Finitism is Easy to Prove

[Anonymous No.16113638]

Allowing for infinities gets you a logically contradictory system where you can prove anything simultaneously true and false.

After all, what's 1 equal?
1=(½+½)=(⅓+⅓+⅓)=(¼+¼+¼+¼)=....
And so on.

If we accept actual infinities, then eventually we arrive at:
1=[(1/∞)+(1/∞)+(1/∞)...]

What's two equal?
2=(1+1)=[(½+½)+(½+½)]=[(⅓+⅓+⅓)+(⅓+⅓+⅓)+(⅓+⅓+⅓)]...

Allowing for actually infinites eventually we arrive at
2={[(1/∞)+(1/∞)+(1/∞)...]+[(1/∞)+(1/∞)+(1/∞)...]}

However, if you double infinity or if you half it, it doesn't become bigger or smaller. If you cut an infinity in half the two halves are both the same quantity as the whole was before. If you double an infinity it's the same quantity as it was before.

So {[(1/∞)+(1/∞)+(1/∞)...]+[(1/∞)+(1/∞)+(1/∞)...]}=[(1/∞)+(1/∞)+(1/∞)...]

Therefore 2=[(1/∞)+(1/∞)+(1/∞)...]
But earlier we saw that 1=[(1/∞)+(1/∞)+(1/∞)...]
Getting us the conclusion 2=1.

We can do the same to show anything, like 500=1, -24=1000, 6>∞, and so on.

So actual infinities lead to logical contradictions, telling us that they are logical impossibilities.

So finitism is true.

[Anonymous No.16113641]

>>16113638
It’s no coincidence that all paradoxes are the result of actual infinities or self-reference.

[Anonymous No.16113681]

The reference to truth and falsity depends on infinities. Truth has to persist infinitely in time and space.

[Anonymous No.16113690]

>>16113681
> Truth has to persist infinitely in time and space.
what is relativity

[Anonymous No.16113692]

>>16113681
There hasn't been an infinite amount of time though. The world started to exist at a certain point and the successive addition of years will never make it infinitely old, it will always be some finite number of years old

[Anonymous No.16114221]

>>16113690
what are physical constants
>>16113692
it doesn't matter, triangles will always be three sided as long as there is space to manifest them, etc

[OP No.16114230]

>>16114221
How does that mean there must be actual infinities though? If anything, the fact that contradictions (like non-three-sided triangles) are eternally false means the eternal nonexistence of infinities if the OP is correct

[Anonymous No.16114244]

>>16114221
Truth is merely a function of belief. It’s possible for organisms to have contradictory beliefs. Evolution simply doesn’t select for it so often. But even in humans, cognitive dissonance leads to contradictory beliefs. You could theoretically breed an organism that believes there are triangles that don’t have three sides. And you can’t argue that the organism is “wrong.” Perspectivism.

[Anonymous No.16114924]

>>16113638
>metaphysical retard(aka finitst, aka mathematical zeteticism) divides by zero for the first time
go back to preschool, you mouth-breathing failure

[Anonymous No.16114933]

>>16114230
non-three-sided triangles are not triangles, this is a problem of definitions, also infinities exist in a circle when you intend to make them perfect enough

[Anonymous No.16114935]

>>16114244
no you can't, you would run into contradictory self inconsistencies, you can't mess with foundational definitions and then assume to derive any meaningful conclusion from such shaky foundations

[Anonymous No.16114941]

>>16113638
>However, if you double infinity or if you half it, it doesn't become bigger or smaller
Doesn't follow.

[OP No.16115273]

>>16114924
See now how the infinitist argues! A simple, easy demonstration of a fault with the lazy eight and his vents anger, yet anger which, as Shakespeare said, is full of sound and fury signifying nothing. As vacuous as infinity itself.

>>16114933
>non-three-sided triangles are not triangles, this is a problem of definitions
Right exactly - some things like that have a contradictory definition that's easy to see ("non-three sided object with three sides" in that case) but others can be more hidden, like actual infinities, which boil down to "completed uncompletable process".

>also infinities exist in a circle when you intend to make them perfect enough
That doesn't seem to be so to me - a circle is a shape, and shapes are a structure of physical objects, so any object with this structure, maybe in ink maybe in chalk, will be made of a finite number of molecules and atoms. On a computer it will be made of a finite number of pixels.

[OP No.16115286]

>>16114941
It's very odd (indeed it does lead to contradictions) but it is correct! An often given example is all of the the counting numbers (1, 2, 3...) and all of the even numbers (2, 4, 6...).

You might think one of these should be bigger than the other, with there being half as many even numbers as there are counting numbers. And for any finite extension of these, you're absolutely correct. But when you make them infinite, you'll find that you can always pair every even number up with every counting number; you'll never run out of even numbers to pair with counting numbers. So they must be the same size.

Since what it really means is "all endless processes have endless steps".

[Anonymous No.16115295]

>>16115273
>See now how the infinitist argues!
you have been spamming your zeteticistic shit on /sci/ for over a year(that i know of), and think you are owed dialogue?, eat shit and die, filth

[OP No.16115306]

>>16115295
Finitism is the opposite of something zeteticistic. It seeks to ground mathematics in actual reality. Reality contains no actual infinities.

[Anonymous No.16115369]

>>16115286
>An often given example is all of the the counting numbers (1, 2, 3...) and all of the even numbers (2, 4, 6...).
One-to-one correspondences of sets and set cardinalities aren't the only way to conceive infinity, especially as a number to calculate with.

[Third Poster No.16115395]

>>16115369
Another way to visualize it. Say we have an acute triangle with area ∞. If you bisect it with a line from one vertex to the midpoint of the opposite side and separate the halves, what is the area of the resulting right triangles?

[Anonymous No.16115464]

>>16115395
∞/2, i.e. an infinite quantity which, when added to itself, gives the original infinite quantity, ∞.

[OP No.16115592]

>>16115464
Is ