sci/15349364.txt

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--- 15349364
Uhm, real number bros? I just figured something out and it's not looking good for us.

Every proof is a finite string over a finite alphabet. Therefore, the set of all proofs is countable. There are uncountably many real numbers though. That means we cannot have a proof of existence for each real number.
--- 15349478
>>15349364 (OP)
>Every proof is a finite string over a finite alphabet. Therefore, the set of all proofs is countable.

Cool. Come up with a way to enumerate every proof, ie come up with a sequence or some other bijection between your list of proofs and the natural numbers.
--- 15349482
>>15349364 (OP)
the set of all theorems of an axiomatic system with real numbers is uncountable..
--- 15349823
show me one, literally one, meaningful mathematical paper that doesn't need to use real numbers to save 20 pages of redundant and obviously intuitive information.

If you cant do this, your entire identity and personality is based on a non-issue that no one really cares about and you should go back to your meaningless crackpot blog.
--- 15351652
>>15349364 (OP)
Correct, we can't prove the existence of each real number individually.
We can't even name each real number individually. There are some real numbers that we can't even refer to.
But we can prove that the set of real numbers as a whole exists from the ZFC axioms, in particular, because of the power set axiom.
You hear a lot about people rejecting the axiom of choice and pseuds rejecting the axiom of infinity, but not a lot of people question the power set axiom even though there are valid reasons to do so: https://en.wikipedia.org/wiki/Impredicativity
It's also perfectly valid imo to not worry about it because we can't even speak about the real numbers we can't even refer to.


>>15349478
Dude...

small correction: The alphabet of first-order logic is actually (usually) not finite.
There are usually countably many variable symbols (e.g. x_1, x_2, x_3, ...).

Size of the alphabet = countable
Size of proofs of length n <= product of n copies of the alphabet = finite product of countable sets = countable
Size of proofs of any finite length = union of proofs of length n for all n <= countable union of countable sets = countable


>>15349482
If you're using a formal system with uncountably many symbols or that allow infinitely long statements, then I'm afraid you're retarded.
--- 15351744
>>15349823
Most of combinatorics and a non-trivial portion of algebra only deals with finite sets.
--- 15352099
>>15351652
>There are some real numbers that we can't even refer to
For example? Like neighborhood of infinity schizo numbers?
--- 15352114
>>15349364 (OP)
>mfw there is no actual definition of what an infinite set is
>the definition of "real" numbers is based on infinite sets
Did we get too cocky mathbros???
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>>15352114
There is no definition of set. This is the ultimate midwit filter. Only naive set theory Chads are winning at math while midwits get caught in language games.
--- 15352187
>>15349364 (OP)
[math]\forall x \in \mathbb{R}, ~x~\mathrm{exists}[/math]

thats only like 10 letters long
--- 15352190
>>15349823
This apu is high quality, I'm saving it for personal and recreational use.
--- 15352191
>>15349364 (OP)
Some proofs are extremely long (see: Fermat's Last Theorem). I always wondered if there are some things that can't be proven/disproven in practice, because the minimum length of proof is infinite.
--- 15352219
>>15352190
All yours my friend
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>>15352187
That's not a proof and it's wrong. Existence is not a property.
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>>15352191
I think Kolmogorov's complexity might have something to do with it
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>>15352191
>I always wondered if there are some things that can't be proven/disproven in practice, because the minimum length of proof is infinite.
The hard problem of consciousness is an example.
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>>15352099
You do realize that to give you a specific example of a real number that I can't refer to, I would have refer to that specific real number.
When I mean "can't refer to" I really mean "can't refer to".
There are only countably many English sentences but there are uncountably many real numbers.
Even if I try to use other languages, there are more real numbers than possible 4chan replies (as long as the length if finite).
Even if I add an image to my reply, there are more real numbers than possible 2 dimensional pixel colour configurations (as long as the size/resolution of the monitor is finite and each pixel can only be one of finitely many colours).

