>infinite sets
>some infinities are bigger than other infinities
>undefinable real numbers are real
why is math infected with unfalsifiable nonsense? Is that why this board is called Science AND Math? Because modern math isn’t actually scientific?
Math is an abstraction. You can debate whether or it's still *real*, but if you believe the scientific model encompasses all of reality, then math doesn't exist, much like god and consciousness
>unfalsifiable nonsense
It is falsifiable, see cantors diagonal proof. Also falsifiability is a pretty bad metric in science anyway
>modern math isn’t actually scientific
It never was. Math isn't based on empirical observation
You can’t falsify cantor’s proof in the real world. It’s just scribbles on a piece of paper. The proof can’t even be verified by a computer because it has hidden assumptions. By the way, the definable numbers are countable. Cantor’s proof relies on assuming there exist undefinable numbers, which cannot be represented by a finite definition. For example, there are numbers like 0.6205842038…. that can ONLY be expressed by their infinite sums, which means we will never be able to interact with them whatsoever. They aren’t even necessary to complete the reals, since the closure property is defined by finite addition, not infinite addition. You can only attain an undefinable number from a definable number with an infinite, undefinable sum. The problem with Cantor’s proof is that the diagonal number is undefinable, even if every number in the list is definable, because the mapping is not well-defined.
Wow, you’re wrong in several ways.
pathetic.
the diagonal argument is equivalent to turing's proof for the halting problem
which means that if you hold that cantors's proof is actually false, that you must then hold that "H"(or however you know the machine involved in turing's proof as) does exist, would that be your position?, or will you grace >us with a pretty display of mental acrobatics?
The diagonal proof requires undefinable numbers and is therefore useless. If every number in the list is definable, then the construction of the diagonal number must be undefinable, since if it were definable, it would be in the list at a precise location, since its definition would be finite and the list could be arranged increasing in size of the definition. So there exists a 1:1 mapping between the natural numbers and definable real numbers (the numbers that we can actually interact with).
No it doesnt
>So there exists a 1:1 mapping between the natural numbers and definable real numbers (the numbers that we can actually interact with).
So you're saying there does exist a definable bijection f : N -> R?
There is a definable bijection f : N -> A where A is the set of algebraic numbers of solutions to countably long polynomial equations with integer coefficients.
I'm not a computability theorist, but some people smarter than me have used the same arguments people use for the algebraic numbers being countable to demonstrate that the set of all computable numbers are countable (though mathematicians tend to be pretty wishy washy about computability theory being legitimate as an extension of algebra).
I’m not saying that there is a definable 1:1 mapping, but there is a mapping between N and R that covers every real number. All you do is convert every real number to a finite definition, then arrange the definitions in order of their size. You could also eliminate repeats (Ex: 2+2, 4, 2^2, 5-1, etc. as all these are definitions of the number 4). Now if you eliminated all the repeats, then the mapping would be 1:1. But the process of eliminating the repeats is what makes the mapping hard to define. But even if you don’t eliminate repeats, then it’s not clear how to define how the definitions increase in size. But theoretically it could be done. But if you were to try to predict the real number that maps to, say, the natural number 918, then you would seemingly have to iterate through the whole list to figure it out. It’s messy but it would indeed be a 1:1 mapping.
The problem with the diagonal number is that if it were definable, then it would be in the list at a precise location. Therefore it would define itself to be different than itself, a contradiction. So at BEST the diagonal number is undefinable, if not completely non-existent (I’m not so sure that the definable numbers do not exhaust all possible sequences of decimals)
Nta but this statement below is the problem:
> All you do is convert every real number to a finite definition.
You definitionally cannot do this, even within computable numbers. For example, simple irrational numbers like sqrt(2) literally cannot be represented via a finite length decimal expansion and can only be made finite via representation in polynomial space (which isn't an expansion of any rational base).
>sqrt(2)
x^2 - 2 = 0
This is a definition.
That's polynomial space. That isn't a finite expansion with some finite rational base.
This still has the same problem because there's n solutions to each n-degree polynomial you'd specify in polynomial space, meaning you will not only have unique expansions but you also won't even have unique numbers.
Even in this case, +sqrt(2) and -sqrt(2) would have the same representation in this "alphabet" despite being different numbers. This is a much worse problem than having potentially multiple representations for the same number.
The list could be in English:
“The positive square root of two”
“The smallest number such that the number cubed minus three equals zero”
etc.
Even in math notation I’m sure you could do this too
> Even in math notation I'm sure you could do this too.
If you can find a way to represent any arbitrary finite real number in a finite number of bits, you have a fortune waiting for you in silicone valley.
Defining a number linguistically via a description in English is not the same as an actual numerical expansion to this number. Your English definition of the number cannot be used for any functional transformations, and as a result it ceases to be a meaningful number.
> Every number is computable and therefore definable.
This is just flat out not true. In fact, the non-computable numbers outnumber the computable numbers so heavily that there's an entire field of study devoted towards minimum distortion representations of numbers which cannot be directly computes/represented via a binary expansion.
Every combination of digit should be stored in a substring of irrationals or a sum thereof. Give me my paycheck.
You not only haven't answered the question, I don't think you even understand what the problem is.
> Every combination of digit should be stored in a substring of irrationals or a sum there of.
What irrationals and how do you represent them? Your "substring" will be made of individual entries which must come from some formal language to have a meaningful representation. There is no such formal language which can represent irrational numbers in a finite number of D-ary numbers where D is some positive integers. Again, you should actually try to spend some time learning about how things work before you speak ignorant bullshit.
If you can produce errorless finite length encoding of any real values number, you'd win a novel prize with no problems. You can't because it's not possible without handwavy bullshit that can't actually have a numeric representation with a properly defined alphabet and algebraic closure under functional operations.
[math]
sqrt{3}
[/math]
Stay mad.
Okay, so what is that? What is sqrt(3) as a number with an expansion in some form?
Also, what would you call the set of all numbers such that x-y = sqrt(3)? Is this set finite in your view for some reason? Even if we take your moronic conclusion at face value, how does that lead you to the idea that all sets are finite?
it's the solution of x^2 = 3
>Also, what would you call the set of all numbers such that x-y = sqrt(3)?
The set of all numbers such that x-y = sqrt(3) is equivalent to the set of all numbers such that x-y = sqrt(3).
When you embrace infinity, repeating the question is the solution.
Can you tell me what e + π + sqrt(2) is?
> Can you tell me what e + π + sqrt(2) is?
What exactly are you asking?
When we ask things like "what is 2 + 3", we except an answer like 5 since it's a simpler/canonical way to describe 2 + 3.
But as far as I know, there are no simpler ways to describe e + π + sqrt(2).
In what way do you want me to simplify e + π + sqrt(2)?
> Defining a number linguistically via a description in English is not the same as an actual numerical expansion to this number. Your English definition of the number cannot be used for any functional transformations, and as a result it ceases to be a meaningful number.
So? Each expression would still uniquely represent a real number. That doesn’t affect my argument in the slightest.
Okay, so your argument is "we can define all numbers in some way, even if we can't define them as numbers which can actually have operations performed on them."
Any other genius arguments or are you done being completely moronic?
> even if we can't define them as numbers which can actually have operations performed on them
you haven’t explained why this matters. I think you’re just trying to show off that you’re a computer nerd and know so much more than everyone else. That has nothing to do with my argument. There is a list that contains every definable real number, whether or not you can perform operations on the definitions themselves.
The reason I'm bringing up computational /numerical representation of the number is because that's literally where the concept of a number being "definable" comes from. A number is definable within a language if and only if there are some sequence of operations in that language which can uniquely produce this number.
For a number to be a number (as opposed to a word in a language) there has to be some kind of mathematical operations you can perform on it and get numbers of a similar type. You can define sqrt(3) as "the positive solution to the equation x^2-3=0" but you cannot represent this solution via a finite number of digits in any mathematical/numerical structure. However, you still need to define this numerical representation via some operational structure on "similar" numbers.
As an example, the counting numbers/natural numbers are numbers which are closed under addition. Any two natural numbers added together produces another natural number.
In your way of thinking, having some "definition" of the number is sufficient to it being a number, even though every number is potentially an isolated island where the method of definition of that number potentially may only work for that specific number. As a result, it ceases to be a number as there's literally nothing algebraic or arithmetic you can do within this "list" for the super majority of your numbers (which can at most be described linguistically or via the solution of some infinite lenght polynomial equation).
All of this is besides the point that the set of "decidable" numbers is definitionally infinite, meaning your whole beef with infinite sets is even more moronic.
>silicone valley
>all this other trash
You have to be 18 to post here.
Now someone might say
>but the diagonal number is in a higher language!
Then I say that it is still definable in that language. So I simply create a list that contains numbers that are definable in any language, the only requirement is that the definitions are specific, defining a unique number.
My god computer science has broken your brain. Please, for the love of God, take an actual math course before you start making moronic statements about continuous spaces like the real numbers.
Just think about what it would mean if you were correct. If you were correct, it would be possible to do lossless compression of any real number in a finite number of bits. Don't you think that computing and digital communication would be a little different if this were true?
Why do you bring up computing? In computing, every number is computable and therefore definable. No undefinable numbers included
>then it’s not clear how to define how the definitions increase in size
Just define the size of a definition to be the number of characters used to express it.
But if you really can't define any such list, then why are you okay with a 1:1 mapping between N and R existing which can't be defined, but you aren't okay with real numbers that aren't defined?
> Just define the size of a definition to be the number of characters used to express it.
Sure, but the order is not specified among definitions with the same character size. I imagine that you COULD do this, but you would have to know precisely which characters are available to use in the definitions. For example, all 26 characters in the English language.
> But if you really can't define any such list, then why are you okay with a 1:1 mapping between N and R existing which can't be defined, but you aren't okay with real numbers that aren't defined?
I actually think the mapping would be definable, but even if it weren’t, there would still exist a 1:1 mapping. For example, you could randomize the order of the definitions, and then it wouldn’t be definable, since declaring the mapping to be random does not actually specify the exact order of the mapping in a finite set of characters. You would have to iterate through the whole list to understand the mapping, similar to how you would have to represent an undefinable number by an infinite sum. But now I’m starting to question if undefinable numbers can even “exist.” Suppose there is some infinite sum in the Platonic realm. Then how do we know there isn’t some finite definition that corresponds to that exact infinite sum?
>For example, you could randomize the order of the definitions, and then it wouldn’t be definable, since declaring the mapping to be random does not actually specify the exact order of the mapping in a finite set of characters.
Could you though? Wouldn't the randomization not exist, because it isn't definable?
“The number generated by rolling dice 7 times and adding the results”
This is not technically definable since the definition is not unique. But any possible result would still be a real number
So in this analogy, the randomization is like rolling dice infinity many times?
yep. Or perhaps rolling dice a finite amount of times to determine the ordering of the mapping. But going back to what I said earlier: suppose that you wanted to create a decimal number by rolling dice an infinite amount of times. Presumably, the resulting number would be undefinable. But how do we know that it wouldn’t correspond to some finite definition by accident? The same is true for the “undefinable” mapping: what if the mapping is still definable, we just don’t immediately know the definition?
>suppose that you wanted to create a decimal number by rolling dice an infinite amount of times
I thought you were OP and didn't believe in infinity
It’s true that we could get rid of actual infinities and lose nothing of value, but for the sake of shitting on those who still accept infinite sets and Cantor’s proof, I assume their existence.
blah blah blah
I will end this once and for all. Suppose that you create your list of real numbers. And then you define the diagonal number in such a way that it cannot be found anywhere in the list (for example, if the list is in English then we define the diagonal number in Spanish). Since the higher-language diagonal number does not reference itself, its definition is valid, and it outside the list. Now let us create all possible Spanish diagonal numbers by changing the order of the English list. Now we have two lists: English real numbers and the Spanish diagonal numbers formed by the infinite variations of the English list. Then you say, but we can create diagonal numbers based on the Spanish list, or we can combine the English and Spanish list and then create another list in a different language. You can repeat this process to infinity. But what do you get? A countably infinite set of countable infinitely sets of real numbers, which as we all know, is therefore a countable set of real numbers! You could also simply choose to include all possible languages in the original list, at which point it would be obvious that all real numbers are already in the list. So the list would be defined thus: the set of finite strings that, when interpreted according to the rules of that language, uniquely define a real number. And there is no possible way to define a diagonal number since the definition would be in the list, a contradiction.
