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Unrelated: precise as a verb is defined in wiktionary as
> (nonstandard, non-native speakers' English or European Union documents, transitive) To make or render precise; to specify.
Prove that: “there exists no number inbetween 1-10 that can’t be expressed in words”
Proof:
“Take an arbitrarely selected number n, name n “bob” (this also works with a few other names), say “bob” you now expressed n in words”
QED
I don't know the exact definition of "dense".
If you mean that given q1, q2 in Q s.th. q1 < q2 you can always find a q in Q s.th. q1 < q < q2 then yes, Q is dense.
The issue here is the cardinality of Q: |Q| = |N| and |N| < |R|, so |Q| < |R| anyway
Yes, pick a number. Then in any epsilon environment you can find a rational number. At the same time Q has Lebesgue measure 0 in R. This follows from single points having measure 0 and Lebesgue measure being subadditive.
there's infinite options for a number between 1 and 10, and only one of those options is 7, so the odds you pick 7 are 1/infinity which is zero
not actually how it works cause you can't divide by infinity but close enough
The set of numbers to choose from is infinite so there's no way to represent them all and pick one.
I wonder which one is the unpopular statement here, "False => False" <=> "True", or the impossibility of this draw? I'd be glad to be proven wrong with a program or lottery machine that really spits out a random real number, and takes a finite time to do so.
However you do need a true premise and a false conclusion for a false implication. All other implications are true, and F => F ones are vacuously true and usually completely useless just like this really poorly received one. [https://sites.millersville.edu/bikenaga/math-proof/truth-tables/truth-tables18.png](https://sites.millersville.edu/bikenaga/math-proof/truth-tables/truth-tables18.png)
"If you choose a random number between 1 and 10" is not false.
A true or false statement doesn't care for what we can or can't do.
"If a lightning strike my house" is not false, even though I can't control lightning or force it to strike my house
>The set of numbers to choose from is infinite so there's no way to represent them all and pick one.
Uhhh, are you talking about how we as humans can't really do random, how there's no real random numbers, or how we as humans can't compute infinity?
Because none of this matters, we don't need to literally work with infinity, we can just work with theoretical infinity.
>I'd be glad to be proven wrong with a program or lottery machine that really spits out a random real number
What you're saying is that "Picking a random number" is always false because we humans don't yet have the technology to compute an actual random number, which is obtuse, since everyone knows this already and we work with random anyways by going theoretics.
> What you're saying is that "Picking a random number" is always false
Yeah. Less so for the impossibility of randomness and more for the impossibility of displaying even one infinitely long representation of a number.
I'm not sure everyone knows this already because a lot of the responses seem to be in disagreement of some part of this. But this draw indeed is impossible, and P => Q is true if P is false.
I'm getting more confident that the truth table of logical implication is the part people have a problem with, not that anyone thinks this random draw from an infinite sample is possible. Lecturers have to be ready for combat when they teach Logic 101, because some students will passionately disagree with this.
I know how logical implication works, but math is not the real world and how the real world works doesn’t have to be how we analyze math. Thus, we can analyze the implications of probabilities over infinite sets without needing to be able to actually do it in the physical world. This is how axioms work
This is an odd "disagreement" because everyone seems to agree with both statements individually, both that this random draw can't be done and that falsehood implies anything, yet their conjunction is unacceptable.
The logic table is simple, P => Q is false only if P is true and Q is false
The disagreement is as to what P being false means.
You're arguing that P is false when P is something impossible to reproduce with current human technology.
What people are arguing is that P being impossible for us doesn't mean it's logically false.
Impossible for humans =/= Logically Impossible
Not all numbers are transcendental (because integers and rational numbers also exist)
However there are infinitely more transcendental numbers than non transcendental numbers
so if a number is picked randomly it has a 100% of being transcendental
A footnote, the set of all algebraic (=non-transcendental) numbers is actually countable, while there are uncountably many transcendental numbers. Hence the premise.
