digits be recorded or written down for even just one measurement, let alone scads of them?

            Thus, I conjecture that not even the probability of a single (continuous) observable or variable need be globally > or = 0—and, indeed, why not dispense as well with the stipulation that it be normalized to unity (i.e., that the “total probability” = 100%)?  What if sometimes the attempt to measure a quantity didn’t result in anything that could be represented by a number, or that, though it was in principle a number, it couldn’t be measured or calculated (like Chaitin’s Omega—the “probability” that a computer program loaded with a specific input would ever stop and not get caught in an infinite loop [the Turing “halting problem”])?

            This brings me to my concept of “ineffables”—things that cannot, even in principle, be expressed by measurable or calculable “numbers” (isn’t the number concept itself an ineffable itself—WHAT IS a “number” [or, similarly, WHAT DOES it “mean” to “mean”?].  Mind you, I’m not looking for definitions in terms of other words—the attempt to describe an infinite reality with a paltry 100,000 or 300,000 words [the English language, the language with the most extensive vocabulary by far on earth] is really pathetic, and, ultimately, a vicious circle [all the words in a dictionary must be defined by other words—or by “pointing” to something: when you point to something, a dog will assume you’re trying to draw attention to your finger, not the object being “pointed” to—and, how do you point to “love” or “growth”?]).

            Ineffables that we can mention, besides Chaitin’s Omega, are Cantor’s “Absolute” (infinity), which he associated with God, and consciousness, not to mention “color” (I mean the raw, conscious experience of a color, not its wavelength or frequency—I very much doubt that telling a congenitally blind person that the wavelength of blue light is 4000 Angstroms would in any way convey what a sighted person experiences as the sky).

            What of the phenomenon of synesthesia, where a person might associate colors with music or numbers?  And who says that what I experience as blue, if I could experience it through your sensory system, I might not experience as teal or red—or the note C-flat (synesthesia again!)?  Science can describe only the “average,” “typical” case—these machines that can determine what a person is thinking about fairly accurately describe only what is common in the experience—what can be described in the public sphere.  To what is atypical or unique in an experience—the private, subjective, world—science will forever be blind.  Study your Nietzsche!—these things have been known a long time!

            Mind blowing!

            This is another thing that makes my continued interest in black hole theory look bland by comparison, no?

            The notion that “empirical facts” can always be reduced quantitatively to numbers is hereby completely exploded.  And the Hawkings of the world are not concerned with matters of “mere” philosophy!  They’re missing half the fun!  Why DO we perform our experiments if not precisely to interpret them and adjust our pardi pris to what they are trying so desperately to tell us about how the world really is—do we continue to stick to our dogmas in defiance of proper understanding of the evidence?  Are we just to be content with the literal results of our piddlin’ experiments conducted in the sterile, controlled environment of the laboratory and not venture a guess as to what might be happening in the “wild?”

            My feelings towards religion are mixed—part Hobbesian materialistic, quantitative atheist/tea kettle agnostic (Russell’s phrase), part Berkeleyian fundamentalist idealist-theist.  There are NO certainties in life—but how much education is required to understand this!  I would venture that the entire purpose of education (as opposed to the vocational training that the modern university system has devolved into) is precisely to depart some sense of just how small the collective knowledge of Man is (Feynman once famously calculated that every document ever produced—most of it the dross of phone books, suspense novels, restaurant receipts, X-rated movies, &c; the true knowledge is a very small percentage of even this tiny amount—could be stored on an ideal memory device of approximately one cubic centimeter—one seventeenth of a cubic inch!), let alone what is known to a single individual, be he a sports news anchor or the highest of artistic geniuses.

            Dogma is never pretty nor correct. How arrogant—let alone unimaginative—is a person to think that he is in possession of even a single atom of the “Truth!”

            As a banner in a church I attend states, “God is too big to fit into one religion.”

            COEXIST!!

[vexwerewolf]

Disclaimer: I am not a mathematician, but I will attempt to explain this to the best of my understanding.

