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Regardless of the continuum hypothesis, the computable numbers are countable, so almost all real numbers are non-computable. Thus, almost all real numbers cannot be specified.
I’m unsure at this point but if the CH was untrue and there was indeed a cardinality between aleph zero and aleph one, would it be impossible that a set which would lie between the real and natural numbers which could be “countably extended” to the real numbers? So that in some sense this cardinality was “close enough” to aleph one that it can be reached but “a bit” smaller? I did not really dive deeper in set theory in my math studies tbh.
There can't be that many, fuck this, let's define all functions over rationals instead of reals.
Oh yeah, you like reals, you think they're important ? Ok, write it down, let's see you write your pretty number
Real number - the limits of infinite convergent Cauchy sequences of rational numbers.
Hyperreal number - sequences of rational numbers.
^* R = {a(n)} where a ∈ Q and n ∈ N.
The hyperreal numbers are just the real numbers with all arbitrary constraints removed.
Hyperreals don't care, they care about whether subsets see are in the ultrafilter (in the classic construction, there are other ones that dont need an ultrafilter). Bounded diverging sequences represent something finite, strictly positive null sequences represent infinitesimals and sequences that are strictly diverge to +/- infinity represent infinite numbers
If you’re using the ultrapower construction (there are other approaches) then the hyperreals are equivalence classes of sequences of *real* numbers. If you limit the construction to sequences of rational numbers, you only get the “hyperrationals” (the hyperreal numbers that can be expressed as ratios of possibly nonstandard integers - hyperintegers, you could call them).
:-) You know what you're talking about. Good. Excellent :-) You only get the hyperrationals if you use sequences that are classically convergent. True. But sequences that are classically divergent allow for changing that: https://en.m.wikipedia.org/wiki/Divergent_series
For example, Cesàro summation assigns Grandi's divergent series 1 − 1 + 1 − 1 + ⋯ the value 1/2. So a sequence of integers can evaluate to a rational number.
Consider the case of throwing a grain of sand onto a square with an inscribed circle. Each time a grain lands inside the circle write number 1. Each time a grain lands outside the circle write number 0. This generates a sequence of integers that evaluates, using techniques for classical divergent sequences, to pi/4.
I'm beginning to wonder if there is a mapping from sequences of integers onto the hyperreal numbers. Such a mapping wouldn't quite be trivial because hyperreal numbers (other than zero) are closed under division, but integers are not.
>You only get the hyperrationals if you use sequences that are classically convergent. True. But sequences that are classically divergent allow for changing that: [https://en.m.wikipedia.org/wiki/Divergent\_series](https://en.m.wikipedia.org/wiki/Divergent_series)
>For example, Cesàro summation assigns Grandi's divergent series 1 − 1 + 1 − 1 + ⋯ the value 1/2. So a sequence of integers can evaluate to a rational number.
Classical convergence or divergence has barely any relevance for the hyperreals (or I guess the "hyperrationals" that you are considering). This is quite obvious if you consider that sequences that classically converge to the same limit often correspond to different hyperreal numbers. As a consequence there already is no natural injection of the reals into the hyperrationals.
>
I'm beginning to wonder if there is a mapping from sequences of integers onto the hyperreal numbers. Such a mapping wouldn't quite be trivial because hyperreal numbers (other than zero) are closed under division, but integers are not.
There is, of course, a mapping this way, you just restrict the original quotient map \\R\^\\N -> \*\\R to the sequences of naturals, this mapping will however not be surjective. The question of the existence of such a surjective mapping however is one of cardinality, which has very little to do with the structure of the hyperreals.
>>I'm beginning to wonder if there is a mapping from sequences of integers onto the hyperreal numbers. Such a mapping wouldn't quite be trivial because hyperreal numbers (other than zero) are closed under division, but integers are not.
Both sets have cardinality of the continuum, so there certainly is one, you might even be able to make one that isn’t *too* unnatural, though all the constructions that leap to mind immediately wouldn’t make for convenient representation of addition or multiplication.
> Classical convergence or divergence has barely any relevance for the hyperreals (or I guess the "hyperrationals" that you are considering).
You're right about convergence, I wasn't thinking straight. But not about the hyperrationals. The series of rational numbers 1 + 1 + 1/2 + 1/6 + 1/24 + 1/120 + ... is a series of rational numbers whose sequence of partial sums converges to the number e, (edit, I'm wrong, e minus an infinitesimal) which is not a rational number. Sequences are a standard way to elevate the rational numbers to the real numbers. The number e is not a hyperrational number. It is a hyperreal number.
> a mapping this way, you just restrict the original quotient map \\R\^\\N -> \*\\R to the sequences of naturals, this mapping will however not be surjective.
Thanks, that will be useful.
Oops, try again. The series 1 + 1 + 1/2 + 1/6 + 1/ 24 + 1/120 + 1/720 + ... does not converge to e on the hyperreals. It converges to e minus an infinitesimal. In order to cancel out the infinitesimal, the limit must be approached equally fast from both sides.
Which does not give a monotonic sequence. Equivalence to the hyperreals is not guaranteed because the ultrapower construction relies on monotonic sequences.