> Suppose you create your list of real numbers.
Everything else after that is fricking moronic and can be ignored. You can't list the reals. It's literally the whole point of the diagonalization argument. Anywhere that you would try to make your list for the "next real number" there are an infinite number of real numbers in between. That's literally the whole reason they are called uncountable because you can't list them.
But I just showed that they are countable.
The original list is List 1
So the numbers are 1.1, 1.2, 1.3, and so on. The higher-language list is list 2.
So the first diagonal number created by the original list is 2.1 etc.
When you incorporate all the lists as a whole, they are countable.
So your first list goes 1.1, 1.2, 1.3, ... , 1.10, 1.11, 1.12, ... And continues this way?
This is definitely a way to list all of the rational numbers between 1 and 2. The problem is that it will never actually include sqrt(2), as an example. At best you could argue sqrt(2), and all of its rational multiples and elements in the same equivalence class between 1 and 2 would be excluded.
The same will be true for every other irrational number between 1 and 2. So even in your first list, you can't make it countable.
No. The first list is represented by 1.
Each number in the list is what is appended to 1. The 34th element in list 1 is 1.34. The 56th element in list 2 is 2.56. Each list is countable, and there are countably infinite lists, so the total union of the real numbers is also countable. And this is even assuming that the diagonal number’s construction is valid, and that we can’t just create a list that contains all definable numbers such that there is no higher language.
The problem is that the size of the "list" which contains every element starting with 1 isn't countable. So you have a countably infinite number of uncountable lists!
The first list isn't countable unless you are explicitly willing to exclude irrationals from that list. The irrational numbers will never be included in those lists. At best you could say the irrationals would be limit points in that list, but they'll never be included.
Every single mapping is injective. It is countable by virtue of the fact that the naturals are corresponding to some subset of the reals.
> The first list isn't countable unless you are explicitly willing to exclude irrationals from that list.
If the first list is defined in English, then this would include irrationals. “The ratio of a a radius to the circumference of a circle” would be irrational.
> It is countable by virtue of the fact that the naturals are corresponding to some subset of the reals.
Yes, and they are a strict subset. It is not possible to establish a 1-1 correspondence between the naturals and the reals specifically because your list structure cannot include irrationals. There are ways of making countable structures for algebraic numbers (at the cost of arithmetic closure) but you can't make a countable structure which includes the reals. It's literally impossible.
Defining a number in English doesn't give an arithmetic structure to which it can be used as a number.
You are running in circles because you don't wish to admit that you are wrong, and don't understand how set size functions. Uncountability is a genuinely difficult concept to engage with. There's nothing wrong with admitting you don't understand it yet.
> Defining a number in English doesn't give an arithmetic structure to which it can be used as a number.
so? It still refers to a unique number. Math notation is just another language that can be translated to English. The math language itself originated from man-made languages. Are you admitting that real numbers are countable but only in certain languages? Then we agree lmao
If you cannot perform elementary operations on this representation of a number, it definitionally isn't a number. "Number words" are in reference to a number, they are not numbers themselves because there are no arithmetic/elementary operations you can perform on a number word (meaning it isn't a number itself, but only a reference to a number).
Even if we accept the idea that an English representation of a number is itself a number (which it isn't) that doesn't imply that there are countably many of them. In fact, it necessitates that there are uncountably many of them because the alphabet used has an uncountably many number of permutations.
> If you cannot perform elementary operations on this representation of a number, it definitionally isn't a number
Then Cantor’s diagonal number isn’t a number. Thanks for playing!
Congratulations, you have literally discovered the problem with the idea of the reals being countable. The whole reason the reals can't be countable is that you can't enumerate the diagonal in the set S.
The fact that the diagonal cannot be included within your countable set of strings of binaries in S is literally the point of his argument.
So what if we constructed a diagonal number using a 1:1 mapping between the naturals and the rationals? According to you, such a construction is possible, but also, according to you:
> If you cannot perform elementary operations on this representation of a number, it definitionally isn't a number.
So which is it? You can’t have your cake and eat it too
You can perform a diagonalization on the rationals (and by extension the algebraic numbers) you can't on the reals.
The diagonal for the rationals will always exist. It cannot exist in the enumerated sets on the reals. You don't have to take my word for it, try it yourself.
Hold on though. You literally said
> If you cannot perform elementary operations on this representation of a number, it definitionally isn't a number.
So the diagonal number is definitionally not a number.
The diagonal not being included in the enumerated set is literally the contradiction in the "proof by contradiction" for the assumption that the reals can be counted.
Yes, you can't perform anything on the diagonal, but we know it can exist because it can be constructed. If you could perform bit-wise operations on it, then it could be included in the enumerated set. That it cannot be a number included in the set is literally the whole point of the argument.
> Yes, you can't perform anything on the diagonal, but we know it can exist because it can be constructed
So… you take back what you said earlier? You can’t perform operations on its representation and yet it IS a number?
Are you just getting a laugh out of being dense or do you genuinely not understand? If you genuinely don't understand and I'm just doing a poor job of explaining, I can try a different strategy, but I want to know if you actually have a genuine interest in understanding (rather than solely sticking to your guns).
> Even if we accept the idea that an English representation of a number is itself a number (which it isn't) that doesn't imply that there are countably many of them. In fact, it necessitates that there are uncountably many of them because the alphabet used has..
lmao. You can’t catch a break
You should actually look into this more, because this is only true for the set of _finite_ strings of a finite alphabet.
The set of countable strings of a finite alphabet is uncountable. You cannot enumerate all of them. It isn't possible. It is only possible if you force the length of the strings to be finite.
The set of all strings of a finite alphabet whose length is less than N is countable for any finite N. This is not true in the limiting case as you allow N to diverge.