In high-level maths you have these things called sets, which are collections of different things. Sets vary greatly in size, from extremely small finite sets (your basket contains ten apples) to very large but still finite sets (your galaxy contains a hundred billion stars) to very very very large but still finite sets (your body contains seven octillion atoms) to absurdly, incomprehensibly large but still ultimately finite sets (Graham's number, the upper bound on a possible answer to a problem in Ramsey theory, is so large it required a new form of mathematical notation just to express).

By definition, all finite sets are countable, and trivially so - they contain a finite number of things, so eventually, given enough time (though it may take a while for larger sets - see above), you will be able to count all items in a finite set. However, it is really funny to annoy mathematicians by describing a "finite, uncountable set." Try it some time.

You can also get infinite sets. Infinite sets, much as the name suggests, do not have a limit on the number of things that are in them. It may seem counter-intuitive, but some infinities are "larger" than other infinities - we call these countably infinite and uncountably infinite.

Countably infinite sets have what is called a "one-to-one correlation with natural numbers," which is to say that if you started counting at 1 and went all the way to the end of numbers (which set theory... pretends is a thing that exists, sometimes, when it's necessary?), you could assign exactly one element of the set for each number you counted. Obviously, the set of all natural numbers is an countably infinite set, which is a great example of a thing being a great example of itself.

Uncountably infinite sets do not have a one-to-one correlation with natural numbers. That is to say, if you were to start counting at 1 and went all the way to the end of numbers, there would be infinitely more items than you could assign natural numbers to. For example, between 1 and 2 there are an infinite number of possible decimals ("real" numbers), so expansive that no matter how many of them you listed, there would always be at least one that was not on the list, because between 1.1 and 1.2 there's 1.11, 1.12, etc, and between those there's 1.111, and then 1.111, 1.1111, 1.11111 and so on. The set of all non-natural numbers is infinitely larger than the set of all natural numbers, which is itself already infinite.

Sets have something called "cardinality," which is the number of elements in the set. For finite sets, cardinality can always be trivially expressed as... the number of elements in the set. For sets with extremely expansive membership (like, say, upper bounds on answers to problems in Ramsey Theory) the number may be difficult to write down but it is still ultimately a number.

The cardinality of infinite sets cannot be expressed as a number because infinity is not actually a number. Thus, when expressing cardinality for infinite sets, mathematicians developed something called the Aleph number - using the Hebrew letter ℵ.

ℵ₀ or Aleph-null is the cardinality for the set of all natural numbers - that is (by most definitions) all non-negative numbers above zero (yes, Tumblr mathmos, I know ISO 80000-2 includes zero, but please don't make me get into that). It is a countably infinite set, and this is trivially provable by thinking of the largest number you can think of and then adding one - as Day9 would say, you can always find more by inspection.

Aleph-null represents, to put it inexpertly, the smallest possible infinity. All infinities other than Aleph-null are by definition either the same size or larger, because if there are less elements in a set than there are natural numbers then that set has an upper bound and is thus finite, and if it has more, then it does not have a one-to-one correlation with natural numbers.

Therefore:

Yo momma so fat her weight is a transfinite cardinal

Explaining the joke makes it funnier

When you are having a very normal day at the Secret Anomaly Research Facility

[recreationaldivorce]

Wait what’s the math thing your new url means!

The aleph numbers are a sequence of numbers representing different sized infinities, more specifically the size of infinite sets. Think about how the number of integers is infinite, but the number of all numbers is also infinite, but there are more numbers than integers, i.e. one infinity is larger than the other.

Aleph null, or ℵ0 (that 0 should be in subscript but idk if tumblr allows for subscript), represents the size of the set of all natural numbers (natural numbers are counting numbers, so like 1 2 3 4); it represents countable infinities, basically. If I say the number 25, you can tell me the next natural number, which is 26. If I say the number 25 and ask for the next number though, not the next integer or natural number, you can't say that; it's not 25.1 because 25.01 exists, and so on. But natural numbers are countable because you can say the next number. A set has cardinality (size) ℵ0 if it maps one-to-one with all natural numbers.