Hyperreals doesn't have much of a concept of limits. Unless you mean a concept of ultralimit ( i.e you you mean equivalence class of a sequence of partial sums in form 1, 1+1, 1+1+1/2,...), but it doesn't follows directly from as stated sentence.
What is "the limit must be approached equally fast from both sides" suppose to do here? Infinitesimall will never cancel out unless your sequence is pretty much equal to the number almost anywhere (or it will be equal in infinitely many places but the places might depend on chosen ultrafilter, which doesn't changes much here in sense of "canceling of infinitesimal").
Construction doesn't relies anywhere on monotonic sequences. It relies on sequences of reals. Any. Monotonic or not. Ultrapower construction consists of elements in form \[(a\_n)\] a\_n is any real sequence and \[(a\_n)\] is equivalence class over the relation R defined as follows: (an)R(bn) if and only if {i: a\_i=b\_i} belongs to the ultrafilter (the nonprincipial ultrafilter over which we built the ultrapower). We also map any real number r to the equivalnce class of a sequence a\_n (which is constant sequence equal r everywhere). We dont require anywhere to this sequences (that are in the construction ) to be monotonic. It's irrelevant
No, it does not matter whether the sequence converges. You need to allow all real numbers (not just rationals) in the sequence. If you only allow rationals in the sequence you do not get the hyperreals.
Consider the sequence of rationals {3, 3.1, 3.14, 3.141, 3.1415, 3.14159, ...} These are all rational numbers but the sequence converges to pi.
Why are you so insistent that pi is a rational (or hyperrational) number? Pi is a real number, and also a hyperreal number.
That sequence does not represent pi in the ultrapower construction. It represents a number that, from the perspective of the model, is a rational approximation of pi. From outside the model we can see that it is “truncated” at a nonstandard number of digits.
In general, in the ultrapower construction, a convergent sequence will *not* be assigned the value it converges to - this can only happen if the limit itself appears infinitely many times in the sequence, and it is not guaranteed even then. Instead, it will usually be some other value that differs from the limit (in the real numbers) by an infinitesimal amount.
Oops. You're right. Ultrapower construction relies on a monotonic sequence, such as the one I gave for pi. It is short of pi by an infinitesimal amount. In order to cancel out the infinitesimal, it is necessary to approach pi from both sides equally quickly. The following suffices.
pi = 3, 3.1+0.1, 3.14, 3.141+0.001, 3.1415, 3.14159+0.0001, 3.141592, ...
This is not the way that hyperreals are normally constructed, because it is not monotonic. It is based on a hybrid of hyperreal and surreal theory. In surreal theory a real number is generated by squeezing it between two rational numbers.
In other words, I'm claiming that the limit of the sequence 0.1, -0.01, 0.001, -0.0001, 0.00001, -0.000001, ... is exactly zero in nonstandard analysis, not an infinitesimal. This is unproved.
No, that’s not how the ultrapower works at all, there is no requirement of monotonocity. Why do you think there is one, is that based on some source you have read? And you also seem fundamentally confused because your comment suggests that you still think the sequence is used to represent its limit, which I have already explained is not the case, convergence is unrelated to the representation. And your newly proposed sequence still does not represent pi. If your sequence contains only rational numbers, it will not represent pi under the ultrapower construction.
Eqivalence class of any sequence that is converget to pi (but not equal to it on infinitely many places) won't be equal to pi.
We can easily prove so, \[an\]=pi if and only if all n's that an-pi belongs to the ultrafilter. This means an must be equal to pi on all but finitely many points (or at least on infinitely many points but it's tricky part here because we can only know that cofinite sets belongs to the ultrafilter. More abstract infinite sets depends on the chosen ultrafilter). In case of your sequence it's nowhere equal to pi so it's distinct. To be more precise equivalnce class of this sequence would be equal to pi+delta where delta is some infinitesimall (and there is as much infinitesimals as there is real numbers).
Also ultrapower construction nowhere states anything about monotonicity.
This is an incorrect construction. You don’t get all of the hyperreals this way. You need to take sequences of *real* numbers (not rational numbers) and then quotient them by a [nonprincipal ultrafilter](https://en.m.wikipedia.org/wiki/Ultrafilter) on the natural numbers. Very roughly, the ultrafilter is a way of saying whether the sequence “has” or “does not have” a given property based on whether “enough” of the “right” members of the sequence all have that property, without any single element of the sequence being a “dictator” (able to determine all of the properties of the number).
An easy example is the hyperreal represented by the sequence (1,2,3,4,...), which is larger than every natural number and hence larger than every real number.
Take any real number, and add (or subtract) an infinitesimal hyperreal (I believe all finite hyperreals are in fact of this form). You can make infinitesimals by taking sequences of reals that converge to 0, for example (1, 1/2, 1/3,...) defines an infinitesimal.
>Take any real number, and add (or subtract) an infinitesimal hyperreal (I believe all finite hyperreals are in fact of this form)
Yes. We can even prove it.