The whole argument is about only including definable numbers which are represented in finite strings. If you need an infinite universe just to DEFINE a number, what use is that number?
> The whole argument is about only including definable numbers which are represented in finite strings.
This may be true if you are only concerned about discrete mathematics. If you are working on systems which you have no reason to believe can be meaningfully discretized without losing information (a.k.a. everything tracking real signals in meat space like voltage, RF, position/velocity/momentum), then you must develop your process to handle continuity.
> If you need an infinite universe just to DEFINE a number, what use is that number.
For you personally, probably not all that useful. There are entire disciplines within computer science and discrete mathematics which never have to deal with the notion of continuity and its implications.
Unfortunately, most of the use of mathematics within science and engineering are not solely focused on computer implementations. They have to interface with analog reality where we have absolutely no reason to believe there is a proper discretization. They have to interact with process uncertainties where we have no reason to believe there's a finite frequency power spectral density.
As a result, you are FORCED to interact with continuity if you want to try and solve these problems. You need to deal with measures of continuous sets. You don't have a choice. It might be easier if you could avoid it, but you can't.
there has never been a single instance in which any problem in computing or science involved undefinable numbers. Zero. Zilch. Nada.
Continuous amplitude random variables and minimizing the distortion when you sample them is literally the backbone of every single communications technology we use today. The whole point is to minimize distortion when you transmit and receive signals transmitted over continuous media with continuous distortion.
Similarly, Newtonian physics on continuous objects underlie the fundamentals of every single infrastructural project, every bridge, every highway, every building on Earth. Using a continuous model of force transmission to identify singular points of maximum strain are literally how modern infrastructural and weather-proofing designs function.
You are so profoundly wrong it can only possibly come from ignorance or a lack of humility (or both).
Literally none of this has anything to do with undefinable numbers. The irrational numbers that we work with are not undefinable.
You’re coping because you contradicted yourself
> Literally none of this has anything to do with undefinable numbers.
Yet it does. It has to do with functions over intervals, which by definition include an uncountable number of undefinable numbers. When you maximize or minimize a continuous function over an interval, almost all of the solutions will not be enumerable with finite digits/bits without distortion under any alphabet which can be translated to binary.
You may not understand it, but every single field which does integration over a measure defined on a continuous interval is including an uncountable number of undefinable numbers in the process.
> You're coping because you contradicted yourself.
Where is the contradiction? Can you explain what the contradiction is?
> You may not understand it, but every single field which does integration over a measure defined on a continuous interval is including an uncountable number of undefinable numbers in the process.
says who? How the frick do you prove this? What exactly would it mean for this to be FALSE? You do know we had calculus before Cantor, right?
> Where is the contradiction? Can you explain what the contradiction is?
You said numbers can’t be defined in English because you can’t perform operations on them, but you allow an exception for diagonal constructions, which rely on English to be constructed.
Integration was first rigorously defined by Riemann before Cantor. Even the Riemann integral was defined by the limiting cases of an infinite partition of the interval under the assumption of continuity (meaning you can keep dividing the interval endlessly and there's no bottom). The whole point of the set theorists was to develop a theory of integration which could account for these circumstances where by geometry we knew an area could be formulated but we couldn't produce it via finite summation without limiting. Even in the original Newtonian formulations of integration they assumed a Cartesian continuity.
> You said numbers can't be defined in English because you can't perform operations on them, but you allow an exception for diagonal constructions.
Here's the Cantor argument for uncountability of the set x in [0,1]:
1) Assume that we can enumerate every possible number by some sequence of 0, 1 coefficients multiplied by powers of (1/2). The endpoint 1 can be specified by all coefficients being 1, and all vice versa for 0.
2) Produce a sequence of s_n where each s_n corresponds to exactly one unique infinite permutation of these 0,1 coefficients.
3) If all of the numbers between [0,1] can be enumerated in this way, then there is some finite N after which we can stop for every single number (as is the case for all rationals). Any number which can be uniquely specified by some finite N number of digits is included in the set and then operations can be performed on them.
4) Take your list of s_n from s_1 onward, and reverse the bit for the nth place. Call this number s. If this s is included in the set, then it is enumerated and we can stop at some finite N and perform bit-wise arithmetic on it.
5) Arrive at the contradiction that there is never a finite N digit where the number s terminates. Thus s cannot be included, meaning we cannot ever perform arithmetic on it in this representation.
6) However it is in between [0,1], contradicting 1).
There is no logical mechanism to add up two real numbers, it is not even computable.
The continuum is sophism, not based in logic, but in words and concepts existing in your head, that's all it is.
> There is no logical mechanism to add up two real numbers, it is not even computable.
What's your point? There are logical mechanisms to approximate the sum of any two real numbers to any precision you'd like (assuming you've described those two real numbers in a way that they themselves can be approximated to any precision we'd like, which is most if not all the real numbers we actually care about).
Why do you need infinite precision? How would you even display the result of such a computation?
All approximations are rational numbers, doing so you're not working with real numbers but rational numbers and in essence two integers.
Rational numbers does not require the continuum. It's true that Newtonian physics requires the continuum, i.e. incomputable and illogical mechanisms, as demonstrated by Zeno.
>All approximations are rational numbers
Correct, but getting those approximations is another process.
For example let's say I want to approximate sqrt(2) + π to the nearest thousandth.
I could first approximate sqrt(2) to the nearest ten thousandth (which is a process of its own based on the definion of square roots), then I could also approximate π to the nearest ten thousandth (which is a different process of it's own based on the definition of π), and then I can add these two rational approximations.
These real numbers contain the rules on how to get those approximations of arbitrary precision, and these rules are less constrained that what you can accomplish with rational numbers.
However, this is just for computable real numbers. I think definable but uncomputable real numbers can still be useful as pathological examples, but I'm fine with getting rid of undefinable numbers. You literally can't tell me a single number that you'll miss.
An AI bot typed that post.
Its impossible for a schizo to type all that ignorant word salad.
I love replies like this. It reminds me that most people (at least on IQfy) don't have the capacity to read more than one sentence or lookup words that have well established technical definitions.