You can then have aleph-one (ℵ1), which is the cardinality of all countable ordinal numbers (ordinal numbers are numbers used for ordering things, e.g. 1st, 2nd, 3rd, 4th). And the cardinality increases with ℵ2, ℵ3, ℵ4, etc. ℵω (aleph-omega) is the least upper bound of all aleph numbers with indices not exceeding ω. Wikipedia has a more mathematical definition here but I tried to put it in words.

I wanted alephnull bc it's my ao3 username and whoever had alephnull deactivated anyway so it's not like they're even on tumblr. But tumblr is holding onto the url and won't release it >:(

[sunnycosmology]

Infinity

What is infinity and is this a ultimate destination for everything? Infinity is the term that is widely use in mathematics, physics, astronomy, cosmology and in many other statistical field. But still our curiosity towards infinity and its some hidden applications become more and more as we are not hiding our presence of mind.

But how infinity assist us to solve our biggest mathematical equations and also understand this chaos is the big challenge for us? First, we have understand infinity is not a physical or non-physical object but is a term widely uses in mathematics and in physics.. One can cancel infinity with infinity and ultimately the consequence is again a infinity. Infinity also assists in many mathematical equations that helps us to solve our ‘Integral’ as well as ‘Differential’ solutions and the infinity is a chapter of “infinite Series”.

However, Infinity is just a perception for our uncertain mind where we all only know about its use but never knows ‘Is there anything beyond infinity’, again it a term not an object so as to perceive its realm but infinity is still a mystery as how limited minds can solve the biggest mystery of nature? Smaller infinities concept was given by “George Ludwig Ferdinand Cantor” where his cardinal numbers showed a difference between “Infinity” and “Smaller Infinity” (described a term “Aleph Null*).

We use term infinity and solve our equations, yet some are unsolvable and some are solvable, but we are sure ‘Infinity’ definitely is not only a term of symbol but a method of medium.

*Aleph Null: is the standard cardinal number of N (Natural Numbers) sometimes called set of  real numbers and the set of cardinality is called “denumerable’.

Image Source: Google Images

[mewbusi]

Ok, so to the best of my knowledge/me researching the determination of whether or not infinity + infinity is larger than infinity depends on if your infinity is a cardinal or ordinal.

Ordinal infinity (or the smallest infinite ordinal) is designated by a ω, and cardinal infinity is regarded as aleph-null, aleph-nought, or aleph-zero. Which is represented by this symbol.

Ordinal numbers are one generalization of the concept of a natural number that is used to describe a way to arrange a (possibly infinite) collection of objects in order, one after another. 

Essentially ordinal infinity is the number that comes after every natural number (identifying itself as the set of all natural numbers, and itself by definition uncountable (at least with our current natural numbers)). However the really cool thing about this infinity (which is the typically used infinity for things like calculus and algebra) is that ω+1 is a larger number than just ω, and even up to ω^ω^ω etc...

Essentially an ordinal number of 3 is represented by the set {0,1,2}. And ω is the set of all finite numbers.

So then what about cardinal infinity? Well unlike ordinal numbers which describe order of a collection cardinal numbers instead describe the number of elements of a set, without caring about the order. So something like the set {5, 2, 4} has a cardinal number of 3. ω has a cardinality of aleph-zero, but so does ω+1, and every version of this ordinal infinity. As such no matter how you screw with aleph-zero using other aleph-zeros you will not have a larger infinity. However there are larger alephs with aleph-ones, twos, and so on. 

Yes this does exist.

Technically both you and your kids are right, simply depending on your determination of which infinity you are using.

Some Sources: Cardinal Number Arithmetic Ordinal Number Arithmetic A place that explains infinity better than me Wikipedia Infinity Wikipedia Ordinal Infinity Wikipedia Cardinal Infinity

Son 1: I bet you infinity that it will.

Son 2: I bet you infinity infinity that it won't.

Son 1: I bet you infinity infinity infinity that it will.

Son 2: I bet you infinity infinity infinity infinity-

Son 1: Well I bet you infinity infinity infinity infinity infinity infinity infinity infinity infinity infinity infinity infinity infinity infinity infinity infinity infinity infinity-

Me, tired, busting out of my room: ADDING INFINITY ON TOP OF INFINITY DOESN'T MAKE INFINITY BIGGER.