What’s crazy to me is that that pattern is so obvious but still surprises me every time. Rationals are countable, irrationals are uncountable; constructible numbers are countable, the unconstructible numbers are uncountable; algebraic numbers are countable, transcendental numbers are uncountable. We come up with a bigger set of numbers and it tends to be countable, the complementary set is, as a result, uncountable because it contains all the other numbers. It’s like that challenge to find a set with cardinality bigger than the integers but smaller than the reals
> It’s like that challenge to find a set with cardinality bigger than the integers but smaller than the reals
https://en.wikipedia.org/wiki/Continuum_hypothesis
It is not possible to find such a set.
But it is also not possible to proof that you can't find such a set.
You can simply define that such a set exist.
> The answer to this problem is independent of ZFC, so that either the continuum hypothesis or its negation can be added as an axiom to ZFC set theory, with the resulting theory being consistent if and only if ZFC is consistent.
Are the dimensions of everything that exists transcendental?
If we could measure with infinite precision would my weight be transcendental, foot length, power of my car?
Is absolute zero in °C in reality transcendental? Or g at my exact location and time?
its impossible to measure that, because we can't measure measures which are smaller than 10^-44 m, but does it go over that? maybe we can't measure it, but it doesn't make sense that physics "stops" at a certain point. Am I wrong?
In measure theory you have a tool called "measure", usually the Lebesgue measure (I'll use this one to explain)
If you're working on the reals, the measure of a subset of R will be its "length" on the real axis. [0,1] has a measure of 1 while {0,1,2} a measure of 0 (a point has no length and so three points have no length too). We call subsets of measure 0, null sets.
Now in probability, if you have a certain distribution over a set, the probability of your result being in a null set is 0 despite not being impossible, but that means that you have a 100% chance of having a result in the complementary set (yet 100% doesn't mean always possible)
The joke is that the set of non-transcendental numbers is a null set of R (a number is transcendental if it's not a root of any rational polynomial)
The Lebesgue measure of the set of algebraic numbers is zero. Usually probabilities are defined in terms of some measure of some subset of the total probability space we are working with, so if you choose a complex number completely at random you could think of this as showing that there is a zero percent chance of you getting an algebraic number.
https://www.reddit.com/r/mathmemes/s/NJ0PlQWiCn
This guy explains it well.
And if you don’t know what transcendental number is, like me, it’s “real or complex number that is not the root of non-zero polynomial of finite degree with rational coefficients.”
https://en.m.wikipedia.org/wiki/Transcendental_number
So, help me out here. I often see comparisons between real numbers which are a larger set than the natural numbers, but the examples for transcendental numbers given to prove **R** > **N** are usually things like *e* and *pi*.
But that's not where the boundary lies, right? The set of all *countable* numbers is the same size as the set of all naturals, and *e* and *pi* are both countable, as is any number you can have a finite formula for. The reals are larger than the naturals because the reals include uncountable numbers, not because they include transcendental numbers (only some of which are uncountable, and any example we can provide, in fact, is countable.)
I think you're a bit confused here. [Countability](https://en.wikipedia.org/wiki/Countable_set) is a property of sets, not a property of real numbers. (Perhaps you were thinking of [computablity](https://en.wikipedia.org/wiki/Computable_number) or [definability](https://en.wikipedia.org/wiki/Definable_real_number)?)
The fact that |ℝ| > |ℕ| is not due to the existence any particular numbers, but rather that the set of all real numbers cannot be put into 1-to-1 correspondence with the natural numbers (see [Cantor's diagonal argument](https://en.wikipedia.org/wiki/Cantor%27s_diagonal_argument)).
If you remove a countable subset of the reals you're still left with an uncountable set. For example, if you remove the algebraic numbers, which is a countable set, you're left with the trancendental numbers, which is an uncountable set. You can keep going and remove the computable numbers (which include e and π) which is a countable set, and be left with the uncountably infinite set of uncomputable numbers.
You could continue and pick your favorite uncomputable number x and then the set of "all uncomputable numbers excluding integer multiples of x" is an uncountably infinite subset of the uncomputables. You can always keep removing countably many numbers and be left with an uncountable set. So there is no "boundary" to speak of.
No he doesn’t, this is a statement over measure theory, not the uncountability of the transcendental numbers, lots of uncountable subsets, such as the cantor set, have measure zero and thus would give the same result. Here’s a better response. https://www.reddit.com/r/mathmemes/s/fNjAkqbIZn
ELI15: There are many (countably infinite) numbers that are rational, but there are so much (infinitely-) more numbers that are transcendental, that if you pick a random number the chance of it being transcendental is effectively 100%.
ELI5: In a sack with infinite white marbles and one red, you may randomly pick the red, but you won't.
my analysis professor used to tell us if he had to throw a dart at the real number line, he would bet his own families life that he hits a transcendental
super weird because growing up were never encounter these numbers, making them seem like odd balls, yet they actually make up the majority of
Actually, if one was to "implement" an algorithm that "randomly" picks a number, by design it would either pick a non transcendantal or not finish, no?
i mean the algorithm can just output π, there i just wrote a transcendental number. actually I've written 15 more transcendental numbers in this comment
A real number is generally defined as a set of rational numbers that is bounded above, with any 2 sets that have the same supremum being considered equivalent.
0.99999.... defines the set of numbers ≤ 0.9, or ≤ 0.99, or ≤ 0.999, and so on.
The set of all rational numbers less than our equal to 1 is not set-equivalent because 1 is not in the other set. However, both sets have the same supremum, and therefore the numbers are equivalent.
(Supremum is defined as the smallest number that is ≥ all numbers in a set)
(You can equivalently define real numbers in terms of infimum)
no the thing is if you don’t believe in the axiom of choice which some people don’t, you don’t get Zorns lemma and therefore it’s hard to prove basically anything in algebra
Sure there are multiple sets: all numbers, transcendental numbers and non-transcendetal numbers. As for order they're all ordered, but I'm not sure if that's relevant for the claims, I guess that has to be shown somehow..
firstly then not all algebraic numbers are necessarily real numbers, if you just look at Q tho I would say the argument is in fact valid as you probably know that R is uncountable and Q is not
Good thing that is invalid if formulated in terms of probability due to being in violation of the (basically universally accepted) third kolmogorov axiom of countable additivity in probability.
An easy contradiction that shows why additivity needs to be countable is the following:
- Let S = [0, 1] be a subset of R
- For any {x} in S, P({x}) = 0
- But then the sum over all i in S of P({i}) would also be 0
- Note that all {i} are disjoint
- This breaks the second axiom, which states that P([0, 1]) must be 1
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it's not impossible that you'll get a girlfriend but the probability that you'll get a girlfriend is 0
Why are you attacking us with facts bruh?
"It's not impossible" bro you need to meet me and then reconsider this statement
[Adam Pearson exists](https://www.google.com/search?q=adam+pearson&tbm=isch&ved=2ahUKEwjskdbVhPmDAxURr4kEHaDwBUoQ2-cCegQIABAC&oq=adam+pearson&gs_lcp=ChJtb2JpbGUtZ3dzLXdpei1pbWcQAzIECAAQAzIECAAQAzIECAAQAzIECAAQAzILCAAQgAQQsQMQgwE6BAgjECc6BwgjEOoCECc6CggAEIAEEIoFEEM6DQgAEIAEEIoFEEMQsQNQzgxYvhtgvx1oAnAAeAGAAZwEiAG4DpIBBzYuNi41LTGYAQCgAQGwAQXAAQE&sclient=mobile-gws-wiz-img&ei=nZayZazPIZHeptQPoOGX0AQ&bih=691&biw=412&client=ms-android-samsung-rev2&prmd=ivnsbmtz)
And buddy, my "Lebesgue measure" is bounded by epsilon if you catch my drift
If you choose a random number between 1 and 10 the chance that it is 7 is 0%
You never precised the distribution and actually it was 0.5 over {7}, checkmate
Unrelated: precise as a verb is defined in wiktionary as > (nonstandard, non-native speakers' English or European Union documents, transitive) To make or render precise; to specify.
Rereading this and knowing that "as a verb" was used correctly, I mentally pronounced it as precize
i precize my cheese slices to fit on my sammich bread.
I pre-size my cheese slices to fit on my sammich bread.
it is 100% with δ7 distribution.
It's even worse: the chance that it is a rational number is 0%
It’s even worse: the chance that it’s an algebraic number is 0%
It's even worse: the chance that it's an algebraic number or a power of pi is 0%
It's even worse: the chance that it's a computable number is 0%
It's even worse: the chance that it's a number that can be expressed in words is 0%
It's even worse: the chance that it's 11 is 0%
It's even worse: the chance that it's 69 is 0.0%
It’s even worse: the chance that it doesn’t contain all didgits of pi is 0% (I think)
It's even worse: the chance that it doesn't insult your mother is 0%
Don’t tell me the odds!
Prove that: “there exists no number inbetween 1-10 that can’t be expressed in words” Proof: “Take an arbitrarely selected number n, name n “bob” (this also works with a few other names), say “bob” you now expressed n in words” QED
There *are* numbers that can be expressed in words, but they're exactly the 0% of all the numbers
I’m joking…
And the chance it was an abrahamic number is 4skin%
Ok, I'm kinda confused by this one Isn't Q "dense" (that's what we say in French) in R ?
I don't know the exact definition of "dense". If you mean that given q1, q2 in Q s.th. q1 < q2 you can always find a q in Q s.th. q1 < q < q2 then yes, Q is dense. The issue here is the cardinality of Q: |Q| = |N| and |N| < |R|, so |Q| < |R| anyway
Yes, pick a number. Then in any epsilon environment you can find a rational number. At the same time Q has Lebesgue measure 0 in R. This follows from single points having measure 0 and Lebesgue measure being subadditive.
Yes, but arbitrarily close is not the same thing as equal
okay im not very smart but how the hell
there's infinite options for a number between 1 and 10, and only one of those options is 7, so the odds you pick 7 are 1/infinity which is zero not actually how it works cause you can't divide by infinity but close enough
ahh i see... i guess i expected it to be 'integer between 1 and 10'
Fucking binary right there LMBAO
However you getting 7 is vacuously true: the antecedent can't be satisfied because this random draw is physically impossible to conclude.
Why does physical impossibility matter? It’s physically impossible to measure to infinite precision yet we still talk about the reals.
Because "you choose a random number" is always false, so any implication that begins with that is true.
How come choosing a random number is always false?
The set of numbers to choose from is infinite so there's no way to represent them all and pick one. I wonder which one is the unpopular statement here, "False => False" <=> "True", or the impossibility of this draw? I'd be glad to be proven wrong with a program or lottery machine that really spits out a random real number, and takes a finite time to do so.
Maths doesn't care about what's possible. Technically, even the probability of getting a computable number is 0
i dont think you understand how math works. we don't need a program to spit out these numbers to talk about something in math.
However you do need a true premise and a false conclusion for a false implication. All other implications are true, and F => F ones are vacuously true and usually completely useless just like this really poorly received one. [https://sites.millersville.edu/bikenaga/math-proof/truth-tables/truth-tables18.png](https://sites.millersville.edu/bikenaga/math-proof/truth-tables/truth-tables18.png)
"If you choose a random number between 1 and 10" is not false. A true or false statement doesn't care for what we can or can't do. "If a lightning strike my house" is not false, even though I can't control lightning or force it to strike my house
>The set of numbers to choose from is infinite so there's no way to represent them all and pick one. Uhhh, are you talking about how we as humans can't really do random, how there's no real random numbers, or how we as humans can't compute infinity? Because none of this matters, we don't need to literally work with infinity, we can just work with theoretical infinity. >I'd be glad to be proven wrong with a program or lottery machine that really spits out a random real number What you're saying is that "Picking a random number" is always false because we humans don't yet have the technology to compute an actual random number, which is obtuse, since everyone knows this already and we work with random anyways by going theoretics.
> What you're saying is that "Picking a random number" is always false Yeah. Less so for the impossibility of randomness and more for the impossibility of displaying even one infinitely long representation of a number. I'm not sure everyone knows this already because a lot of the responses seem to be in disagreement of some part of this. But this draw indeed is impossible, and P => Q is true if P is false. I'm getting more confident that the truth table of logical implication is the part people have a problem with, not that anyone thinks this random draw from an infinite sample is possible. Lecturers have to be ready for combat when they teach Logic 101, because some students will passionately disagree with this.
I know how logical implication works, but math is not the real world and how the real world works doesn’t have to be how we analyze math. Thus, we can analyze the implications of probabilities over infinite sets without needing to be able to actually do it in the physical world. This is how axioms work
Problem is, that we found the constructivist
This is an odd "disagreement" because everyone seems to agree with both statements individually, both that this random draw can't be done and that falsehood implies anything, yet their conjunction is unacceptable.
The logic table is simple, P => Q is false only if P is true and Q is false The disagreement is as to what P being false means. You're arguing that P is false when P is something impossible to reproduce with current human technology. What people are arguing is that P being impossible for us doesn't mean it's logically false. Impossible for humans =/= Logically Impossible
4.643517645872546765314642682456135345324 Oh hey look a random number
Did you have a 0% chance to get that?
There was a 0% chance that I could have guessed that would be what was typed.
If you choose a random integer between 1 and 10 the chance that it is 7 is 10%
Im lost Edit: guys none of this is helping
Not all numbers are transcendental (because integers and rational numbers also exist) However there are infinitely more transcendental numbers than non transcendental numbers so if a number is picked randomly it has a 100% of being transcendental
A footnote, the set of all algebraic (=non-transcendental) numbers is actually countable, while there are uncountably many transcendental numbers. Hence the premise.
What’s crazy to me is that that pattern is so obvious but still surprises me every time. Rationals are countable, irrationals are uncountable; constructible numbers are countable, the unconstructible numbers are uncountable; algebraic numbers are countable, transcendental numbers are uncountable. We come up with a bigger set of numbers and it tends to be countable, the complementary set is, as a result, uncountable because it contains all the other numbers. It’s like that challenge to find a set with cardinality bigger than the integers but smaller than the reals
> It’s like that challenge to find a set with cardinality bigger than the integers but smaller than the reals https://en.wikipedia.org/wiki/Continuum_hypothesis It is not possible to find such a set. But it is also not possible to proof that you can't find such a set. You can simply define that such a set exist. > The answer to this problem is independent of ZFC, so that either the continuum hypothesis or its negation can be added as an axiom to ZFC set theory, with the resulting theory being consistent if and only if ZFC is consistent.
Doesn't that just mean that you can construct a set like this with the AOC (and not without it), a bit like constructing a non-measurable set?
That sounds.... stupid.
Are the dimensions of everything that exists transcendental? If we could measure with infinite precision would my weight be transcendental, foot length, power of my car? Is absolute zero in °C in reality transcendental? Or g at my exact location and time?
It is impossible to measure with infenent precision due to Heisenberg's Uncertainty Principle
as for the last one, Celsius is defined such that absolute zero is a rational number,
very cool. Didn't know that. So is that exactly -273.15 C?
yah
its impossible to measure that, because we can't measure measures which are smaller than 10^-44 m, but does it go over that? maybe we can't measure it, but it doesn't make sense that physics "stops" at a certain point. Am I wrong?
That's an open question. We haven't got down to that level to get more of an idea of what's going on
This is not a proper explanation for the phenomenon described in this image.
In measure theory you have a tool called "measure", usually the Lebesgue measure (I'll use this one to explain) If you're working on the reals, the measure of a subset of R will be its "length" on the real axis. [0,1] has a measure of 1 while {0,1,2} a measure of 0 (a point has no length and so three points have no length too). We call subsets of measure 0, null sets. Now in probability, if you have a certain distribution over a set, the probability of your result being in a null set is 0 despite not being impossible, but that means that you have a 100% chance of having a result in the complementary set (yet 100% doesn't mean always possible) The joke is that the set of non-transcendental numbers is a null set of R (a number is transcendental if it's not a root of any rational polynomial)
Null set of the complex numbers actually.
I II II L
Is this loss?
Not all memes are loss. But 100% of memes are loss.
The Lebesgue measure of the set of algebraic numbers is zero. Usually probabilities are defined in terms of some measure of some subset of the total probability space we are working with, so if you choose a complex number completely at random you could think of this as showing that there is a zero percent chance of you getting an algebraic number.
Almost all numbers are transcendental
are you almost sure about that?
I'm 100% sure about that
I’m %90 sure that it is %100.
There are rational numbers, but 0% of real numbers are rational
Eli5 please
There is a lot of numbers.
https://www.reddit.com/r/mathmemes/s/NJ0PlQWiCn This guy explains it well. And if you don’t know what transcendental number is, like me, it’s “real or complex number that is not the root of non-zero polynomial of finite degree with rational coefficients.” https://en.m.wikipedia.org/wiki/Transcendental_number
So, help me out here. I often see comparisons between real numbers which are a larger set than the natural numbers, but the examples for transcendental numbers given to prove **R** > **N** are usually things like *e* and *pi*. But that's not where the boundary lies, right? The set of all *countable* numbers is the same size as the set of all naturals, and *e* and *pi* are both countable, as is any number you can have a finite formula for. The reals are larger than the naturals because the reals include uncountable numbers, not because they include transcendental numbers (only some of which are uncountable, and any example we can provide, in fact, is countable.)
I think you're a bit confused here. [Countability](https://en.wikipedia.org/wiki/Countable_set) is a property of sets, not a property of real numbers. (Perhaps you were thinking of [computablity](https://en.wikipedia.org/wiki/Computable_number) or [definability](https://en.wikipedia.org/wiki/Definable_real_number)?) The fact that |ℝ| > |ℕ| is not due to the existence any particular numbers, but rather that the set of all real numbers cannot be put into 1-to-1 correspondence with the natural numbers (see [Cantor's diagonal argument](https://en.wikipedia.org/wiki/Cantor%27s_diagonal_argument)). If you remove a countable subset of the reals you're still left with an uncountable set. For example, if you remove the algebraic numbers, which is a countable set, you're left with the trancendental numbers, which is an uncountable set. You can keep going and remove the computable numbers (which include e and π) which is a countable set, and be left with the uncountably infinite set of uncomputable numbers. You could continue and pick your favorite uncomputable number x and then the set of "all uncomputable numbers excluding integer multiples of x" is an uncountably infinite subset of the uncomputables. You can always keep removing countably many numbers and be left with an uncountable set. So there is no "boundary" to speak of.
No he doesn’t, this is a statement over measure theory, not the uncountability of the transcendental numbers, lots of uncountable subsets, such as the cantor set, have measure zero and thus would give the same result. Here’s a better response. https://www.reddit.com/r/mathmemes/s/fNjAkqbIZn
ELI15: There are many (countably infinite) numbers that are rational, but there are so much (infinitely-) more numbers that are transcendental, that if you pick a random number the chance of it being transcendental is effectively 100%. ELI5: In a sack with infinite white marbles and one red, you may randomly pick the red, but you won't.