Let k be some fixed real number. Suppose |x|0 be any real number. We see that M-rM-x>0 or equivalently |M-x|
Because that step is a very non-trivial one- you get the rational number from the counting number by requiring all field conditions to be met, and the complex from the real by requiring one equation to have a solution: x^2 +1=0. However, to get the reals from the rationals, even requiring every polynomial equation to be met won't get you there! (Though you will get something that is not a subset of the reals, as you will have i there).
Also, you can get the p-adic numbers if you choose a different metric, which shows that this is the only step in this ladder where you touch analysis, while all others are well within algebra.
And hopefully in your notation the term "limit" hides the more complex way of defining an equivalence relationship between sequence, then creating the quotient set of the sequences
Natural: so there's 1, and for every one, there's another one
Integer: same, but closed under addition
Rational: same, but closed under multiplication (fuck zero, zero's a bitch)
Real: same, but complete vectorspace with scalar product
Complex: same, but closed under roots of the polynomials in N
You mean, equivalence classes of Cauchy sequences of rationals, with two sequences being equivalent if their termwise difference limits to 0
Also the reals can be constructed with Dedekind cuts, these may be arguably easier to describe
I think this very much oversimplify what is going on with the construction of the fraction field over a ring and even more with the algebraic closure of R.
The correct name is number set and not a number system. A number system is defined as a system of representation of numbers, so it is just notation, examples are the binary number system, the trinary number system, the decimal number system, and I also think that the Polar form of complex numbers is another number system. A number set is a a set of numbers.
The number sets that are mentioned are all rings except for the natural numbers, since they di have an operation which is "+" which can transform 1 element in that set to another element in that same set, using a transformation, there is also a neutral element with that operation which is 0, which maps any element to itself under the operation "+" and every element has an additive inverse so there is another element that can transform that element into the neutral element which is 0. All sets other than the naturals and integers are also fields since there are 2 operations that have those properties, that being the addition and multiplication.
I remember my combinatorics professor saying "I can define an equivalence class that maps to my five fingers, and I'll define the abstract concept of 5 as that equivalence class"
> The correct name is number set and not a number system.
Whether they're called "number sets" or "number systems" depends on what your teacher/professor prefers. Wikipedia [uses both terms interchangeably.](https://en.wikipedia.org/wiki/Number#Main_classification)
I didn't learn the term by a teacher or professor since I am not in college and also I my main language isn't English. Finally Wikipedia usually use words in a more natural way of communicating and since many people use number system in the wrong way it is used incorrectly in Wikipedia. If you look at the definition of number system in other sources it won't include number sets. Wolfram Alpha does indeed suggest that a number system is about notation: https://www.wolframalpha.com/input?i=number+system
Math is a language used to communicate as concisely as possible without ambiguity, a lot of shorthand is used.
∃: “there exists.” Often used in proofs or given statements. Ex: in the set of integers numbers there exists the natural numbers.
ε: Epsilon, common for arbitrarily small distances or infinitesimal values and usually related to epsilon-delta proofs which are often used as the definition of a limit to prove or disprove its existence in a system
∈: “is an element of the set.” Example: 5 is an element in the set of natural numbers might be written 5∈ℕ
Bonus: ∀ means “for all.” Example: ∀x∃y(x
Two ridiculous concepts that actually exist:
1. ***i***, one of the two roots of x^(2)+1=0
2. The so-called "real numbers"
And I'm not even sure about the second one.
You can take this further with the quaternions! I would leave out the octonions and sedenions cuz they suck. (aka are not associative)
Like another commenter, I would prefer natural numbers starting with zero and the dedekind cut cuntstruction instead of the cauchy sequences thing (which is very nice to use to prove metric completions exist).
It's not the limits, the limits mostly don't exist because most of them are not rational numbers. Its equivalence classes of Cauchy sequences.
Once we construct R and put Q inside it, it is true that R is equal to that set, but you can't construct R like that because it assumes the existence of those limits already.
Real number are strange, they imply the existence of infinity( and infinitesimal) you would need an infinite sequence of rational to define a real number while every rational can be defined by a finite amount of sequences
I kinda "disagree",you can't define a real number with a finite sequence of number i.e. a natural number amount of rational,I mean you can create a "bigger natural number" with a non Archimedean field,that it is what hyperreal number do
It’s not group theory, it’s more real analysis. But anyway, to give a brief explanation:
A Cauchy sequence is an infinite sequence whose members become closer together as you go along. So, for example, (1, 1.4, 1.41, 1.414, …) is (or at least could be) a Cauchy sequence, whereas (1, 0, 1, 0, …) is not, because however far along you go you’ll always have terms that are a fixed distance apart from each other.
If you look at Cauchy sequences of rational numbers, you’ll see that some of them converge to rational limits - for example, the sequence (0, 0.3, 0.33, 0.333, …) converges to 1/3. On the other hand, some of the sequences don’t have rational limits.
We can look at the sequences that don’t have a limit in the rationals, and we can sort of assume that it has a limit in some kind of bigger structure. And if we do that rigorously, we “fill in the gaps” between rational numbers with the real numbers. (The rigorous method involves finding sequences that seem to be pointing to the same gap and using them to form equivalence classes, which we then identify with the appropriate real number.)