Son 1: Yeah but it makes it go faster.

Me: ....

[lipshits-continuous]

I hope you don't mind my addition cause there are a few things I'd like to say /lh :)

So we can actually construct at least countably many infinite cardinals! The first is aleph-0 which is the cardinality of the naturals, the integers and the rationals. A set with this size is called countable! (More formally, a set X is countable if there exists a bijection between X and the natural numbers). You can then construct the rest of the aleph numbers for finite cardinals inductively! (Note I've not studied this construction in great detail). What is neat is that each aleph number is strictly smaller than the next!

Things start to get a bit weird when you ask which aleph number is the cardinality of the reals. The answer is that it's undecidable within the usual framework of set theory (ZFC). The famous continuum hypothesis is that aleph null is the cardinality of the real numbers. However, this cannot be proven nor disproven in using ZFC! However, what can be proven is that the cardinality of the reals is 2^(aleph-0)!

Now I'd like to touch on the point about there always being a number between any two real numbers. This is a property called density! The example you gave demonstrates the density of the rational numbers within the reals. However this alone is not enough to show that the cardinality of the reals is strictly greater than aleph-0. Despite the rationals being dense in the reals, there are less of them than there are reals, which is quite counterintuitive at first. You need to use cantor's diagonalisation argument.

The construction you mention about adding a point at infinity to the 2d plane is really interesting! When we consider the 2d plane to be the complex numbers, this is called the Riemann Sphere or the complex projective line (line because it is a 1 dimensional complex manifold). I'm pretty interested in projective spaces so it was nice to see it mentioned!

Tell me everything about infinity.

Oh, a very loaded question! All right. Let's start with the sizes of infinity!

Roughly speaking, there are two sizes of infinity; or, in proper terminology, "cardinalities." (There is some debate, as I recall, over whether there are more sizes of infinity. But we know there are two.) The first cardinality is the same size as the integers, which are the positive and negative whole numbers; essentially tick marks going in a line forever. 1, 2, 3, and so on; and in the opposite direction, -1, -2, -3, and so on.

The second cardinality is the same size as the real numbers, which are all the numbers that most people use on a daily basis; think, instead of tick marks, a line, and every place on that line is a number. No matter how close two places on that line are, there's always another number in between them. So you have 2.5 and 2.6, but between them you have 2.55 and 2.5932, and infinitely many more.

The concept of infinity, of course, gets a bit weird once you move into more than one dimension; it's easy enough to point in the same direction as a line and say, "that goes to infinity," but once you have multiple dimensions, is it meaningful to talk about a negative or positive infinity? Oh, and adding and subtracting get weird once you start adding infinitely many things together. Addition loses commutativity (e.g. 5+3 = 3+5), which still blows my mind, even though I've seen the proof for it. It's the kind of thing that keeps me up at night.

Generally mathematicians get around the multiple-dimensions problem by using the modulus of a number, which is the distance from the number to zero, measured using our dear old friend the Pythagorean Theorem: so, if your point is at 4 in one dimension and at 3 in another, you use the Pythagorean theorem to get 5, and then you consider that point to be 5 away from zero. (This would be easier to explain if I had a chalkboard!) Then, infinity is sort of a circle that surrounds the whole plane; or, if you think of your 2d plane like a flat circle, if you folded it up into a ball, infinity would be all the points at the very top of the ball, and zero would be the point at the bottom. (Obviously this gets weirder if we have three dimensions, but you get the idea.) Okay, so that's a quick introduction to infinity from a mathematical perspective. I think there were also some physics questions re: the expanding universe and spacetime? I'm happy to write a bit about that, too, but I think it belongs in a separate post! So if you have questions about that, please let me know and I'll try to share what I do know! Disclaimer: while I DO know more math than the average person, I have essentially a bachelor's-degree level of knowledge in math. I think everything I've typed out is correct, but I may very well have missed some details! Dear readers, please feel free to correct me if you have greater knowledge than I.

Edit: also, I should have mentioned that the pythagorean theorem only works in a euclidean space. But I feel like going into non-Euclidean stuff is a bit past the scope of (this) tumblr post.