I shouldn’t have even mentioned identifying the elements as 1.1, 1.2, etc. The point is that each list is countable and there are countably infinite lists. What I was trying to say is that instead of 1 mapping to the real number, 2 mapping to the real number, etc. its 1.1 mapping to the first, 1.2 mapping to the second, and so on. The second list maps 2.1, 2.2, etc. to the reals
But your proof relies on the axiom "there are no undefinable numbers", which most Cantor believers do not accept
The real numbers are defined using the closure property, which is finite. So the real numbers are complete using just the definable numbers. The closure property does not state that you must be able to add a random infinite sum of decimals to a real number and produce another real number. But even if it did, there’s no way of guaranteeing that the newly produced number wasn’t already in the list in the first place. And even if you want to ignore the logic of all of this, and assume that there actually could exist undefinable numbers in the platonic realm, the fact is, they are completely useless to us and only spawn problems that are a waste of time, like the Continuum hypothesis.
I have no idea what you mean by "closure property" in this context. The reals are defined such that they have the completeness property (Cauchy or Dedekind) which absolutely does state that you can stick infinitely many extra decimal places at the end of a terminating decimal and get a new real number. Yes, this means real numbers can have an infinite amount of information in them. Yes, this is mathematically necessary and useful because completeness is that important. No, R is not countable.
> Yes, this means real numbers can have an infinite amount of information in them.
Prove that there exists a real number that cannot be uniquely represented by a finite definition. For example, pi has infinite information, but it can be defined in various ways in a finite amount of characters. If every real number is definable, then the argument still holds.
But even if a real number cannot be reduced to a finite representation, then it is useless as you would need an infinite universe just to express it, let alone try to interact with it.
>I actually think the mapping would be definable, but even if it weren’t, there would still exist a 1:1 mapping.
So you're ok with assuming the existence of mathematical entities without a definition.
only to prove that infinitards are moronic, yes. Even after you grant their assumptions, they find a way to churn out more bullshit
Where's your response to
Is does s have a value between [0,1] even though we cannot properly enumerate it? If it doesn't have a value between [0,1], why not?
English isn’t a formal language. Sorry sweaty, but nothing you said made any sense
I accept your concession.
>If every number in the list is definable,
Do you mean "If every definable number is on the list"?
mh, i rate your acrobatics a 3.2 out of 20, i've certainly seen better, but i did like how you outright dodged the fact that you now have declared to hold that the machine "H" exist and so it is possible to determine if any program halts
Proof by contradiction means one of the assumptions is false.
The incorrect assumption is that infinite sets exist.
ah, argumentum ad "nuh-uh", classic finitist move
If we do not have infinite sets, then we cannot have infinite tape.
If we do not have infinite tape, yes H exists.
For any program state of N bytes, there is an enumeration of size 2^N that lists every possible state and whether or not the machine will halt.
>You can only attain an undefinable number from a definable number with an infinite, undefinable sum.
No, sqrt(2).
>problem with Cantor’s proof is that the diagonal number is undefinable
No, you just apply Pythagoras thereom to the base unit to find the unit square's diagonal equals sqrt(2) which is irrational because diagonals can not be fully expressed in terms of the original square unit.
sqrt(2) is most definitely definable.
That fact that you wrote it down and I knew what number you're talking about kinda suggests that there's a definition...
Note that the set of natural numbers has the same cardinality as the set of algebraic numbers (which include all the roots).
But people seem to misunderstand Cantor's result.
First of all, it doesn't use the assumption of undefinable numbers, it simply uses the assumption that every set has a power set (one of the ZFC axioms).
Second, the existence of undefinable numbers is the result itself! The fact that any definition in any sensible language (formal or natural) needs to use finitely many symbols means that the set of all definitions has at most the same cardinality as the set of naturals. So we have that the cardinality of definitions = cardinality of naturals < cardinality of reals, so that means that there are real numbers without definitions.
> it simply uses the assumption that every set has a power set (one of the ZFC axioms).
infinite power sets are full of infinite undefinable sets. They are meaningless
> infinite power sets are full of infinite undefinable sets.
Correct, as mathematically and validely proven by Cantor.
> They are meaningless.
I agree, but this is ultimately a philosophical position outside the scope of math. Look up predicativism.
i dissagre
thank god for making finitist's as a joke
>Zeno.
>in anno domino's 2024
come on now
>First of all, it doesn't use the assumption of undefinable numbers, it simply uses the assumption that every set has a power set (one of the ZFC axioms).
Common misconception. Without some form of Separation, Power Set is not strong enough to prove Cantor's theorem.
>You can’t falsify cantor’s proof in the real world
Mathematical proofs and scientific proofs are not the same things and don't have the same burden.
A mathematical proof only means that you can formally derive a result from the presupposed axioms and inference rules.
Even if the assumed axioms or inference rules are not "true", the proof is considered valid and the result is considered true within that formal system.
The point of a mathematical proof is not to know what's objectively true in this reality, it's to explore the consequences of the logical system in question.
Whether that logical system system reflects reality or not is a question outside the scope of math. It's up to philosophy and/or science to decide that.
However, note that the words "bigger infinities" don't appear in any formal system that I know of. "Bigger infinities" is someone trying to give meaning to a formal result. The MEANING of a formal result is 100% philosophically debatable.
There's a fundamental difference between the cardinality of the natural numbers and the cardinality of the real numbers as shown by Cantor's proof. But in my opinion, calling one infinity bigger than the other is just people failing at using old words to describe a new concept that was never thought of before (which happens a lot in math).
>You can only attain an undefinable number from a definable number with an infinite, undefinable sum.
This is incorrect. Your weakness is that you are limiting yourself to DIGITS as representing numbers. Use a number line and the 'undefinable' number is clearly identified.
Of course, the reason you are still somewhat correct - even though you don't realize it - is because within the natural world infinite concepts can NEVER exist, which is why we cannot represent them accurately. Math and reality ARE equal - math is NOT an abstraction. Math *IS* the definition of reality. Math is all there is, not the other way around.
"undefinable' numbers are unreal because math and reality are equivalent, not because we cannot represent them.