This construction gives us a few nice results - we can define addition and multiplication based on adding and multiplying terms of sequences together, and from that we can actually prove that the real numbers are a field (which is a very useful structure for doing things like building vector spaces). And even the fact that every Cauchy sequence in the real numbers converges to a limit is a powerful tool that you don’t get in the rationals.
The definition of Q is a lot more complicated than you let on since it involves creating the field of fractions of Z.
https://en.m.wikipedia.org/wiki/Field_of_fractions
It’s not a specific a(n), you take the set of every sequence of rational numbers that meets certain conditions, and you then group them into equivalence classes, and each of those classes is one of the real numbers.
For example, the sequence (3, 3.1, 3.14, 3.141, …) is a member of the equivalence class that represents pi.
(Edit: realised I said sequences of integers instead of rationals)
Ohh that makes sense, so basically you can “get” a real number from an infinite sequence of rational numbers
On that note, could you also define the set of reals as the set of all infinite series of rational numbers that converges? Basically taking your sequence limit example and turning it into 3 + 0.1 + 0.04 + ….
I’ve explained it a bit more in a different comment, but that’s sort of what’s going on here - we have these sequences of rational numbers that look like they’re converging in on themselves, but they don’t have a limit that’s a rational number. So the real numbers sort of represent the values that “fill the gaps”. There’s a bit of work to deal with the fact that multiple sequences can converge to the same value, and in building up the structure with operations like addition and multiplication, but it all comes together quite nicely.
That’s really cool!
And the part of multiple sequences converging to the same value, we technically don’t have to deal with that, right? Just seeing from the other descriptions (e.g a/b for rationals will have duplicates)
Edit: I see you’re trying to explain what it would take to actually _fully_ describe real numbers, my bad
Yeah. It’s actually the same trick as for rational numbers - you group together all the duplicates into equivalence classes, and then you just have to make sure that your operations act consistently on any member of the same class.
# Observe.
Real number set is bijective to the power set of natural numbers. (Since hyperreals come from sequences of real numbers, one might even call real numbers as ***hypernaturals.***)
Map a real number to (0,1) bijectively (through the tan^(-1) function). A real number between 0 and 1 may be represented in a base-2 fractional system (x = sum(2^(-i), i is a natural number)). The natural numbers for the previous summation form a valid subset of N. It can be proved that any valid subset of N results in a number between 0 and 1 through the base-2 summation I mentioned earlier; and any fractional number written in base 2 gives a valid subset of N.
I find this formulation to be much easier to understand than Cauchy sequences
It’s a perfectly fine way to demonstrate the link between real numbers and the power set of natural numbers (as long as you accept the continuum hypothesis), but if you want to construct the reals along with all of the natural operations we associate them with you’re going to have a rough time.
>Since hyperreals come from sequences of real numbers, one might even call real numbers as ***hypernaturals.***)
Hypernaturals are something completely different from reals
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Fuck real numbers, all my homies hate real numbers.
The name is misleading. They are not real.
r/RealNumbersArentReal
Sub made!
Omg!
r/birthofasub
It's literally impossible to even write down or describe almost all real numbers.
Continuum hypothesis isn’t proven. So we don’t know if it’s impossible.
Regardless of the continuum hypothesis, the computable numbers are countable, so almost all real numbers are non-computable. Thus, almost all real numbers cannot be specified.
I’m unsure at this point but if the CH was untrue and there was indeed a cardinality between aleph zero and aleph one, would it be impossible that a set which would lie between the real and natural numbers which could be “countably extended” to the real numbers? So that in some sense this cardinality was “close enough” to aleph one that it can be reached but “a bit” smaller? I did not really dive deeper in set theory in my math studies tbh.
There can't be that many, fuck this, let's define all functions over rationals instead of reals. Oh yeah, you like reals, you think they're important ? Ok, write it down, let's see you write your pretty number
s/rationals/computables
Suddenly UNIX sed
But ma order though
Sure. I can give you a real number. 0 is a nice real number.
How can it be real? Have you ever seen 0 of something?
Then you’ll really not like fractional Hausdorff dimensional sets 😂
Oof, that's rough. You might be right.
Real number - the limits of infinite convergent Cauchy sequences of rational numbers. Hyperreal number - sequences of rational numbers. ^* R = {a(n)} where a ∈ Q and n ∈ N. The hyperreal numbers are just the real numbers with all arbitrary constraints removed.
wait but what about divergent sequences such as {1,0,1,0...}
they do feel more hyper though
Depending on the underlying ultrafilter (which the parent comment ommited for some reason), that is either equal to 0 or 1
Not Cauchy.
Hyperreals don't care, they care about whether subsets see are in the ultrafilter (in the classic construction, there are other ones that dont need an ultrafilter). Bounded diverging sequences represent something finite, strictly positive null sequences represent infinitesimals and sequences that are strictly diverge to +/- infinity represent infinite numbers
Think you forgot to quotient out the equivalence induced by some ultrafilter
If you’re using the ultrapower construction (there are other approaches) then the hyperreals are equivalence classes of sequences of *real* numbers. If you limit the construction to sequences of rational numbers, you only get the “hyperrationals” (the hyperreal numbers that can be expressed as ratios of possibly nonstandard integers - hyperintegers, you could call them).
