/sci/ - Science & Math

finitist fags get off my board

>>9444759
>The axiom of infinity cannot be derived from the rest of the axioms of ZFC, if these other axioms are consistent. Nor can it be refuted, if all of ZFC is consistent.

can't stop listening to snow patrol lads

>>9444759
>finitist fags get off my board
Why the apeirophobia?

I don't understand why people wanted such Axiom.
Was it just because it made some proofs shorter or something?
I'm fine with the idea of a set of all natural numbers... but things like transfinite numbers, or sets with cardinality bigger than of the set of all natural numbers, does it serves any practical purpose at all?

>>9444789
Dont question infinityfags. If they were smart enough to give you a good answer they'd be smart enough to not believe "a lot idk" suffices as a number which could be used in math.

If you want to be a faggot about foundations look at foundations that do not rely on ZFC. Especially type theory related ones because bonus points, computers can easily check their logic.

>>9444814
hyperreal brainlets are just as bad as finitists

>>9444754
>implying these random ancient cavemen symbols have any actual meaning

>>9444765
Is there something wrong with that? That’s exactly why it’s an axiom.

>>9444843
>implying "meaning" exists

>>9444849
New axiom: you're a faggot. Can't refute this, because it's an axiom. Consistent too.

>>9444789
Why are you such a brainlet faggot that needs "purpose" for something rather than just accepting a logical objective truth is true regardless of it's engineering "purpose".
Scientists and engineers are fucking brainlets.

[math]\sum_{n=1}^{\infty} \frac{x}{(x+1)^n} < \sum_{n=1}^{\infty} \frac{(x+1)}{(x+2)^n}[/math] for every partial sum.
There are truly an infinite amount of unique numbers that are read as "0.999..." and they all have relative greater than or less than values compared to each other.

[math]0.\bar{9} < 0.\bar{9} < 0.\bar{9} < 0.\bar{9} < ... < 1[/math]

I mean, if you believe infinity is a valid concept and it's usage in mathematics doesn't need to be corrected, you'll just have to live with the fact that [math]0.\bar{9} \neq 0.\bar{9}[/math], as prescribed by infinite property.

>>9444865
I have no idea what your point is.

>>9444850
>implying

*saves mathematics*

>>9444883

>>9444901
Is this your first time attempting an argument? Why is this so hard for you?

>>9444882
It would make sense for [math]\infty \neq \infty[/math] since [math]\infty \stackrel{+}{-} n = \infty[/math]. Mathlets just never gotta around to firing enough simultaneous neurons to piece together whether they wanted to treat infinity like a number or treat it by its own arbitrary rules. Get off the fence brainlets.

>>9444914
Define number

>>9444876
Because the existence of transfinite numbers and set of cardinality bigger than of the natural numbers surely don't look like "logical objetive truth" for me. Rather, they almost look like a logical falsehood. So i wanted to know if they at least serve a practical purpose.

>>9444903
Is this your first time responding ti bait? Why is this so hard for you?

>>9444789
>I'm fine with the idea of a set of all natural numbers...
That's literally what the axiom in pic says.

>>9444882
[math] \sum_{n=1}^{\infty} \frac{x}{(x+1)^n} < \sum_{n=1}^{\infty} \frac{(x+1)}{(x+2)^n} [/math]
But that's incorrect. It's should be [math] \leq [/math] , not [math] < [/math]

>>9444882
>x = 0.999....
>10x doesn't equal 9.999... because there is a zero percent chance i've referenced the same exact 0.999..., given there are infinitely many of them with non-equivalent values
>y can equal 9.999... though
>y-x = 9.999...a - 0.999...b
>y-x = 9
> x = 0.999...
> y = 9.999...
?????????????? How domes i prove 0.999 = 1 den?

>>9444939
Yes, but when you use other axioms with it (like the power set axiom), it result in things that I'm not fine with.

>>9444947
For every partial sum, it is never equal. For every partial sum, it is always less than. It is not incorrect. The incorrect statement is to say they are equal in some loose misinterpretation of final summation when there are literally infinite proofs claiming otherwise.

>>9444928
something which can be used in arithmetic.
for example, n+1 = n+1

infinity fails this outright. infinity+1 = infinity.

>>9444950
3/3=1
1/3 X 3 =1
1/3=0.333...
0.333... X 3 = 1 = 0.999...

>>9444968
How does infinity fail the test? inf+1 = inf+1.

>>9444968
>infinity fails this outright. infinity+1 = infinity.
What's the issue?

>>9444958
What's wrong with the power set axiom?

>>9444887
>

>>9444980
(You)
>>9444982
it means infinity is not a number and is no more valid a concept to math than the word "many". How much is many?

