/sci/ - Science & Math

[Anonymous No.10833599]

What is the proof that numbers are infinite?

[Anonymous No.10833603]

there is an infinite amount of them
qed

[Anonymous No.10833606]

>>10833603
Thread/

[Anonymous No.10833625]

>>10833599
logic

[Anonymous No.10833632]

>>10833599
There's only 10 numbers in a decimal system. Prove me wrong, protip you can't.

[Anonymous No.10833633]

>>10833599
Hint: you only need axioms 1, 2, and 4 to prove it.

[Anonymous No.10833648]

>>10833599
True by definition.

[Anonymous No.10833751]

Numbers beeing infinite is an axiom, meaning it is assumed to be true without needing a proof. https://en.wikipedia.org/wiki/Axiom_of_infinity

[Anonymous No.10833759]

>>10833633
False. I need axiom 3 to exclude the possibility of 0 being its own sucessor.

[Anonymous No.10833772]

>>10833633
Do people write these things unironically?

[Anonymous No.10833779]

>>10833599
the logic is this: for every number you'll give me, i can give you a larger number. that's one of the definitions abstractly used in calculus

[Anonymous No.10833790]

>>10833599
just use proof by contradiction

theorem: numbers are infinite

suppose n is the largest number
n + 1 > n
therefore numbers are infinite.

[Anonymous No.10833837]

To prove is

[math] \forall m. \, \exist n. \, \exist d. \, (\neg (d = 0) \land m+d=n) [/math]

Prove in anything like agda will be

[math] \lambda m. \ \langle m+1, \langle 1, \langle refl , refl \rangle \rangle \rangle [/math]

where it depends what the compiler makes of 1=0 and x=x internally.

[Anonymous No.10833843]

To prove is

[math] \forall m. \, \exists n. \, \exists d. \, (\neg (d = 0) \land m+d=n) [/math]

Prove in anything dependently typed like Agda will be

[math] m \mapsto \langle m+1, \langle 1, \langle \top , refl \rangle \rangle \rangle [/math]

where it depends what the compiler makes of 1=0 and x=x internally.

[Anonymous No.10833859]

To prove is

[math] \forall m. \, \exists n. \, \exists d. \, (\neg (d = 0) \land (m+d=n)) [/math]

(For all numbers m, there's an n and a nonzero d such that m+d=n, i.e. n comes properly after m)

The proof in anything dependently typed like Agda will be a map from any m to all the ingrediences, e.g. d=1 will work, so

[math] m \mapsto \langle m+1, \langle 1, \langle \top , {\mathrm{refl}} \rangle \rangle \rangle [/math]

where it depends what the compiler makes of 1=0 and x=x internally. Those languages usually have Peano arithmetic recursion evaluated at compile time, so the matrix of the proposition is essentially free.

[Anonymous No.10833934]

>>10833633
Can't you just use 2 and prove by induction
t.engineer who took one discrete math class

[Anonymous No.10834003]

>>10833633
You actually need 3 as well, otherwise Z/2Z = {0,1} (integers modulo 2) is a model.
>>10833934
>Can't you just use 2 and prove by induction
If you just use 2 and induction then the trivial Z/1Z = {0} is a model (with 0 as its own successor).

[Anonymous No.10834008]

>>10834003
Actually without 3 even the trivial Z/1Z satisfies the axioms.

[Anonymous No.10834024]

look at Euclid's proof of infinite primes, it's not very complicated

[Anonymous No.10834037]

>>10833599
Diagonalization

[Anonymous No.10834050]

>>10833599
How did Nikola Tesla age so horribly

[Anonymous No.10834055]

>>10833599
a priori

[Anonymous No.10834313]

>>10833859
>m⟨m+1,⟨1,⟨⊤,refl⟩⟩⟩
what notation is that anon?

[Anonymous No.10834648]

>>10833599
n+1 > n
QED

[Anonymous No.10834727]

>>10833599
I've fucked your mother countless times and I ain't stoppin'

[Anonymous No.10834740]

>>10833790
In that case n+1 is not a number since n is the largest number, so the comparison operator is meaningless.
Attempting to increment the largest number is just stupid.

[Anonymous No.10834776]

>>10833599
>What is the proof that numbers are infinite?
For any positive integer n, n! is divisible by the positive integers from 2 to n (its definition to begin with is literally just that it's the product of these same integers).
e.g. 5! = 120 and 120 is divisible by 5 (24), 4 (30), 3 (40), and 2 (60).
n! + 1 is NOT divisible by ANY of the positive integers from 2 to n (again it's basically defined up front in a transparent way for this conclusion to follow since it's simply the product of those same integers but with a remainder of 1 making it not divisible).
e.g. 5! + 1 = 121 and 121 is not divisible by 5 (24 with a remainder of 1), 4 (30 with a remainder of 1), 3 (30 with a remainder of 1), and 2 (60 with a remainder of 1).
So now we get to conclude for any positive integer n that n! + 1 must either be:
A) A prime number or
B) A composite (non-prime) number divisible by some new prime number that wasn't one of the factors of n!
5! + 1 falls into the A category (121 is prime).
4! + 1 (25) falls into the B category (25 is divisible by 5, and 5 is not one of the positive integers from 2 to n that factor into n! when n=4).
In either the case of A or B what we've established is there is always another new prime number every time you evaluate n! + 1.
This means there are infinite prime numbers.
This means there are infinite numbers.

[Anonymous No.10834781]

>>10834740
It's called proof by contradiction, anon.
You assume the opposite of what you're trying to prove and then show how it *doesn't* work.
The fact n+1 would be a contradiction if you assume n is the largest number is the entire point.

[Anonymous No.10834805]

>>10834776
Suppose that the naturals contain multiple unities.

[Anonymous No.10834815]

>>10834781
I'm saying that the statement n+1>1 is incorrect, so you haven't shown a contradiction.
Let's say we assume there is a largest number, n.
I might argue that in that case n+1 is just not defined. So where's the contradiction?

[Anonymous No.10834921]

>>10834815
>I might argue that in that case n+1 is just not defined
That doesn’t really affect whether or not there is a largest number n. You’ve just given it the property that S(n) is undefined. Remember that this started with the assumption that there exists the largest number n. So S(n) not existing is already assumed.

[Anonymous No.10834975]

can an infinite amount of numbers fit in the space of experience only if the space of experience is also infinite?

[Anonymous No.10835027]

0. Infinite can be achieved through the addition of any other 2 numbers.
1. Start with zero. Zero is a number that represents nothing or the absence of numbers
2. Add zero to itself and it equal zero
3. You can add as many zeros as you want, let's say an infinite amount of zeros in a consecutive series and the outcome will equal zero.
4. You can, therefore, reach the outcome of zero an infinite number of ways thus proving numbers to be infinite.

[Anonymous No.10835040]

>>10834975
An uncountably infinite set of numbers can fit within a finite space, the interval (0,1) for example, on the number line, if that's what you're asking.

[Anonymous No.10835074]

>>10835040
appreciate the reminder of that example, i actually don't know what i'm asking

[Anonymous No.10835079]

>>10833632
you have webbed fingers

[Anonymous No.10835082]

>>10834975
yes i think there is a maximum capacity for information in the phyisical universe ..
so i looked it up and this guy looks to say so , pretty nutty stuff desu

[Anonymous No.10835083]

>>10835027
>. You can add as many zeros as you want
that's how your iq was counted

[Anonymous No.10835100]

>>10835082
this is in tune with what i was pondering - reading about this cat rn, thanks anon

[Anonymous No.10835238]

>>10833599

because for every number a, there's always an a+1.

[Anonymous No.10835273]

>>10834313
Just a tuple with 4 elements.
Used langle tangle instead of round brackets and didn't introduce/use the abbreviation (,,) for (,(,)).
Refl is the common name for the proof of the reflexivity of =.
Top is True

[Sage No.10835275]

>>10833599
Every number is either composite, prime, 0 or 1.
We know there are an infinite number of primes and that all composite numbers have unique prime factorization (disregarding the actual order you put the numbers in while multiplying). Hence, while there are only two non-prime/non-composite numbers, there is an infinite number of combinations of primes to give us an infinite number of composite numbers. The proof for negative numbers pretty much follows immediately since to get them, you just subtract a positive composite number from zero. Not that hard of a proof all things considered. In fact, it's a good question for an intro to proofs HW.

[Anonymous No.10835278]

If by "number" you mean "natural number", then numbers are finite, by definition of the notion of "finite".

[Anonymous No.10835289]

>>10833599
>Dumb answer
Cuz any number I can state, I can always write +1 after it.
>Actual answer
They are not, because there's a limit to the underlying logic that drives the universe. You require a means of representing a number. You require a machine that can interpret that representation and map its meaning somehow. The complexity of the machinery to do this, the time the process takes, and the amount of material available, is finite.

There's even a limit to how many numbers any given machine can pseudo-random generate in the lifespan of the universe. It can't just invent a place value system with a large base (massive number of symbols) then keep adding the biggest symbol, because it probably has finite time in the lifepsan of the universe. It can't generate every possibility for the same reason.

Anyway. No. Odds are there is no infinity.

[Anonymous No.10835349]

>>10835278
Sets can be finite, not numbers.

The statement makes sense at best with something like the Neumann model.

[Anonymous No.10835354]

>>10835289
>>Actual answer
m-muh monkey with pebbles answer

[Sage No.10835356]

>>10835275
BTW, I know it's obvious, but for the sake of being complete, we mean a member of either Z or N when we are talking about numbers. There are easy proofs for other number systems, but I'm assuming you mean those ones when you say "number".

[Anonymous No.10835382]

>>10833633

[Anonymous No.10835505]

>>10835382
I fucking lost it hard

[Anonymous No.10835509]

>>10834050
Contracted an STD from his pidgeon gf

[Sage No.10837204]

>>10834050
Slavs have shitty genes when it comes to aging desu

[Anonymous No.10837400]

>>10835289
>the universe
>a means of representing
>a machine that can interpret
>material
Real world physical limitations don't define mathematics you fucking pseud retard.
NONE of the numbers exist in the real world. They're abstractions, not physical things. Mathematics isn't a branch of physics.

[Anonymous No.10837457]

Numbers are made up you fucking retards, they dont really exist they are a mental image of something, so i suppose as far as your mind goes, so do numbers you dumb fucking fucks

[Anonymous No.10837526]

>>10834815
Then you don't understand how the successor function works, it's defined on all numbers and it's result is always another number. No one cares about what seems reasonable in natural language to you, what actually matters are the consequences of the axioms and definitions.

[Anonymous No.10837606]

>>10833603
based

[Anonymous No.10837621]

>>10833599
1) n is a number (by definition)

2) n+1 is a number (by definition of addition)

3) for each number there is at least one more (from 2)

4) there is no upper bound on the number of numbers (from 3)

5) numbers are infinte (from 4)

[Anonymous No.10837626]

>>10837621
nigga where in the definition of addition does it say n+1 is a number

[Anonymous No.10837728]

>>10837626
not him and it's not on the definition of addition, but you can prove that n+1 = S(n), for the successor function where closure is part of the axiom. It's true.

[Anonymous No.10837735]

>>10833599
Suppose that your mom is gay and that your asshole is full of fuck. Then you're a faggot fuck fucking fuck get fucked up your fucking asshole you fucking fucked faggot fuck.

QED

[Anonymous No.10837754]

>>10837735
hmmm you bring up some good points there

[Anonymous No.10837762]

>>10837735
Assume traps are gay (a good assumption). Thrm: If a problem is easy to solve it is NOT easy to check if a given solution is correct. Proof: ask the question traps are gay, we find the answer converges to yes by assumption. However, given the answer of yes we plug it in backwards and find that it is very easy to verify. If there were any chance our assumption was incorrect it would not be EASY to know if traps are gay or not. A clear contradiction with basic results in complexity theory which find traps to be gay. Because we have so little chance of traps not being gay our original thrm is false, but the converse is true.

Therefore, P=NP QED.

[Anonymous No.10837772]

>>10837762
Negation* not converse

[Anonymous No.10837815]

>>10837735
Little known fact: homosexual men crave the sexual company of other men.

[Anonymous No.10837855]

>>10833599
>What is the proof that numbers are infinite?
The proof that the cardinality of a line segment is infinity is a sufficient proof of the existence of an infinite quantity of unique numbers.