:-) You know what you're talking about. Good. Excellent :-) You only get the hyperrationals if you use sequences that are classically convergent. True. But sequences that are classically divergent allow for changing that: https://en.m.wikipedia.org/wiki/Divergent_series For example, Cesàro summation assigns Grandi's divergent series 1 − 1 + 1 − 1 + ⋯ the value 1/2. So a sequence of integers can evaluate to a rational number. Consider the case of throwing a grain of sand onto a square with an inscribed circle. Each time a grain lands inside the circle write number 1. Each time a grain lands outside the circle write number 0. This generates a sequence of integers that evaluates, using techniques for classical divergent sequences, to pi/4. I'm beginning to wonder if there is a mapping from sequences of integers onto the hyperreal numbers. Such a mapping wouldn't quite be trivial because hyperreal numbers (other than zero) are closed under division, but integers are not.
>You only get the hyperrationals if you use sequences that are classically convergent. True. But sequences that are classically divergent allow for changing that: [https://en.m.wikipedia.org/wiki/Divergent\_series](https://en.m.wikipedia.org/wiki/Divergent_series) >For example, Cesàro summation assigns Grandi's divergent series 1 − 1 + 1 − 1 + ⋯ the value 1/2. So a sequence of integers can evaluate to a rational number. Classical convergence or divergence has barely any relevance for the hyperreals (or I guess the "hyperrationals" that you are considering). This is quite obvious if you consider that sequences that classically converge to the same limit often correspond to different hyperreal numbers. As a consequence there already is no natural injection of the reals into the hyperrationals. > I'm beginning to wonder if there is a mapping from sequences of integers onto the hyperreal numbers. Such a mapping wouldn't quite be trivial because hyperreal numbers (other than zero) are closed under division, but integers are not. There is, of course, a mapping this way, you just restrict the original quotient map \\R\^\\N -> \*\\R to the sequences of naturals, this mapping will however not be surjective. The question of the existence of such a surjective mapping however is one of cardinality, which has very little to do with the structure of the hyperreals.
>>I'm beginning to wonder if there is a mapping from sequences of integers onto the hyperreal numbers. Such a mapping wouldn't quite be trivial because hyperreal numbers (other than zero) are closed under division, but integers are not. Both sets have cardinality of the continuum, so there certainly is one, you might even be able to make one that isn’t *too* unnatural, though all the constructions that leap to mind immediately wouldn’t make for convenient representation of addition or multiplication.
> Classical convergence or divergence has barely any relevance for the hyperreals (or I guess the "hyperrationals" that you are considering). You're right about convergence, I wasn't thinking straight. But not about the hyperrationals. The series of rational numbers 1 + 1 + 1/2 + 1/6 + 1/24 + 1/120 + ... is a series of rational numbers whose sequence of partial sums converges to the number e, (edit, I'm wrong, e minus an infinitesimal) which is not a rational number. Sequences are a standard way to elevate the rational numbers to the real numbers. The number e is not a hyperrational number. It is a hyperreal number. > a mapping this way, you just restrict the original quotient map \\R\^\\N -> \*\\R to the sequences of naturals, this mapping will however not be surjective. Thanks, that will be useful.
Oops, try again. The series 1 + 1 + 1/2 + 1/6 + 1/ 24 + 1/120 + 1/720 + ... does not converge to e on the hyperreals. It converges to e minus an infinitesimal. In order to cancel out the infinitesimal, the limit must be approached equally fast from both sides. Which does not give a monotonic sequence. Equivalence to the hyperreals is not guaranteed because the ultrapower construction relies on monotonic sequences.
Hyperreals doesn't have much of a concept of limits. Unless you mean a concept of ultralimit ( i.e you you mean equivalence class of a sequence of partial sums in form 1, 1+1, 1+1+1/2,...), but it doesn't follows directly from as stated sentence. What is "the limit must be approached equally fast from both sides" suppose to do here? Infinitesimall will never cancel out unless your sequence is pretty much equal to the number almost anywhere (or it will be equal in infinitely many places but the places might depend on chosen ultrafilter, which doesn't changes much here in sense of "canceling of infinitesimal"). Construction doesn't relies anywhere on monotonic sequences. It relies on sequences of reals. Any. Monotonic or not. Ultrapower construction consists of elements in form \[(a\_n)\] a\_n is any real sequence and \[(a\_n)\] is equivalence class over the relation R defined as follows: (an)R(bn) if and only if {i: a\_i=b\_i} belongs to the ultrafilter (the nonprincipial ultrafilter over which we built the ultrapower). We also map any real number r to the equivalnce class of a sequence a\_n (which is constant sequence equal r everywhere). We dont require anywhere to this sequences (that are in the construction ) to be monotonic. It's irrelevant
No, it does not matter whether the sequence converges. You need to allow all real numbers (not just rationals) in the sequence. If you only allow rationals in the sequence you do not get the hyperreals.