I never liked Cantor's diagonal proof. In particular, the process of creation always assumes numbers with same digits always have next digit to differ by (for the diagonal number construction), particularly with repeated construction.
I.e.
...3456 -> ...34560
...3456 -> ...3456(1)
At finite sizes it's a reasonable assumption.
Here's attempt of a counter-proof.
Let's consider a simplified Cantor (Bant-or). Form a set consisting of 0, then continue creating numbers as follows: for each number, compare digits and append a 9 at the end of the number's representation. This will ensure the number will differ from each by at least "a" digit, which is the cornerstone of Cantor's proof.
Initial numbers would therefore be
1- 0
2- 0.9
3- 0.99
4- 0.999 and so on.
Mind you, they will all essentially be of form 1-(1/10)^n and <=1. Let us proceed with this construction.
The final entry of a filled infinite-sized set would be
aleph-0 - 1 (a number whose every enumerable digit is 9 =0.(9)=1)
Attempting creation of another number fails, as it attempts to dig into a digit number above aleph-0 through a successor. Therefore the construction fails. Therefore you cannot necessarily create a new member of a set by creating a number which differs from all other members of this set by a digit.
(aleph-0)-1=aleph-0
and you are using cardinals wrong
how much are you paid to constantly make these moronic threads?
I got mad because I lost a chess game online so I thought I would create this thread to “win” at a different game. So far I’m quite pleased
>Is that why this board is called Science AND Math? Because modern math isn’t actually scientific?
I mean if your definition of scientific is "something deduced from performing experiments in the real world" then yeah it isn't
The fundamental problem with Cantorians is that they assume that the faulty premise in the diagonal proof is that the list can be made, instead of the premise that the diagonal number can be constructed. We assume that every real number in the list, right? So it’s possible that the diagonal number is impossible to construct, as it implicitly uses self-reference to define a number different than itself, a contradiction
you are not owed logical congruity
like damn its obvious some are bigger than others maybe you shouldnt be on here if youre under 18
QED
Even if you assume the existence of infinite sets, it makes more sense to compare them by density, not the total amount. The rationals are more dense then the naturals, the reals are more dense than the rationals. But they’re all countable
they STILL can't find the well ordering of the real.
they just don't care about rigour, only tradition, and social clout.
What if we just got rid of powerset axiom? Also excluded middle.
i do math myself but math is not a science
it just is a "system" to better understand our physical world and something that we as humans "constructed"
in my opinion is the saying, that math is the language of the universe, is complete bullshit
why is math infected with unfalsifiable nonsense?
Because we are still have monkey brains and fish eyes.
Of course its nonsense. Quite a few people are aware of that, but they are a minority. The rest are just parrots who defend it vociferously because anything that challenges their "set in stone" world view upsets and frightens them.
The thing is though we dont have anything better to replace it with. Like being stuck with Newtonian physics while being aware that somewhere out there is the theory of relativity, but lacking an Einstein to discover it.
What we have works for our calculations, close enough. For many people close enough is good enough. That's forgivable in applied sciences, but for a discipline like Mathematics which is supposedly absolute, it is unforgivable.
What we have at the moment are people tying themselves in knots trying to deal with the inconsistencies that arise from current mathematics. Instead they should be focused on why these these inconsistencies occur in the first place and why the principles of mathematics are fundamentally flawed.
Okay that's my piece OP. Now watch as the parrots begin squawking.
>being stuck with Newtonian physics
You mean classical physics which are still overwhelmingly more relevant to the building and maintenance of civilizaiton than relativity?
Classical physics are really not all that useful beyond very simple kinematics interactions.
Most practical physics is either statistical (electric charge propagation, thermodynamics, combustion and energy exchanges) or chaotic. Neither of these require the extremes of generalized relativity for massive objects traveling at absurd speeds or quantum mechanics, but still are more sophisticated than Newtonian kinematics/dynamics.
Hear that whooshing sound above your head?
This so much. One of the few intelligent posts so far. The thing I cant get over over is the insane arrogance of many people today who think we have got it all figured out. In a thousand years people are going to look back on these times and view them as barely distinguishable from the Middle ages.
I don't think anyone who actually works in mathematics believes that we have it all figured out.
One thing we have figured out is that there are more numbers between (0,1) than can be put in a countable list. How to properly describe the reals and working on uncountable sets is very difficult.
>infinite sets
yes, and?
>some infinities are bigger than other infinities
you don't understand what "bigger" means
>undefinable real numbers are real
The are well-defined. Look up Cauchy construction.
>Because modern math isn’t actually scientific?
Yes, math is a humanities not a science.
Cauchy sequences can be defined for actual real numbers though. Not the case with undefinable numbers (if they even “exist”). I’ve yet to see proof that there exists some infinite decimal number that does not have some sort of finite definition, whether that be an infinite sum or Cauchy sequence or solution to some equation etc. All of the numbers that we work with and that we have encountered have finite definitions, and therefore belong to a countable set.
Look up Chaitin constants
I'm a mathematician and I can do whatever the frick I want and only god can stop me.
Frick you and your homosexual empiricism.
Math was never science, never will be, and that's a good thing.
Frick science, hail Math
if this board allowed only sub 75IQ people to post this would be the sticky
Take the square root of two.
1.41421356237....
The decimal expansion is infinite. It never ends, it never repeats.
Now think about what this means. We have taken two fundamental mathematical concepts. A length of one unit and the square. But when we attempt to calculate the distance across the diagonal we find it is undetermined. We can never precisely calculate its value.
Imagine if you will starting to measure from one point of the diagonal to the other. You begin but as you approach the other end you find it gets always closer but never quite reaches the end. And yet it must do, for we can extend the diagonal line beyond the boundaries of the square.
The question I find appealing is this: Why? Why is a calculation arising from two fundamental mathematical constructs undetermined?
Yet if we take the same fundamental unit of length 1 and multiply it to construct a square with sides of 3 and 4 we find the diagonal is absolutely precise. Precisely 5. No more, no less. How interesting.
With respect to indeterminable values the same applies to circles. Again a fundamental mathematical construct. Doesn't really get much simpler than that. But it also doesn't matter what we do, the value of pi can never be precisely calculated, its exact value vanishes off into infinity. Now why is that? I find it an interesting question.
