Relatively computable

[Anonymous No.15139260]

Just to be clear: we are all in agreement that even if (You) have
>muh "definition of computability"
we can still *informally compute* with your noncomputability proofs that specific real numbers are noncomputable, using diagonalization arguments, transfinite recursion, AC, &c.
in other words, they're only "noncomputable" from the point of view of your primitive CS brains bound to weak and feeble finite bodies, while us mathchads send thunderbolts down like Zeus

[Anonymous No.15139580]

This trick relies on the fact that there are parts of math that in theory have not yet been formalized, even if it appears that this is somehow not the case and "there is nothing to discover" we can still recall artificial formalization boundaries as definitions of computability, retaining "artificial lack of formalization" as a quality
however, that possibility is quite remote as there is plenty of real lack of formalization when people try to play games where they are allowed to continuously redefine their favorite words to mean whatever they want
this only stops when someone develops an "abstract theory of definitions of computable" and says what abstract rules a definition of computable and a proof of non-computability should follow
now there is a standard language for computing with "non-computable quantities" on the basis of qualities of the _proofs_ that the quantities are not computable
so it is a matter of axioms and logic outright and a meta-mathematical concern, not part of "ordinary math" per se

[Anonymous No.15141162]

>>15139260

[Anonymous No.15141276]

>>15139260
Do you have a question or do you want to make a point? I don't know which of the two it is.
If you have a question, state the question.

[Anonymous No.15141306]

This is a follow up on transcendental numbers I take it.
The exact value of a real number is usually not required, numbers like pi are extremely rare.

Also you are imagining yourself as sending down a thunderbolt, you don't actually have the power to send down a real thunderbolt.

[Anonymous No.15141366]

>>15139260
Isn't computability only defined in terms of a turing machine? All you need is another model of computation.

[Anonymous No.15141382]

>>15139260
meds
now

[Anonymous No.15141433]

>>15139260
when you do forcing you realize that there is no way to distinguish between the situation that the universe of sets your talking about actually has uncountable sets, from the one where the whole universe is itself countable and all bijections between objets and substets of your N are in fact outside the afforementioned universe or said othewise, that you are unable to actually provide them (and this by assuming that universes do exist at all, which you will never be sure because of Gödel 2nd incompleteness theorem). See also "generic extensions" (how they are built) and the "reflection principle" from Montague.
You should realize that from the point of view of a finistic creature whose horizon is finite, all infinite objets have the same size (you'll never probe more than a finite amount of objects in your finistic life).

[Anonymous No.15141440]

These are the parts of math we don't want to formalize. Noncomputability is closedly linked to free will. Free will is also noncomputable. We wouldn't want to eliminate free will by formalizing it, by demoting it into the realm of algorithmic math.

[Anonymous No.15141577]

>>15139260
Has nobody here done any actual computability theory? Am I the only person on /sci/ who has done oracle constructions, definable forcings, and priority arguments. A set of natural numbers and equivalently a real number being non computable is not that mysterious.

[Anonymous No.15141653]

>>15141433
Suddenly...CHUANG TZU
>Section THREE - THE SECRET OF CARING FOR LIFE.1
>YOUR LIFE HAS A LIMIT but knowledge has none. If you use what is limited to pursue what has no limit, you will be in danger. If you understand this and still strive for knowledge, you will be in danger for certain! If you do good, stay away from fame. If you do evil, stay away from punishments. Follow the middle; go by what is constant, and you can stay in one piece, keep yourself alive, look after your parents, and live out your years.
https://terebess.hu/english/chuangtzu.html

[Anonymous No.15141669]

>>15141577
Hello Sirs!
Let me interest you in a variety of computer software solutions for tracking and batch applying definitions of computability to sentences of first-order logic!
Student edition only $19.95! Operators are standing by! Basic enterprise software seats available starting at $4,995! Fully loaded corporate licensed version $249,995 per year, up to 500 seats, includes 24/7 technical support.

[Anonymous No.15141727]

>>15139260
A-B-C-D-E-F-G-H-I-J-K-L-M-N-O-P-Q-R-S-T-U-V-W-X-Y-Z

But why are they in that order tho...

Maybe we should keep all the transcendentals in one place since they're so weird.

[Anonymous No.15141736]

>>15141306
>The exact value of a real number is usually not required, numbers like pi are extremely rare.
Non-computable numbers are those who's digits you can't compute. Pi isn't such a number.

The prime example of a non-computable real is "0." followed by the binary digits on n'th position defined to be 1 if the n'th Turing machine eventually halts.
This is provably a real number in the unit interval which provably has some digit position which is classically either 0 or 1 but whos value there can't be obtained

[Anonymous No.15141801]

>>15141736
So we can write down
> (*) a Turing machine that halts iff ZFC entails a contradiction in FOL
That means one of those digits encodes whether or not ZFC entails a contradiction.
So what you're saying suggests that there are real numbers that are "too fine for ZFC" in the sense that ZFC can't detect them or rule out their existence because ZFC can't provably determine the value of the n'th digit, where n is chosen to be the number of the Turing machine (*).
So you can "compute" or "calculate" with this number in a certain primitive and highly abstract sense, but you can't actually evaluate the value of its binary digits and be satisfied that ZFC entails a proof that you evaluated it correctly. You need some other theory to support a proven finding of the value of the binary digits.

[Anonymous No.15141840]

>>15141801
Sorry, ZFC has no problem with existence and uniqueness of the binary digits themselves, it's just that no ZFC proof could can tell you the correct explicit value of the n'th digit. You need a calculator that follows some other theory in order to get a proof of an explicit value for the n'th digit.
The ZFC calculator can still "calculate" with this number in an abstract sense as an unknown quantity.

[Anonymous No.15142084]

>>15141801
>>15141840
I suppose yes to what you say, but I'd probably not phase this in terms of ZFC. Not in set theory, and certainly not in set theory with choice. If you got a set theory with choice, the choice functions in general can't be evaluated either.
You'll have to be more concrete about your talking about "too fine". Most reals in ZFC will be of the uncomputable sort in the above sense, because in fact - worse - by the "tea table argument", most numbers in ZFC are not even definable, i.e. "write-downable".
https://en.wikipedia.org/wiki/Definable_real_number

Aside: The digits of the sequence I stated can't be evaluated already in an arithmetic. If the Church-Turing thesis holds, and we're interested in computability, why import that above ballast.
Let I in the naturals be an index, S in {2, 3} be a sign (2 for minus and 3 for plus), N in the naturals be the numerator and D in the naturals without 0 be the enumerator. Then you can code indexed rationals as naturals e.g. via
2^I · 3^S · 5^N · 7^D
(idx=80, rational=-2/11) = 2^80 · 3^2 · 5^2 · 7^11 \approx e^80
By unique prime decomposition, you can find the index and the rational.
In this way you can define Cauchy sequences in the rational (representatives of Cauchy reals) as subsets of the natural numbers. The second order arithmetic quantifier
[math]\forall S\subset {\mathbb N}.[/math]
thus lets you formulate theorems of the reals.

If ya interested, I took my example directly from a collection of results in computability theory or computable analysis. In particular, if you telly ourself you're just going to restrict your mathematics to computable reals, then it turns out you're forced to give up completeness
https://en.wikipedia.org/wiki/Specker_sequence

>You need a calculator that follows some other theory in order to get a proof of an explicit value for the n'th digit.
The other theory will have the same problem with any of its own enumeration of Turing machines and the corresponding number.

[Anonymous No.15142157]

>>15139580
Well you dont even have a real definition of set theory as it is pending on the intuition that we have of what a set should be and what properties it should have. Also we dont even have a definition of "definition" or "to define something" and its not hard to reach a contradiction supossing that there exist such thing as a definition of "defining" and to avoid those contradictions you have to exclude properties that nobody is willing to take out and at the same time call that thing a definition. Its like you saying "its a natural number if and only if..." and then puking a finite list of axioms, its either calling some aleph natural numbers, saying that they should not have a property that they clearly should have or accepting that formalism has its boundaries.

[Anonymous No.15142161]

>>15141366
you are retarded. try give me a model of computation that can decide a family of languages strictly larger than decidable languages (the languages a turing machine can decide)XP JD

[Jealous No.15142168]

I agree.

[Anonymous No.15142550]

Yes, it is true that some real numbers are considered to be non-computable using traditional definitions of computability, such as the Turing machine model. However, it is also true that these numbers can still be "computed" informally using methods such as diagonalization and transfinite recursion. It is important to note that these methods rely on assumptions and definitions that may not be formalized or may not be considered standard in the field of computation. Additionally, it is also important to note that while these methods can be used to "compute" non-computable numbers, they may not always yield meaningful or useful results. The field of computation is constantly evolving, and new developments in logic, axioms, and formalization may change the current understanding of computability and non-computability.

[Anonymous No.15142565]

>>15142550
Did the OpenAI ChatGPT write this?
So a computer is saying
>"I can define computability to be whatever I want it to be."

[Anonymous No.15142599]

>>15141736
What are those uncomputable real numbers useful for ? What you're describing can be described as [math] \frac{1}{10^n} [/math] where n is the position on which the turing machine halts, but that's not even a real number it's always going to be fixed length.
Something like [math] \sum_{k=n}^{\infty} \frac{1}{10^k} [/math] will actually be a real, also noncomputable.

[Anonymous No.15142600]

>>15139580
Higher than the average Ivy league grad, but still not very impressive.

[Anonymous No.15142603]

>>15142157
Not bad at all.

[Anonymous No.15142608]

>>15142168
>neighborhood of infinity
>"real" number

[Anonymous No.15142609]

>>15142600
>>15142552
that thread did a /his/ on me and I'm kinda annoyed
I mean, i've been on the
>History of IQ is about using numbers to rationalize lynching
bandwagon for a long time, so the sordid anti-historical condition of /sci/ is *well established* in my mind
and I fuckin' HATE niggers
but you won't admit that IQ is about killing 'em in this really nasty way
I have mental conflict about this subject...seriously /his/ need to focus on the fact that the history of IQ is all about killing niggers and somebody needs to make hella bank on this
get the word OUT muttafukka
so exasperating

[Anonymous No.15142611]

>>15142550

[Anonymous No.15142613]

>>15142609

[Anonymous No.15142617]

>>15142084

[Anonymous No.15142619]

>>15142600
>>15142603
That thread is all about how that website says that demand generation ad copy has genius IQ.
That website is financially invested in an anti-historical approach to IQ.
That website will say that I'm an idiot for bringing up the historical realities behind IQ.
I do not like that website.

[Anonymous No.15142621]

>>15142611
>>15142613
>>15142617
>"I will give you high numbers if you get people to use me."
oh boy...
isn't there some ST:TNG where some hot babe hijacks the Enterprise this way??

[Anonymous No.15142620]

>>15142619
seethe midwit

[Anonymous No.15142625]

>>15142620
I hate niggers so goddamn much...I would fucking lick the toes of the first nigga to write a paragraph of a book slayin' anti-historical muttafukkahs like yourself...you realize that we have genius superpredator niggers IN MUTTAFUKKAN CONGRESS NO LESS all because the USA went APESHIT over IQ tests and now the niggers are all SMART see
HOW FUCKING DUMB WUZ DAT, HUH?!?!

[Anonymous No.15142661]

>>15141840
No ZFC proof has ever lead to anything anyone would identify as a real number nor proven their existence. There is no such thing as a ZFC calculator. Rather, the deceitful game played by analysts is that they pretend they can do things that they never could like claim they wrote down of some arbitrary binary sequence they call a "number" and then pretend they can do arithmetic with it when at best any "example" of such a thing is them writing down a finite sequence of 1's and 0's, then writing "..." without explicitly defining what "..." means other than there are allegedly some succeeding digits. As it is, without the specification of the algorithm used to generate the digits in the number, there is no reason to even suppose that there are any following digits. If they are chosen, then the last digit written down is the last digit in the sequence. If they are generated by an algorithm, what's the algorithm? Yet this practice of writing down a finite list of finite lists and writing an ellipsis at the end of each element of the list and writing a vertical ellipsis below the last element of the list somehow, without any reasonable explanation, denotes that this is now an INFINITE LIST OF INFINITE LISTS. And then they all pretend somehow that they each can perform an infinite number of actions all at once (in fact, it's prescribed as an "axiom") when no reasonable person would ever pretend so. This is absolute nonsense! This is PSEUDOSCIENCE! Yet this nonsense is called the "foundation" of modern mathematics. What a ludicrous joke.

[Anonymous No.15142678]

>>15142621
I'm already horny enough Mr. Spock. Don't make it worse.

[Anonymous No.15142684]

>>15139260
I have no clue what you're trying to say. Noncomputable numbers aren't able to be computed regardless of whether you're a mathchad or CScel. Try to compute Chaitin's Constant and get back to me

[Anonymous No.15142685]

>>15142661

[Anonymous No.15142689]

>>15142661
we went over this already in the other thread
it's because students like jokes and nonsense
by the way, have you read pic related
and modern math is about unifying the foundations of education philosophy
https://archive.org/details/AliceInWonderland1951

[Anonymous No.15142692]

>>15142685
he didn't even conjugate the first verb he used properly

[Anonymous No.15143921]

>>15142617
>>15142685
Those help little. Not everybody here is an English native, if you only speak Spanish for work and on the internet, your score here would also be pretty bad.

[Anonymous No.15143937]

>>15141669
Hello sir, good day sir!
Does the fully loaded version of the offer licensing support deployment on proprietary application sarvar, or is only cloud platform integration supported?

Yours truly,
IIT certified architect engineer, Prashad Vandraprasan on the behalf of FAGMAN, Inc.

[Anonymous No.15144756]

>OP: We all agree that computers are allowed to redefine computability whenever they feel like it.
meds
now

[Anonymous No.15144766]

>>15139260
Polite and respectful English use is the only thing holding you back, 4chan. If you're so hell-bent on always being some edge-cases and never the roots or points of unity then *brain is now fumar*