>>9445014
Ironically the only use for infinity in math is proving infinity doesn't belong in math using the very same methods mathematicians use infinity with.
it completely cancels itself out as a concept and has amounted to little more than a wafting fart.

>>9444933
There are proofs that the real numbers don't have a bijection with the naturals, I don't understand why you are so angry at this.

>>9444987
By itself? Nothing.
But when you use it with the infinity axiom you create a infinite set that is "bigger" than another infinite set. And that you can't even define all it's members properly with logic because of how "big" it is.
It seems counter-intuitive and unecessary to me.

>>9445053
My problem is exactly with this concept of "real number". I don't see why you would need a set or even the concept of "real numbers". Computable numbers, or definable numbers, plus some rougue elements perhaps would be enough for anything I can imagine and you still would have a cardinality small enough to have a bijection to the set of natural numbers.

>>9445054
Yea because there is a difference between a countably infinite and uncountably infinite cardinality. Why are you so bent out of shape?

>>9445062
My problem is that I see no practical use for uncountable sets and I don't see them as intuitive/obvious enough for me to accept they "exist" or "shoul exist". They seem counter-intuitive for me and I don't see why we would even want them.

>>9445058
>>9445067
It doesn't matter that you have a problem with the concept or if you can't see a use for it or if they're "un-intuitive" (which they aren't you just need to think about them more), there are proofs that the real numbers exist. If you seriously want to deny the existence of e, √2, π or literally the uncountable infinite amount of real numbers just because they make you angry that math isn't discrete I don't know what to tell you. This universe and it's physical laws are NOT what math is, math is a platonic thing that exists outside of space and its laws are a superset of the laws of physics.
Just by accepting the existence of the naturals and we can build all the other sets from them. How can you deny √2? If we accept the naturals, what happens when I take the root of 2?
>you can't do that
Yes I literally and obviously can and just did, so what now? The number is irrational, it's a real number. How do you deny e and pi and the other transcendental numbers? I really don't get this the finitists have been wrong for hundreds of years and there are proofs this is actually just denying reality.

>>9445054
I don't undestand, the continuum of a line seems counterintuitive to you?

>>9444975
3/3 = 1
1/3 = [math]0.\bar{3}_{\frac{1}{3}}[/math]
1/3 × 3 = [math]0.\bar{9}_{\frac{3}{3}}[/math]
[math]0.\bar{9}_{\frac{3}{3}} = 0.\bar{9}_{1}[/math]
[math]0.\bar{9}+_{1} = 1[/math]
[math]0.\bar{9}_{\frac{n}{n}} < 0.\bar{9}_{\frac{n+1}{n+1}}[/math]
[math]0.\bar{9} \stackrel{=}{\neq} 1[/math]

>>9445112
Sci is never gonna fix it's latex

>>9445112
[math]\frac{1}{3} = 0.\bar{3}_{\frac{1}{3}}[/math]

>>9445071
Dont bring manmade laws of nature into math. Theres also nothing explicitly wrong with infinity, just its implements in math thus far. It needs retuning. In any case, infinity is not a number. Infinity even has a problem of breaking time, for example, something normal numbers cant do. An infinite sum should be completely unsolveable because the work never ends, and any slice of the most recent work accomplished is instantly outdated and wrong by continued partial sums and the eternal future history of partial sums. An infinitely long significand of unique numbers not falling in a pattern can only be proven true in its infinite lack of pattern by doing the work, which if it were truly infinite this work would never end but if it were finite to some insane degree, the work would end. In either case, if the work has not yet ended or cannot end, neither can predict when or if the work will end. Both are equally valid and neither can be invalid. If i told you I was immortal and you asked me to prove it, what could I do? We'd just have to wait a sufficient amount of time before enough has passed to convince yourself I hadn't aged. Same problem here. We can only wait for an answer to if numbers like pi or e have an end to them, but ironically numbers like these present a concept of infinity that is ignored when [math]\infty[/math] is used in place of limits - the concept of time. Infinite sums ignore the infinite time requirement of infinity, and in doing so, ignores the value of infinity outright allowing any little shitter to substitute an answer that fits the bill regardless of proveable authenticity.

>>9445170
>In any case, infinity is not a number.
Wrong.

>>9445207
>wrong

>>9445071
I don't have any problems with e, √2 or π.
You can have all this numbers in a set and it will still be countable.
All numbers that you can define are countable.

What makes the real numbers so much "bigger" than the set of natural number is a huge amount or "numbers" that you can't even define. And that is what I have a problem with.

>>9445222
>What makes the real numbers so much "bigger" than the set of natural number is a huge amount or "numbers" that you can't even define. And that is what I have a problem with.
there's a model where all reals are definable (note that the notion definability is always something that must be external from a model)

>>9445102
No. But do we need all those undefinable numbers to have the properties of a continuum line? The rational numbers already make a line "dense". Computable numbers already make so that most numbers that you would excpect to be in the line like √2 are already there. And if you consider all possible definable numbers, you get even more points in the line, while still being countable. "Between any two points that you can define there are infinitely many more than all definable numbers in the line, so many more numbers that you can't even make a bijection between them" seems counterintuitive for me.

>>9445238
What model?
I wasn't aware of this. If there is a way to define this numbers, I won't have a problem with them anymore. Could you please tell me where I can find about such model anon?

>>9445014
Are you going to answer my question?

>>9445014
>it means infinity is not a number
How so?

>>9445014
>it means infinity is not a number and is no more valid a concept to math than the word "many".
I don't follow your logic, if infinity + 1 = infinity implies infinity is not a number then does 0 * 1 =0 imply that 0 isn't a number?

>>9445170
You can’t even define number and yet you think infinity is not a number. When will you accept that freshman calculus students are not smart enough to argue against the important of the reals?

>>9445170
>Infinite sums ignore the infinite time requirement of infinity
What do you mean?

>>9444995

>>9444959
>something which is true for the finite cases must be true for infinite cases
>no, infinity doesn't follow the rules of finity (by definition)
>b-but the finite cases...

Yes, the partial sums are less than, but we're not talking about partial sums, now are we?
>what is a limit

>>9445311
Are you retarded or do you not understand that for every countable finite n partial sum, of which there are infinitely many, they all evaluate inequal.

Mathlets literally understand infinity so poorly that they cannot keep a consistent concept for it. In one moment its treated by normal number rules, in another moment it obeys arbitrary irrational rules instead.

Garbage game. Not interested in it. If you dont want to use infinity correctly then we're just as well off not using it at all.

>>9445334
>In one moment its treated by normal number rules, in another moment it obeys arbitrary irrational rules instead.
What do you mean?

>>9445334
It is never to be treated by normal number rules; noone claimed that.
It is true that each of the infinitely many partial sums, the relationship is strictly equal, but again, this is not about partial sums. The face that the sequence of partial sums has a limit which is outside the sequence is not a contradiction of anything other than an evidently poor intuition of limits.

>>9444936
It must be very hard to live your daily life. You have my condolences anon. Perhaps you will get lucky and a drunk driver will hit you while you walk down the street.

>>9445247
>But do we need all those undefinable numbers to have the properties of a continuum line?
Yes, we do. Doesn't your intuition say that every Cauchy sequence on the line should converge somewhere on the line?

>>9445071
You actually cannot take the square root of two without some concept of infinity. It requires an infinite process in order to compute.

Why aren't you allowed to take the square root of the negative integers, they're not so far from the natural numbers? Because it produces an imaginary number? The fact of the matter is that both operations take you out of your original set of discourse.

>>9445464
>You actually cannot take the square root of two without some concept of infinity.
What do you mean?

>>9444899
Saved mathematics. RIP, Vladimir Voevodsky!

>>9445464
[math] \sqrt{2} [/math] can be considered/defined as the element [math] x [/math] of the field [math] \mathbb{Q}[x]/ \langle x^2-2 \rangle [/math] or of the field [math] \mathbb{Z}[x]/ \langle x^2-2 \rangle [/math]
You don't need a concept of "infinity" (notion of distance, limits, etc.) to define it, like how you have to with π. It's an algebraic number.

>>9444775
out of curiosity, what does "snow patrol" refer to? can't find anything on urban dictionary

>>9445468
Try taking the square root of 43 and writing it down in decimal form without a calculator. You can't because the operation is basically an approximation that continues infinitely until some limit is reached. You can only reach that limit if the concept of infinity exists. This is also where the finitist problem with irrationals and other infinitely long numbers comes from; you can't actually compute them, or do any operations with them, without some concept of infinity.

Consider doing sums with irrationals. We typically denote irrationals either with specific operators (root 2) or with symbols (e or pi), or with infinitely long decimals. You can't actually ever write out infinitely long decimals, and if you try to add them together that requires and infinite number of computations. If you sum together e and pi, you can't actually simplify it to a single irrational number unless you convert to infinitely long decimals, and you run into the same problem.

Contrast this with natural numbers. We're agreed upon the specifications for these numbers; we use the arabic numerals and we can generate arbitrarily large numbers using this form. If you sum together any two natural numbers, you can produce a single number in a finite number of steps, although it might take a lot of time. There is no computational problem for us.

It's a valid position to take, although it's limiting to a certain degree.

>>9445489
do tell, what's Q[x], if infinity is not allowed?

>>9444754

>>9445489
The x that satisfies those criteria does not exists in the fields you've given, so the set would be null in both cases. You have to extend your field to algebraic numbers in order to actually have the number exists.

Furthermore, even after this extension you still require an infinite number of steps to actually simply expressions involving this number.

>>9445501
>out of curiosity, what does "snow patrol" refer to?
Presumably the band.

>>9445502
>Try taking the square root of 43 and writing it down in decimal form without a calculator.
Does that make it require the "concept of infinity"?

Does writing 1.000... mean that the decimal representation of 1 requires the "concept of infinity"?

>>9445054
>>9445247

Actually, the continuum is not really "bigger". There is just no bijection between the natural numbers and the continuum.
And you can in fact not resolve this dilemma by switching to computable reals - you won't find a computable bijection between the naturals and the computable reals.
And this, however, can't be resolved by saying: "Ok, but there is a classical set-theoretical bijection." because then you look at the computable standpoint from a higher standpoint, i.e. you go to a stronger set theory and act as if there were only computable reals.

>>9445517
>The x that satisfies those criteria does not exists in the fields you've given
What do you mean?

>>9445519
The decimal representation you gave would require the concept of infinity, because it would require an infinite process in order to write down that number.

However, the number 1 does not require this process.

>>9445531
>However, the number 1 does not require this process.
Neither does the square root of 43.

>>9444843
Please be bait. Don't tell me /sci/ can't even read basic notation.

>>9445529
I'm currently reading the notation you've used as "element of the rationals/integers which satisfies this equations."

Based on what operations you allow, you can construct pi with a finite number of steps, just like you can with root 2. For instance, if sin(x) and arcsin(x) are legitimate operations, you can simply write sin(x/2)=1 and solve for pi with two steps.

However, the sin function, like the square root function, requires an infinite number of steps to actually compute.

>>9445549
>>9444843


How about these (below) ancient caveman symbols with new "trips dubs" accoutrements?

>>9445511

>>9445555
jesus christ you're so dumb
just say "ok I don't know sorry" instead of this bullshit

>>9445555
Nice quads.

>>9445538
Ok. Let's do a test. I'm going to let you pick two natural numbers, and I'll add them together and produce a single natural number.

Afterwards, I'm going to produce two irrational numbers, and I want you to write down that number in a single representation.

The rules are that both of us have to use the symbols associated with our respective numerical fields. For natural numbers, I can only use the symbols {0,1,2,3,4,5,6,7,8,9}. I cannot put 0's at the beginning of numbers. I can combine numbers in any order I like.

For the irrationals, you are only allowed to use pi, e, the square root symbol, or any other common operator, and you can also use any decimal representation. You are allowed to use the same digits as above, except you are also allowed to put in a decimal point between your numbers.

I'll wait for your two natural numbers.

>>9445510
As far as only cardinality is concert, it's the set in OP's pic, which is the set of all natural numbers.
There's no mention of "infinity" in it, whatever "infinity" means.

>>9445517
Huh? What do you mean it does not exist?
Also, by x I meant the coset x+<x^2-2>.

>>9445614
OP's pic is the axiom of infinity, genius

>>9445703
That's just its name. All it says is that there is an inductive set (the natural numbers {1,2,...}).
Yes, you can't biject it with a finite set.
In any case, when we talk about infinity we have to be precise about what we are talking.
I believe that the guy I responded to was saying that you can't define sqrt(2) without a limiting process because "It requires an infinite process in order to compute" which is simply incorrect.

>>9445718
That's not what I claimed, I know that √2 isn't a transcendental number, I simply claimed that just with the acceptance of the naturals we can get the reals by taking the √2 which can not be expressed in N Z or Q

>>9445731
what are you even trying to say? if you accept N, you accept Q[sqrt(2)]

>>9445733
√2 is not in Q it's in R, I think I'm misunderstanding what you're saying.

>>9445731
>with the acceptance of the naturals we can get the reals by taking the √2
How does "taking" the square root of 2 "gets" you the real?
>√2 which can not be expressed in N Z or Q
Yes it can, I showed you how: >>9445489
That's also how you can construct i, or any algebraic number for that matter.

>>9445258
Two things to define a number. You can count it, and you can do math with it. You can't do proper division and multiplication on 0 (even though you can do addition and subtraction so w/e) but you can count no things to acknowledge there is zero of whatever you're looking for. By this abbreviation, infinity falls away from even 0. You can't count an infinite amount of things, and you can't perform logical arithmetic on it. It is simply, pimply, dimply not a number. Again, replace "infinity" with "many" and see how far thay gets you in trying to derive finite results for something like infinite sums. Its stupid shit.

When mathematicians use "infinity" i imagine a bunch of shirtless rednecks in lousiana trying to figure how many beers they'll need to buy to hoot and holler all night. "A lot" does not suffice as an accurate result, and it's pure shibrained laziness to concatenate infinite work and assume any finite number result.

>>9445343
Its treated as a normal number in limits, as you can iterate over n finitely to get a finite result but iterating over n infinitely logically does not return a finite result at any time because there is infinite work to be done. That a result is claimed from infinite limits anyway directly implies mathematicians treat infinity like a finite number because retardation.

It really is pick abd choose flip and flop how mathematicians attempt to justify the existence of infinity in maths. It's just fuckin gay and lame.

>>9445749
>When mathematicians use "infinity" i imagine a bunch of shirtless rednecks in lousiana trying to figure how many beers they'll need to buy to hoot and holler all night. "A lot" does not suffice as an accurate result, and it's pure shibrained laziness to concatenate infinite work and assume any finite number result.
You are obviously are an engineering shitter who doesn't even know what [math] \lim\limits_{n \to \infty} a_n = a [/math] means.

>>9445757
>You are obviously are an engineering shitter
This, fucking engineers are so annoying.

>>9445738
that's because you just don't care I guess, you were already shown how to construct Q[sqrt(2)]: >>9445489

>>9445574
Graham's number and the square of TREE(5)

>>9445757
>infinity is a number
>infinity is larger than all numbers
LITERALLY
GET
OFF
THE
FUCKING
FENCE

PICK ONE DEFINITION AND STICK WITH IT

>>9445783
What's the "fence" here? The post was consistent.

>>9445574
>Ok. Let's do a test. I'm going to let you pick two natural numbers, and I'll add them together and produce a single natural number.
R(5,5) and R(4,6)

>>9445769
>This, fucking engineers are so annoying.
Do you need to swear? It really invalidates your post.

>>9445749
>You can count it
Can you "count" pi?

>>9445574
>For the irrationals, you are only allowed to use pi, e, the square root symbol, or any other common operator, and you can also use any decimal representation. You are allowed to use the same digits as above, except you are also allowed to put in a decimal point between your numbers.
For any two irrationals a and b you give me, a+b is a single representation.

>>9445054
Perhaps you would like Kripke-Platek.

>>9445787
If infinity was a number, there'd be a greater number, as all numbers have greater numbers.

Its not consistent.
Mathlets dont seem to understand that because infinity implies multiple concepts, they assume to get to pick and choose which concept to use per usage rather than being smart enough to realize the multiple concepts counter each other, cancel out, invalidate infinity and equate it's history of usage as paradoxical. Infinity is the maths equivalent of fucking zodiac horoscopes. Infinity is completely pseudo-intelligence.

>>9445783
>>infinity is a number
>>infinity is larger than all numbers
Where did I say that?
The only time (pretty much) mathematicians consider infinity as a "number", is here:
https://en.wikipedia.org/wiki/Extended_real_number_line
Which is essentially just a useful shorthand notation.

>>9445806
>as all numbers have greater numbers.
[citation needed]

>>9445806
>If infinity was a number, there'd be a greater number
There is, you consider the set whose cardinality is your infinity, then consider the cardinality of its power set.

>>9445800
KP was not originally formulated as a set theory and it is a gross misinterpretation on the part of barwise to assume so

>>9444882
This is an elegant construction of proofs. No infinitely homoerectus erection penetration to be found in that proof at all.

I dub thee 'not a pooftah'.

7NxiEt 5%817zJ16N j#e`C

>>9445797
You cant count to pi by incrementing arbitary decimal accuracy but sure.
If we use pi's significand as a base
>0.14159
>1.14159
>2.14159
>pi

>>9445938
>infinity - 3
>infinity - 2
>infinity - 1
>infinity

[math]0.\bar{n}[/math] is an invalid concept by itself. Any repeating pattern number "infinitely" can only equal itself at the moment of discovering the answer throgh such a loose attribution of infinity as just an overline. Infinity has time and rate attributed to it, but aren't specifically infinite amount of time or an infinitely fast rate. Take the sums
[math]A=\sum_{n=1}^\infty \frac{9}{10^n}[/math]
And
[math]B=\sum_{n=1}^\infty \frac{1}{2^n}[/math]
for example.
Calculus would tell you these both equal 1 and they equal each other, but there is no specific way to prove it. If you spent eternity comparing partial sums, A would always be greater than B, and neither would logically equate to 1 without rounding at some point, even though that point would be different for A which could round whenever, compared to B which could only round when it has eventually added up to string of 9's instead of looking like ...99917339[math]\rightarrow[/math] at some random test of n.
But say we stretched and redefined how these sums are supposed to work, that they do not require infinite time to compute because we are summing them infinitely fast. By laymans eyes, they would be able to equate to 1 instantaneously via infinite work done infinitely fast, and because it is so fast, we can't see which one completed first even if one might have completed it's work in a relative infinite amount of time faster than the other, so cheating this way with speeds infinitely faster than the speed of light we can extract finite, equivocal answers from the infinite sums.
Thats a really generous way of putting it, but ends up defying even more logic in the process that such astronomically large events could occur in such immeasureably small nothings of time, thus really solves nothing towards addressing issues with the usage of infinity.

>>9445996
So if A > B with infinite time to compare infinite partial sums, constantly proving A > B infinitely many times via comparing these partial sums to the point that it could do nothing but assure with 100% confidence that A > B, why would or should it be logical to assume A = B = 1?
if infinity is a number we can reach by doing work infinitely fast, we'd have proven that (A > B) < 1 defacto. We did infinite work and every infinitisemal comparison validated A > B, despite what calculus claims.

So touching back at the first point of [math]0.\bar{n}[/math] not validating as a number, we can prove this because any arbitrary rounding or A or B will give us approxiamately [math]0.\bar{9}[/math], and i say approxiamately because A will have significantly more 9's than B even if B's final mantissa are 9's and not random numbers. We also know A > B so simply defining both A and B by [math]0.\bar{9}[/math] is actually missing information to the point of assuming [math]0.\bar{9} \neq 0.\bar{9}[/math]

What these values should instead be are results indicative of their work, like
[math]A=0.\bar{9}_{\frac{9}{10}} , B=0.\bar{9}_{\frac{1}{2}}[/math]
Now both have infinite repeating decimals and still validate A > B.

>>9445996
>Infinity has time and rate attributed to it
What do you mean?

>>9445996
We don't have to sum them infinitely fast. We just need an operation that takes a time that doesn't scale with iteration count. An inductive function is one possibility, but it is not the only one.

>>9446015
You are a librarian at the library of infinite books.
The library was struck by an earthquake and all the books have fallen off the shelves.
You can rearrange books at 1 book per second.
How long does it take you to restock all the books back on the shelves.

it takes you a couple seconds to do basic math in your head to solve a single problem. How long will it take to solve an infinite amount of problems.

You are breathing manually.

>>9444789
Any model of ZF without Infinity is necessarily infinite. It's the first consistency/reflection axiom.

>>9445988
All those values are the same value: infinity. You saying "infinity - 1" has no meaning, nor does any such meaning extend into any field of math.
∞ + ∞ = ∞
∞ × ∞ = ∞
∞ ÷ ∞ = ∞
∞ - ∞ = ∞
Once you instantiate infinity in a problem, infinity must be a part of the result, either directly as ∞ or indirectly as a repeating decimal.

heres something else to tease your brain.
let us use the variable x
x + x = 2x
x × x = x[math]^2[/math]
x ÷ x = 1
x - x = 0

We don't know what number x is. It could be any number, but we definitely have that for any number to substitute x, these patterns of results remain true except for infinity if infinity were meant to be treated as a number. Infinity does not even validate to substitute as an undefined variable let alone a number. If x could be any number, the results above will always validate true - and this definition basically says there are an infinite amount of possible numbers you can plug in to substitute for x to validate the results. We can just as easily claim that x can be any number, whether we wish to define it or not, and validate AS a number, even though we all know x is a letter and not a number.

∞ simply doesn't follow the rules of numbers.

>>9446021
In this case, we can create a function

[math]\dfrac{10^n-1}{10^n}[/math]

We can pick an arbitrary number and solve to any number of iterations. This relies on an inductance property.

If we also accept some definitions of limits or the mean value theorem, then we can define this value as one as well.

>>9445808
All of math is useful shorthand notation.

>>9445810
pick a number

>>9446041
>You saying "infinity - 1" has no meaning, nor does any such meaning extend into any field of math.
https://en.wikipedia.org/wiki/Extended_real_number_line#Arithmetic_operations

>>9446021
The point was that infinity is not a number and what it is even includes as a hardcoded feature an aspect beyond arbitrary numbers, being time.
When i say 4, you think 4 right?
Does "four of what?" immediately need to exist as a question to validate the statement "4"? We've merely instantiated an arbitrary number, and for this example the number is just 4.

Now when i say infinity, what do you think? In part i would actually like to know what you think of infinity before you read any further so please write the reply first before continuing.
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.
.
.
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What you should think of infinity is "wow", or "idk", some arbitrary feeling or knowledge attributed that makes you want to question "just how big is it anyway". Thinking of infinity should be a brisk thought where you assume to concatenate an understanding of it so it doesn't blow your mind into gaping slackjawed retardation for the rest of your life, unabled to fully grasp the unending magnitude and scope of it. In any case, whether what you think of infinity or what i think you should think of infinity, just thinking about it has consumed more interest and time than "4" had done.

Now maybe if instead of 4 we thought about 3000000000000000000000, what would you think?
"Threeee... uh"
"How many fuckin zeros is that?"
"Let me count the zeros"
"Fuck i lost my place"
"Thats... 3 sextillion... okay"
Now we have a number that might start to make you think as much as infinity made you to, consuming time to figure it all out or letting your brain acknowledge more than 9 zeros and figure to turn off giving a shit, concatenating the effort internally in a similar way as infinity may have done, but once you figure 3 sextillion it's enough to accept and move on, consuming no greater occupation of your mind relative to other, normal numbers.

>>9446041
>x ÷ x = 1
>It could be any number, but we definitely have that for any number to substitute x, these patterns of results remain true except for infinity if infinity were meant to be treated as a number.
So since 0 doesn't satisfy 0÷0=1, are you saying 0 is not a number?

>>9446113
Now if i say pi, what do you think.
I think [math]\pi[/math]. Maybe i'll try to recite a couple digits as best i can, i onoy make it to 3.14159 and everything is fuzzy after there. Do i need to know pi to some extreme level accuracy to use it? Nope. Would i try to do the work by hand on paper if i wanted to use it? Nope. It doesn't matter too much to wonder where all the numbers of pi go to, but in the end we just end up concatenating an understanding to fit the bill, just like 3 sextillion, without extra consuming interest like infinity.

We can do finite work in finite time to get finite results of necessary value right now, but we cannot do infinite work to ever result a useful answer for any point in the future because there is not infinite time or access to future time from the now to complete such work, conversely you could create a computer program to generate 12-digit random strings of characters ranging from lowercase, uppercase letters and 0-9 that will give you access to over 3 sextillion unique combinations, with this program completed in under 10 minutes. That even though huge numbers are huge, they are still relatively easily workable, and still relatively miniscule to the infinite range of values attributed to infinity

>>9446063
>pick a number
[math] \infty \in \bar{\mathbb{R}} [/math]

>>9446114
0 is poorly used too but whatever. [math]x^0[/math] should equal 0 or x, but instead it equals 1. There are fewer necessary workarounds to get 0 working as a number than infinity so its enough to to say 0 is a number for other reasons that infinity still does not apply to. I've given a few examples so far and even though 0 might fail in a couple cases, it doesnt fail completely like infinity does. You can count no things to result zero. You cant count infinite anythings. You can do some math on 0. Any math on infinity just results infinity. 0 has some kind of identity compared to other numbers which have more of an identity, but infinity has no identity - you could say infinity has zero identity, zero logic, zero real usage, you can play with words all day, but at the end of the day 0 is still just a natural base number necessary in base-10 to craft numbers like 10 or 100 or 0.01, so it still makes itself useful. Knowing there could be an infinite amount of digits in pi is useless information. Pretending to solve and infinite sum in finite time is useless information. Axioms of infinity can be disregarded with hard finite limits used and be good enough to accomplish everything that has been so accomplished without intantiating infinity, which so far in history has been literally everything. We would not have digital audio if not for fourier transforms engineered based on hard limits of data sectors rather than infinity. We can do cos and sin equations quickly and reliably by mapping a result +/- 1. Infinity is not necessary in the slightest and really ahould just be a laymen concept, not an actual thing expected to give finite results in arithmetic.

>>9446123
Infinity is not a number. You get an F.

>>9446133
>There are fewer necessary workarounds to get 0 working as a number
define "number"

>>9446140
you are stuck in an infinite loop

>>9446061
Finally someone who gets it

>>9446111
Really made me think.

>>9446205
>Really made me think.
What's the significance? When you extend from the rationals to the reals, the new elements aren't rational numbers. That doesn't mean irrational numbers aren't numbers.

>>9445549
>thinking that anon's post represents all of /sci/

>>9446144

>>9446144

>>9446063
i+1 :^)

>working in sets and not categories


Why though

>>9446474
>implying category is not a fancy word for set

>>9446484
Bro i know we are friends but like are you on the spectrum.

>>9444882
Limits don't preserve strict inequalities your mouthbreather

>>9446618
Infinity applied as it has been doesnt is whst you meant to say, you shiteating hypoxic niggerlover.
Infinity is fucking worthless in math.

>>9446727
>Infinity applied as it has been doesnt is whst you meant to say, you shiteating hypoxic niggerlover.
Why the racism?

>>9446474
>working in categories and not types

>>9446041
That there may be an "infinite amount of numbers" does not attempt to claim infinity is a number in itself, because the statement "an unending amount of numbers" has the same exact meaning, but we don't try to say the end of the unending numberline has the value "unend".

With the example of the variable x in that post, if x were presented in a way that it needed to be resolved as a finite number as in algebra, how would you define the work to solve it? Would you define it as "unending until a solution is found"? Would the number of steps taken to solve x matter as a numerical value to the solution itself? No no no no no, of course not. We are presented with the algebra equation including x, we need to solve for x, and we'll have our solution for x in a minute after doing the math, then we can define x = n and move on. So if x were a variable in any much more difficult problem, we still would not say we must do infinite work to solve the problem. We know that, even if it takes us many individual steps to solve the problem, if we solve it at all those steps would have been finitely countable. Calculus limits are complete bonkers because none of anything above is implied. Calculus with [math]\infty[/math] ends up parodying math and logic itself through multiple reasons. That we could define an answer from infinite work is the first obvious problem. That an infinite sum like 1/2^n = 1 is another problem in two parts, for beyond 1 the equation explodes into infinity (1 + 2 + 3 + 4...) with continued infinite work, and that the equation itself never even reaches 1 in the first place without explicit rounding, and these two concepts define the convergence to 1 occurs at a finite point, or a rounding occurs at a finite amount of decimal accuracy. What the problem should really read as is [math]\sum_{n=1}^\stackrel{x}{y} \frac{1}{2^n}[/math] where x is the finite variable limit at which convergence might occur at y, the finite decimal accuracy requirement.

>>9446783
That there may be an "infinite amount of numbers" does not attempt to claim infinity is a number in itself, because the statement "an unending amount of numbers" has the same exact meaning, but we don't try to say the end of the unending numberline has the value "unend".

With the example of the variable x in that post, if x were presented in a way that it needed to be resolved as a finite number as in algebra, how would you define the work to solve it? Would you define it as "unending until a solution is found"? Would the number of steps taken to solve x matter as a numerical value to the solution itself? No no no no no, of course not. We are presented with the algebra equation including x, we need to solve for x, and we'll have our solution for x in a minute after doing the math, then we can define x = n and move on. So if x were a variable in any much more difficult problem, we still would not say we must do infinite work to solve the problem. We know that, even if it takes us many individual steps to solve the problem, if we solve it at all those steps would have been finitely countable. Calculus limits are complete bonkers because none of anything above is implied. Calculus with ∞ ends up parodying math and logic itself through multiple reasons. That we could define an answer from infinite work is the first obvious problem. That an infinite sum like 1/2^n = 1 is another problem in two parts, for beyond 1 the equation explodes into infinity (1 + 2 + 3 + 4...) with continued infinite work, and that the equation itself never even reaches 1 in the first place without explicit rounding, and these two concepts define the convergence to 1 occurs at a finite point, or a rounding occurs at a finite amount of decimal accuracy. What the problem should really read as is ∑yxn=112n where x is the finite variable limit at which convergence might occur at y, the finite decimal accuracy requirement.

>>9446147
If you're being rational he's right tho

The real question is why is /sci/ stuck debating topics that have been settled for more than 100 years?

>>9446942
Some questions can never be answered because people have different assumptions.

>>9446135
>what is aleph nul?

made this thread just for a laugh, but glad there's some debate going on. thank you /sci/

>>9444968
Define arithmetic.

>>9445170
>Dont bring manmade laws of nature into math.
>Proceeds to understand mathematical definitions like limits and infinity only based on metaphors involving laws of nature like time.

>>9445238
What do you mean by "definable"? It's literally impossible for there to exist a system with which you can finitely notate any real number - there are countably many finite descriptions.

>>9445501
>>9445538
Holy shit is there any board with a higher concentration of turbo-brainlets than /sci/?

>>9446956
A cardinal number.

>>9447124
Time is not a law. Time is an observation of movement and comparison of relation. It is the underpinning of cosciousness itself. Without time is without movement, and without movement is without space.

Infinity fags should go back to >>>/r/eddit

>>9445170
>Talking about time and work
Everyone stop arguing with him, it'll go no where.

>>9447727
Do you have a problem with time or something?

>>9444776
Have you been saving that word for an occasion such as this, or did you just look it up?

>>9445464
Why not? You can construct line with length square root of 2,so that number definitly exists

>>9444882
>all these people falling for this bait
this thread is great

>>9447130
By Tarski's theorem, definability can only be talked about in the meta-theory, not the theory. And in the meta-theory it's possible that there are only countably many real numbers in your model, even though internally the set of real numbers in the model appears uncountable.

>>9444828
t. Bishop Berkeley

>>9449023
It's objective empirically proveable truth. It's not bait. Do the math.

>>9450066
>empirically proveable truth
>math
Pick at most one.

>>9450354
Both were picked and you were cheated, cry about the unfairness on >>>/lgbt/

>>9444882
The sad part is this could have been proven at any point in time a long time ago but it hadn't been.

Weird how that works.
Are humans literally retarded?
Do humans even validate as human or are they paradoxically better described in entirety as subhuman?

>>9451514
Jokes on you, we used to be called man until being downgraded to hu-man.

>>9451519