Sure, we can slap on a bandaid, the concept of convergence, and walk away whistling a merry tune. Well okay, that works for applications, but it does not answer the original question as to why such such indeterminacy arises in the first place.
>I find it an interesting question.
no you do not, for you are a finitist, which is the the mathematical branch of zeteticism
>The decimal expansion is infinite. It never ends, it never repeats.
maybe but decimal fractions are gay. the continued fraction expansion repeats and is extremely simple, and not gay.
sqrt(2) = [1;2,2,2,…] .
>But when we attempt to calculate the distance across the diagonal we find it is undetermined. We can never precisely calculate its value.
What do you mean by "calculate the value" of a number? How would you "calculate the value" of the number 1? The exact value of 1 is 1 just like the exact value of sqrt(2) is sqrt(2).
You seem perplexed that sqrt(2) doesn't have a representation when using decimal notation. It's not that we can't calculate it exactly, it's just that sqrt(2) is not a whole number multiple of any negative whole power of 10, so of course trying to express it in decimal notation is going to fail.
But that's just means that decimal notation has problems with sqrt(2), it's not sqrt(2) itself that's problematic. For example root notation works perfectly with sqrt(2).
I really don't understand people who accept rational numbers but not algebraic numbers. I guess it's because real numbers are usually presented as the next "level" after rational, but imo the real problematic jump is from definable numbers (still same cardinality as the naturals) to real numbers (meaningless cardinality)
>why is math infected with unfalsifiable nonsense?
Because it has utility; it's USEFUL even if it has no meaningful relation to truthfulness or falsifiability
There could be some profound truths about the Universe found in some purely abstract field of study, but it's unlikely that you'd see a mass of people rushing to study and fund it, unless it was expected to have some utility
STEM advances based on finding useful things (even if they involve unrigorous mental acrobatics); not based on some Noble Pursuit of the Truth.
Somehow I was taught this by an anarchist on a podcast while on a heavy dose of psilocybin mushrooms at 4am rather than in any of the Research or Ethics classes I took in University
(I havent read the rest of the discussion)
its only utility is entertaining math nerds and making them feel special
While I'm willing to accept that; I personally can't think of a way to know ahead of time whether a contribution from abstract math nerds will have significant utility for humans. Sometimes unexpected connections are found between seemingly unrelated fields
Anyway maybe we don't even have to talk about Infinities. Wtf is Zero? It has utility even if it's purely abstract
Zero is clearly useful and can be represented in the real world. I have one apple, I give it to you, now I have zero. You can’t represent infinite sets in the real world. At best you can only have potential infinities. All of our math works without the assumption that actual infinities exist, or that there are more real numbers than natural numbers, etc.
If you wanted to verify that the proof was true in C++, how would you do it? What program would you write and what would it say?
A mathematical proof isn't "true" or "false", it's either valid or not.
As long as you only start with with the allowed axioms and only use the allowed inference rules, your proof is valid.
First you need to chose which formal system (formal language, axioms, and formal logic) you want to work with and then you can implement it in C++.
For example, if you choose first order logic as your formal logic, then you'll need to implement some code that checks if an argument follows the following inference rules properly: https://en.wikipedia.org/wiki/List_of_rules_of_inference
For example, this function checks if an inference is a valid application of modus ponens:
bool isValidModusPonens(string antecedent, string conditional, string consequent):
return conditional == "(" + antecedent + ")(" + consequent + ")"
The other rules from propositional logic are as braindead easy, but the ones for the existential and universal quantifiers can be trickier.
Then you would need to translate Cantor's English/German proof into your system's formal language. This is probably the hardest part.
Finally, you run a program to check that all the steps in the proof follow the system's inference rules properly.
For example, if in the formal version of Cantor's proof it says that line 123 comes from modus ponens on line 3 and line 58, you just run the isValidModusPonens function with lines 3, 58, and 123 as arguments.
Output could be a message that says that all the lines of the proof were properly verified or a message indicating which line doesn't follow the rule it claims to follow.
Before tackling Cantor's proof or writing your own proof checker though, maybe try formally proving that 2+2=4 using the Peano axioms and first order logic.
what could be code to see if it is true "with" certain axioms?
An axiom is just a statement you can write down with no justification.
Just hardcode them in a list somewhere or pass them to your program as launch parameters.
I'll do 2+2=4 because I feel you don't understand what a formal proof is.
I'll be using first order logic with equality and the following formal formulation of some Peano axioms:
a) 1=S(0) ~ definition of 1
b) 2=S(1) ~ definition of 2
c) 3=S(2) ~ definition of 3
d) 4=S(3) ~ definition of 4
e) ∀n, n+0=n ~ addition axiom 1
f) ∀m, ∀n, m+S(n)=S(m+n) ~ addition axiom 2
Proof begins here:
1) 2+0=2 ~ Universal instantiation on axiom e) with n = 2.
2) ∀n, 2+S(n)=S(2+n) ~ Universal instantiation on axiom f) with m = 2.
3) 2+S(0)=S(2+0) ~ Universal instantiation on line 2) with n = 0.
4) 2+1=S(2+0) ~ Equality substitution, replace "S(0)" by "1" in line 3). Valid because of axiom a).
5) 2+1=S(2) ~ Equality substitution, replace "2+0" by "2" in line 4). Valid because of line 1.
6) 2+1=3 ~ Equality substitution, replace "S(2)" by "3" in line 5). Valid because of axiom c).
7) ∀n, 2+S(n)=S(2+n) ~ Universal instantiation on axiom f) with m = 2.
8) 2+S(1)=S(2+1) ~ Universal instantiation on line 7) with n = 1.
9) 2+2=S(2+1) ~ Equality substitution, replace "S(1)" by "2" in line 8). Valid because of axiom b).
10) 2+2=S(3) ~ Equality substitution, replace "2+1" by "3" in line 9). Valid because of line 6.
11) 2+2=4 ~ Equality substitution, replace "S(3)" by "4" in line 10). Valid because of axiom d).