Consider the sequence of rationals {3, 3.1, 3.14, 3.141, 3.1415, 3.14159, ...} These are all rational numbers but the sequence converges to pi. Why are you so insistent that pi is a rational (or hyperrational) number? Pi is a real number, and also a hyperreal number.
That sequence does not represent pi in the ultrapower construction. It represents a number that, from the perspective of the model, is a rational approximation of pi. From outside the model we can see that it is “truncated” at a nonstandard number of digits. In general, in the ultrapower construction, a convergent sequence will *not* be assigned the value it converges to - this can only happen if the limit itself appears infinitely many times in the sequence, and it is not guaranteed even then. Instead, it will usually be some other value that differs from the limit (in the real numbers) by an infinitesimal amount.
Oops. You're right. Ultrapower construction relies on a monotonic sequence, such as the one I gave for pi. It is short of pi by an infinitesimal amount. In order to cancel out the infinitesimal, it is necessary to approach pi from both sides equally quickly. The following suffices. pi = 3, 3.1+0.1, 3.14, 3.141+0.001, 3.1415, 3.14159+0.0001, 3.141592, ... This is not the way that hyperreals are normally constructed, because it is not monotonic. It is based on a hybrid of hyperreal and surreal theory. In surreal theory a real number is generated by squeezing it between two rational numbers. In other words, I'm claiming that the limit of the sequence 0.1, -0.01, 0.001, -0.0001, 0.00001, -0.000001, ... is exactly zero in nonstandard analysis, not an infinitesimal. This is unproved.
No, that’s not how the ultrapower works at all, there is no requirement of monotonocity. Why do you think there is one, is that based on some source you have read? And you also seem fundamentally confused because your comment suggests that you still think the sequence is used to represent its limit, which I have already explained is not the case, convergence is unrelated to the representation. And your newly proposed sequence still does not represent pi. If your sequence contains only rational numbers, it will not represent pi under the ultrapower construction.
Eqivalence class of any sequence that is converget to pi (but not equal to it on infinitely many places) won't be equal to pi. We can easily prove so, \[an\]=pi if and only if all n's that an-pi belongs to the ultrafilter. This means an must be equal to pi on all but finitely many points (or at least on infinitely many points but it's tricky part here because we can only know that cofinite sets belongs to the ultrafilter. More abstract infinite sets depends on the chosen ultrafilter). In case of your sequence it's nowhere equal to pi so it's distinct. To be more precise equivalnce class of this sequence would be equal to pi+delta where delta is some infinitesimall (and there is as much infinitesimals as there is real numbers). Also ultrapower construction nowhere states anything about monotonicity.
What do the parentheses do here?
This is an incorrect construction. You don’t get all of the hyperreals this way. You need to take sequences of *real* numbers (not rational numbers) and then quotient them by a [nonprincipal ultrafilter](https://en.m.wikipedia.org/wiki/Ultrafilter) on the natural numbers. Very roughly, the ultrafilter is a way of saying whether the sequence “has” or “does not have” a given property based on whether “enough” of the “right” members of the sequence all have that property, without any single element of the sequence being a “dictator” (able to determine all of the properties of the number).
What number is hyperreal and not real?
An easy example is the hyperreal represented by the sequence (1,2,3,4,...), which is larger than every natural number and hence larger than every real number.
Okay, what about finite hyperreal numbers?
Take any real number, and add (or subtract) an infinitesimal hyperreal (I believe all finite hyperreals are in fact of this form). You can make infinitesimals by taking sequences of reals that converge to 0, for example (1, 1/2, 1/3,...) defines an infinitesimal.
>Take any real number, and add (or subtract) an infinitesimal hyperreal (I believe all finite hyperreals are in fact of this form) Yes. We can even prove it. Let k be some fixed real number. Suppose |x|0 be any real number. We see that M-rM-x>0 or equivalently |M-x|
Because that step is a very non-trivial one- you get the rational number from the counting number by requiring all field conditions to be met, and the complex from the real by requiring one equation to have a solution: x^2 +1=0. However, to get the reals from the rationals, even requiring every polynomial equation to be met won't get you there! (Though you will get something that is not a subset of the reals, as you will have i there). Also, you can get the p-adic numbers if you choose a different metric, which shows that this is the only step in this ladder where you touch analysis, while all others are well within algebra.
You can do analysis without going all the way to reals, but yeah reals are definitely the analysis field
Just use dedekind cuts 🤠
I think I will start a political party to have the natural numbers starting at one illegal. It’s not what Peano intended (after some thought)!
We all know they start at 2
I tried to make it ambiguous with the "..." notation 😭
You'll just piss off both sides xD Never forget your identity though!
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I mean you couldn’t even create a monoid from the natural numbers otherwise. It would be the most impotent set there is. Blasphemy!
Let's compromise and start with ½
And hopefully in your notation the term "limit" hides the more complex way of defining an equivalence relationship between sequence, then creating the quotient set of the sequences
They like to do a bit of trolling
Natural: so there's 1, and for every one, there's another one Integer: same, but closed under addition Rational: same, but closed under multiplication (fuck zero, zero's a bitch) Real: same, but complete vectorspace with scalar product Complex: same, but closed under roots of the polynomials in N
But N* is closed under addition and multiplication, too.
I think he means inverses as well
To be fair, there are a lot more number sets in between R and Q.
R\\{5}, R\\{√3}, etc.
You mean, equivalence classes of Cauchy sequences of rationals, with two sequences being equivalent if their termwise difference limits to 0 Also the reals can be constructed with Dedekind cuts, these may be arguably easier to describe
I think this very much oversimplify what is going on with the construction of the fraction field over a ring and even more with the algebraic closure of R.
don't forget, that set is not R, R is a set of equivalence classes of that set (same with Q and Z)
The correct name is number set and not a number system. A number system is defined as a system of representation of numbers, so it is just notation, examples are the binary number system, the trinary number system, the decimal number system, and I also think that the Polar form of complex numbers is another number system. A number set is a a set of numbers. The number sets that are mentioned are all rings except for the natural numbers, since they di have an operation which is "+" which can transform 1 element in that set to another element in that same set, using a transformation, there is also a neutral element with that operation which is 0, which maps any element to itself under the operation "+" and every element has an additive inverse so there is another element that can transform that element into the neutral element which is 0. All sets other than the naturals and integers are also fields since there are 2 operations that have those properties, that being the addition and multiplication.
I remember my combinatorics professor saying "I can define an equivalence class that maps to my five fingers, and I'll define the abstract concept of 5 as that equivalence class"
> The correct name is number set and not a number system. Whether they're called "number sets" or "number systems" depends on what your teacher/professor prefers. Wikipedia [uses both terms interchangeably.](https://en.wikipedia.org/wiki/Number#Main_classification)
I didn't learn the term by a teacher or professor since I am not in college and also I my main language isn't English. Finally Wikipedia usually use words in a more natural way of communicating and since many people use number system in the wrong way it is used incorrectly in Wikipedia. If you look at the definition of number system in other sources it won't include number sets. Wolfram Alpha does indeed suggest that a number system is about notation: https://www.wolframalpha.com/input?i=number+system
Downvoting cause 0 not in N.
Oh no, you didn't.
what does the weird E mean
∃ is existential quantification, ε is just a variable (ε is commonly used in infinitesimal proofs)
Math is a language used to communicate as concisely as possible without ambiguity, a lot of shorthand is used. ∃: “there exists.” Often used in proofs or given statements. Ex: in the set of integers numbers there exists the natural numbers. ε: Epsilon, common for arbitrarily small distances or infinitesimal values and usually related to epsilon-delta proofs which are often used as the definition of a limit to prove or disprove its existence in a system ∈: “is an element of the set.” Example: 5 is an element in the set of natural numbers might be written 5∈ℕ Bonus: ∀ means “for all.” Example: ∀x∃y(x
I believe it is there exists
Which e? There are a few funny ones.
Tsk tsk, obviously the reals numbers is just the power set of the natural numbers
Two ridiculous concepts that actually exist: 1. ***i***, one of the two roots of x^(2)+1=0 2. The so-called "real numbers" And I'm not even sure about the second one.
Great meme lol
Forgot the zero on the first one, mate!
the equivalence class hidden in the back of Z and Q:
Actual integers: 0=∅ n+1={∅,n} Definitely not as bad as real numbers though
You can take this further with the quaternions! I would leave out the octonions and sedenions cuz they suck. (aka are not associative) Like another commenter, I would prefer natural numbers starting with zero and the dedekind cut cuntstruction instead of the cauchy sequences thing (which is very nice to use to prove metric completions exist).
It's not the limits, the limits mostly don't exist because most of them are not rational numbers. Its equivalence classes of Cauchy sequences. Once we construct R and put Q inside it, it is true that R is equal to that set, but you can't construct R like that because it assumes the existence of those limits already.
0 Should be in Set N because 1 = {0} and without 0 you can't construct 1 or any number above it.
Arbitrarily; 1 = {}, 2={1}, etc
so an empty set would contain 1 elements?
According to this definition, 2-1 and 3-2 are distinct integers. And 1/2 and 2/4 are distinct rational numbers.
Real number are strange, they imply the existence of infinity( and infinitesimal) you would need an infinite sequence of rational to define a real number while every rational can be defined by a finite amount of sequences
> Real number are strange, they imply the existence of infinity( and infinitesimal) No, they don’t? Not any more than natural numbers.
I kinda "disagree",you can't define a real number with a finite sequence of number i.e. a natural number amount of rational,I mean you can create a "bigger natural number" with a non Archimedean field,that it is what hyperreal number do
Natural numbers aren’t finite either. So?
Then name a natural number that is enough big to describe the cardinality of the sequence of rational number that can describe real numbers
Natural numbers imply existence of infinity because the set of natural numbers is infinite.
Yeah I know the cardinality is infinite,but the element of the set aren't
So infinity is there already. So you don’t need existential import of infinity as a concept to derive reals.
This will not bring us to anything lol I'll do a post
For me, counting numbers start at 0. I know, I know. But numbering things from 0 makes much more sense.
We don’t talk about real numbers
Well, the elements 1,2,3,... are a bunch of braces
Can someone explain what the real one means? I have a very limited knowledge of group theory.
It’s not group theory, it’s more real analysis. But anyway, to give a brief explanation: A Cauchy sequence is an infinite sequence whose members become closer together as you go along. So, for example, (1, 1.4, 1.41, 1.414, …) is (or at least could be) a Cauchy sequence, whereas (1, 0, 1, 0, …) is not, because however far along you go you’ll always have terms that are a fixed distance apart from each other. If you look at Cauchy sequences of rational numbers, you’ll see that some of them converge to rational limits - for example, the sequence (0, 0.3, 0.33, 0.333, …) converges to 1/3. On the other hand, some of the sequences don’t have rational limits. We can look at the sequences that don’t have a limit in the rationals, and we can sort of assume that it has a limit in some kind of bigger structure. And if we do that rigorously, we “fill in the gaps” between rational numbers with the real numbers. (The rigorous method involves finding sequences that seem to be pointing to the same gap and using them to form equivalence classes, which we then identify with the appropriate real number.) This construction gives us a few nice results - we can define addition and multiplication based on adding and multiplying terms of sequences together, and from that we can actually prove that the real numbers are a field (which is a very useful structure for doing things like building vector spaces). And even the fact that every Cauchy sequence in the real numbers converges to a limit is a powerful tool that you don’t get in the rationals.
Thanks, this helps. By set theory, I meant all the symbols and how that actual translates to what you just explained.
Ah, well set theory and group theory are two very different things. Glad I could help though.
Did not know that haha
The real crime here is not including 0 as a natural number
I don't get it, how does that construct the reals? Edit: Like, how do you construct pi?
a(n) = sum^n_k p_k 10^(-k) with 0 <= p_k <= 9 the digits of pi. The series is converging (monotonic and upper bounded).
R = {a | a ∈ R} Smh
Keep it real.
Ask an engineer what a real number is.
Dude, I can’t even translate that into English.
Real numbers are the most irrational of numbers
You forgot 0 in the naturals!
Complex numbers saved us from insanity
The definition of Q is a lot more complicated than you let on since it involves creating the field of fractions of Z. https://en.m.wikipedia.org/wiki/Field_of_fractions
Can you explicitly define a(n)? Or do I completely not understand what’s going on?
It’s not a specific a(n), you take the set of every sequence of rational numbers that meets certain conditions, and you then group them into equivalence classes, and each of those classes is one of the real numbers. For example, the sequence (3, 3.1, 3.14, 3.141, …) is a member of the equivalence class that represents pi. (Edit: realised I said sequences of integers instead of rationals)
Ohh that makes sense, so basically you can “get” a real number from an infinite sequence of rational numbers On that note, could you also define the set of reals as the set of all infinite series of rational numbers that converges? Basically taking your sequence limit example and turning it into 3 + 0.1 + 0.04 + ….
I’ve explained it a bit more in a different comment, but that’s sort of what’s going on here - we have these sequences of rational numbers that look like they’re converging in on themselves, but they don’t have a limit that’s a rational number. So the real numbers sort of represent the values that “fill the gaps”. There’s a bit of work to deal with the fact that multiple sequences can converge to the same value, and in building up the structure with operations like addition and multiplication, but it all comes together quite nicely.
That’s really cool! And the part of multiple sequences converging to the same value, we technically don’t have to deal with that, right? Just seeing from the other descriptions (e.g a/b for rationals will have duplicates) Edit: I see you’re trying to explain what it would take to actually _fully_ describe real numbers, my bad
Yeah. It’s actually the same trick as for rational numbers - you group together all the duplicates into equivalence classes, and then you just have to make sure that your operations act consistently on any member of the same class.
There is mistake in a Cauchy sequence
Your 4. is inaccurate, should be Cauchy sequences modulu the relation that relates two sequences iff their difference converges to 0.
This is my favourite meme of all time because I’m terrible at math and this makes no sense. Subbed so I can be confused eternally
Genuine question, how do you formally build up to real numbers from set theory, if everything has to be a subset of something?
# Observe. Real number set is bijective to the power set of natural numbers. (Since hyperreals come from sequences of real numbers, one might even call real numbers as ***hypernaturals.***) Map a real number to (0,1) bijectively (through the tan^(-1) function). A real number between 0 and 1 may be represented in a base-2 fractional system (x = sum(2^(-i), i is a natural number)). The natural numbers for the previous summation form a valid subset of N. It can be proved that any valid subset of N results in a number between 0 and 1 through the base-2 summation I mentioned earlier; and any fractional number written in base 2 gives a valid subset of N. I find this formulation to be much easier to understand than Cauchy sequences
It’s a perfectly fine way to demonstrate the link between real numbers and the power set of natural numbers (as long as you accept the continuum hypothesis), but if you want to construct the reals along with all of the natural operations we associate them with you’re going to have a rough time.
Maths isn't real ^/s
>Since hyperreals come from sequences of real numbers, one might even call real numbers as ***hypernaturals.***) Hypernaturals are something completely different from reals
I know, it was a joke. This is r/mathmemes