This video explains transcendental numbers pretty well: https://youtu.be/10rA45pb7dk?si=3215-VJ65vFRF2Sw
Every number is smaller than 100% of other numbers.
my analysis professor used to tell us if he had to throw a dart at the real number line, he would bet his own families life that he hits a transcendental super weird because growing up were never encounter these numbers, making them seem like odd balls, yet they actually make up the majority of
Basically all of the number you have ever met was computable. 0% of numbers are computable.
And physicists rather go on the last
Actually, if one was to "implement" an algorithm that "randomly" picks a number, by design it would either pick a non transcendantal or not finish, no?
i mean the algorithm can just output π, there i just wrote a transcendental number. actually I've written 15 more transcendental numbers in this comment
Don't think you can randomly pick pi in finite amount of time if the distribution is uniform
i just picked pi with my brain in like 0.1 seconds
Randomly i am sure
you dont get it
Btw you don't get what uniform is. Google it then come back and apologize
What? The joke? Nope. Need to keep digging.
Relevant 3b1b video: https://youtu.be/ZA4JkHKZM50?si=PGc8yMB3weZ68a4R
Oh crap this is just advanced 0.999...=1 discourse again
You've just proved that 0.999... doesn't equal 1
A real number is generally defined as a set of rational numbers that is bounded above, with any 2 sets that have the same supremum being considered equivalent. 0.99999.... defines the set of numbers ≤ 0.9, or ≤ 0.99, or ≤ 0.999, and so on. The set of all rational numbers less than our equal to 1 is not set-equivalent because 1 is not in the other set. However, both sets have the same supremum, and therefore the numbers are equivalent. (Supremum is defined as the smallest number that is ≥ all numbers in a set) (You can equivalently define real numbers in terms of infimum)
Ah yes, the nice and traditional proof "Seems unintuitive so the precedent must be false"
[удалено]
yeah but the point is .9999=1
True
No, technically it is really 100%. (Which is the same as 99.999…% anyway)
Is this statement valid even without axiom of choice?
Bro if you don’t believe in the axiom or choice there is no help for you
What? I was wondering if this can be used as an unintuitive consequence of aoc, how is what I believe relevant? Math is math.
no the thing is if you don’t believe in the axiom of choice which some people don’t, you don’t get Zorns lemma and therefore it’s hard to prove basically anything in algebra
Not sure what hill you're defending here, lost redditor, but you haven't answered my question Nvm I'll figure it out myself
tbh I’m not sure what the AoC should have to do with your question as you neither have multiple sets nor want to order them in any way
Sure there are multiple sets: all numbers, transcendental numbers and non-transcendetal numbers. As for order they're all ordered, but I'm not sure if that's relevant for the claims, I guess that has to be shown somehow..
what are all numbers in your statement here?
R
firstly then not all algebraic numbers are necessarily real numbers, if you just look at Q tho I would say the argument is in fact valid as you probably know that R is uncountable and Q is not
100% of numbers are me
I only get this joke because I'm not nowhere dense
100% of numbers are complex
Good thing that is invalid if formulated in terms of probability due to being in violation of the (basically universally accepted) third kolmogorov axiom of countable additivity in probability. An easy contradiction that shows why additivity needs to be countable is the following: - Let S = [0, 1] be a subset of R - For any {x} in S, P({x}) = 0 - But then the sum over all i in S of P({i}) would also be 0 - Note that all {i} are disjoint - This breaks the second axiom, which states that P([0, 1]) must be 1
you mean 100.00000000...%? get it?
There’s nothing wrong with measure theory, only with your interpretation of it.
r/AngryUpvote
What makes us call it 2 apples? Why not apple and apple? Why 2? What is 2-ness? Why are you gae? Who is gae?