For a proof checker, you'd need to check if universal instantiation and equality substitution have been applied properly (basic string manipulations and comparisons).
Those are the only two inference rules I needed for this proof.
Then just write code that double checks each line one by one.
The proof goes into detail so that it's easy for the proof checker.
All this just for 2+2=4. So you can imagine a formal version of Cantor's proof is going to be much longer.
Can you just paste the code for the original proof? How many lines would it be
What do you mean by the code for the original proof? The original proof is in German and is about 3 pages.
You can find it by looking up the name of the article: Ueber eine elementare Frage der Mannigfaltigkeitslehre
Feel free to translate it into any formal system of your liking. And then feel free to implement a proof checker for that formal system.
I really feel you don't understand formal systems, so I feel it would be wasted effort if I did it for you.
If you give me a formal proof that 2+3 ≠ 6, it will show me that you genuinely care and understand enough for it to be worth it.
I'll then go ahead and figure out and give you a formal version of Cantor's proof (but it might take a few days depending on how busy I am).
I mean the original proof in the OP
Oh, the original proof makes no formal sense.
Also, there's no formal definition of "definable real number": https://en.wikipedia.org/wiki/Definable_real_number
We can informally work around that though.
Let D = the set of definable real numbers.
If x is in D, then it must have at least one definition. Out of all its possible definitions it must have first definition after sorting by length and then alphabetically.
This shortest definition is a finite sequence of symbols. It can't be an infinite sequence otherwise we can't read it.
ex: sqrt(2) can be defined as the least upper bound of the set {x: x^2 < 2}. So even though the decimal expansion of sqrt(2) is infinite, the definition of sqrt(2) is very much finite.
Also, if x and y are two distinct definable real numbers, then they can't share the same definition.
So there "must" be an injective mapping from D to the set of finite sequences of symbols.
The existence of D and this property of D are the only non-formal results I'll use. If you have problems with it, please formally define what you mean by the set of definable real numbers.
Let f(x) be any injection from D to the set of finite sequences of symbols.
There are bijective mappings between the set of finite sequences of symbols and ℕ.
The exact mapping will depend on your exact set of allowed symbols. I can go into this in more detail if you'd like.
This can be proven formally.
Let g(x) be any bijection from the set of finite sequences of symbols to ℕ
Cantor proved that an injective mapping from R to ℕ is impossible.
This can be proven formally.
It can also be proven formally that the composition of two injective mappings is also injective.
Now we'll do a proof by contradiction with the assumption that D = R:
If D = R, then the identity function, id(x), is a bijection from R to D.
Consider g(f(id(x*~~. That's an injection from R to ℕ. Contradiction.
So D ≠ R.
So you admit that Cantor’s proof works only when you assume the existence of undefinable numbers. Then we agree. I would just say that undefinable numbers are useless and therefore so is Cantor’s theorem.
Cantor’s proof works only when you have the set of all real numbers. (how would a proof involving the set of real numbers work without the set of real numbers existing?)
You can get the set of real numbers from the ZF axioms without assuming the existence of the set of real number explicitly.
The proof I did on top of Cantor's proof shows that not all real numbers are definable. That's the result not the assumption. The existence of undefinable numbers was never assumed. Only the existence of the set of real numbers was assumed/derived without making any assumptions about if they're all definable or not.
I'd argue you find this result useful because now you can use it as an argument as to why ZF is shit and why we shouldn't use it. In particular, why the power set axiom is shit and we shouldn't use it.
>no formal sense.
>no formal definition
..what does that mean?
> no formal sense
I meant I personally can't formalize that argument. The implied inferences don't follow imo. But I can try to read between to lines and try to guess guess what OP meant/thought and add the missing details.
> no formal definition
No standard definition in any formal language.
"Undefinable real numbers are real" is as mathematically rigorous as "circular number are fun".
>But I can try to read between to lines and try to guess guess what OP meant/thought and add the missing details.
Oh. I meant the proof as in something that proves infinite sets or Cantor's diagonalization argument itself.
What could code in C++ be that verifies either of those proofs are true?
>proves infinite sets
Not this but uncountability of real numbers
Write c++ code that verifies that my proof of 2+2=4 is correct (the one in
).
Write a formal proof that 2+3 ≠ 6.
After that I'll help you out with set theory stuff.
If you can't grasp formalism for arithmetic, I don't think you should bother with formalism for set theory.
Also, just to be clear, unless you're working in proof theory or logic, people very rarely care about formal proofs.
Practically all mathematicians just write their proofs in their native language/the university's language but in a way that other mathematicians would be able to formalize the proofs if push came to shove. But that very rarely happens.
Also, proof checkers and formal proofs are rarely done in C++.
Most formal proofs use other languages specifically designed for proof checking.
There's this list of 100 theorems that the different proof checking languages use as "benchmarks".
#63 is Cantor's theorem (no power set is equinumerous with the base set).
Here's the proof in metamath: https://us.metamath.org/mpeuni/canth2.html
Why not just post the code
not him, but i'd assume it's because you can open another tab
>still doesn't paste the code
Infinity is not the same of an eternal number, infinity is it branch wise. Infinity is the branch of the count of sets.
What's the cardinality of a set of 2s, what is the attribute that makes setting possible? That is infinity.
you can't idiot, the halting problem is undecidable
Math was better when it was just used by merchants counting their cattles. "scientists" ruined it. This is why gatekeeping is important.
Damned straight.
It was an arbitrary means of making sure no one ripped you off. But then some raving homosexuals just had to go an screw everything up.
>OH! What if we have a "NUMBER LINE!"???
>Oh OH, Oh I love it when you talk dirty! Why yes, that's fine with me! Now keep fingering my anus!
>OH! Ohhhh! And is it fine if we have these made up things called "irrational numbers!???"
>"Ohhhhhhh! Oh yes that's as fine as your dick in my mouth! OH!"
>"OH GOD yes! Look, and is it FINE to have infinities too b***h?!!?"
>"OMG! I'M CUMMING! YES! YES! That's FINE with me too! Ohhhhhhh!"
Goddamn homosexuals.
>unfalsifiable
Frick off with that popper bullshit
Thoughts about God?