Theoretically, we'll only ever be able to talk about countably many out of the potentially uncountable real numbers.
Practically, we'll only ever talk about finitely many out of the potentially uncountable real numbers.
So by all means, reject the power set axiom and replace it with a constructivist version (or don't replace it at all for all I care).
You'll still (theoretically/practically) be able to prove the existence of any of the countably many (theoretically/practically) describable real number.
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>>15352266
Oh, I see what you mean. You mean "can't refer to" in the sense that, for example, no matter how many digits of pi you write out, there are infinitely many numbers that start with the same digits, some (most) of which can't be uniquely expressed with a finite amount of digits, either. Doesn't that mean that in some ways most of those numbers only exist in an abstract sense because of physical limitations?
--- 15352338
>>15349364 (OP)
Sure you can.
Between any two irrationals is a rational.
Between any two rationals is an irrational.
You can count the rationals.
Build a proof from there.
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>>15352295
> in the sense that, for example, no matter how many digits of pi you write out, there are infinitely many numbers that start with the same digits, some (most) of which can't be uniquely expressed with a finite amount of digits, either
Yeah, kinda similar to that, but a bit stronger.
For example, I can describe some infinite sequences of digits by describing an algorithm that can calculate any given digit (if given enough time).
I could use https://stackoverflow.com/a/5187974 as the definition of the square root of 2.
I could define the square root of 2 as the algorithm itself (as opposed to the result of the algorithm).
It's a finite description that uniquely identifies the real number "square root of 2".
But again, there's only countably many finite strings of characters, so there's only countably many algorithms, so I can only define countably many real numbers that way.
So even if I allow these kinds of definitions, there's some real numbers I can't define.

> Doesn't that mean that in some ways most of those numbers only exist in an abstract sense because of physical limitations?
They only exist in an abstract sense because of theoretical language limitations.
I think that any written language that has any chance of being practical has to consists of only finite sequences of finitely many characters. This forces the set of sentences to be countable.
If you care about physical limitations, then it's even worse, there are only finitely many numbers.
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>>15352392
>If you care about physical limitations, then it's even worse, there are only finitely many numbers.
By abstract I didn't mean they're not, uh, real, but instead only exist in a platonic way.
In any case, thanks for your time.
--- 15352475
>>15349364 (OP)
You don't need a proof for every real number. You, one, can count, and two, use something like induction and functions.
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>>15352475
I don't think anon can
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>>15352475
>You don't need a proof for every real number.
Because the whole thing is nonsensical
>use something like induction and functions.
Based on infinite sets, therefore invalid
--- 15352790
Gödel's Incompleteness theorem, anyone?
--- 15352798
>>15352790
Worthless college kiddy trash spergs out of nowhere and screams
"GoDeL InComPleTnESs AHHHHHHH I'MG GOING INSANNENENENENEEE AHHHHHHH"

Stop discussing mathematics and gtfo this thread worthless piece of shit
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>>15352798
? 
You seem terminally online sir, redeem your anger sir.
--- 15352842
>>15352798
>being this butthurt because someone mentions a theorem
Lmao kid, calm down
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>>15352842
Because a worthless schizo like you doesn't even understands the theorem, DO YOUR HOMEWORK worthless talentless piece of shit and don't even argue!
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>>15352861
>look mom I'm projecting
--- 15354411
Real numbers are actually the imaginary ones ironically. True numbers are discrete. You have one apple or two apples. You have one half of an apple or one third. Shit you could make up a label you have a pi apple. Once you slap a label on it that is a discrete thing the letter is part of an alphabet which is a set of discrete elements. Real numbers are a nice abstraction but many brainlets try to use it as a way of pretending the universe couldn't have had a totally different set of rules more in line with a simulation. Because muh continuous vs discrete so lame. You can't label or look at a continuous thing at a deep level only a high level and at the high level it is still defined by discrete things like f(x) = mx. The formula is itself a discrete representation of a line.
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>>15354411
>math with apples
Brainlet
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>>15354419
It is more grounded in reality and the real source of the concept of a number. Stuff in the world like an apple.
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>>15352790
>Gödel's Incompleteness theorem, anyone?
literally unrelated, fucking brainlet popsci nigger
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>>15351744
ok but what about math past the 12th century?
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>>15352191
>I always wondered if there are some things that can't be proven/disproven in practice, because the minimum length of proof is infinite.
thats the entire point of halting problem, godels theorem, etc. infinite proof is a contradiction. Thats just another way to say there is no proof.
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>>15355598
are you for real?
https://en.wikipedia.org/wiki/Combinatorics#History:
"In the second half of the 20th century, combinatorics enjoyed a rapid growth, which led to establishment of dozens of new journals and conferences in the subject."
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>>15356440
--- 15356461
>>15356440
>>15356459
--- 15357099
>>15356440
>>15356459
>>15356461
Thanks, I'm stealing these.
--- 15357130
LaTeX nonsense
--- 15357168
Traditional Chinese Alphabet be like:
>c
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>>15351652
> If you're using a formal system with uncountably many symbols or that allow infinitely long statements, then I'm afraid you're retarded.

Traditional Chinese Alphabet: >:c
--- 15357620
What about infinite proofs?
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>>15357620
Then you have to get into the philosophy of mathematics, because it becomes important to understand what a proof is.

If you understand a proof as a way of convincing another person, the requirement to read or understand an infinite "proof" seems to disqualify them.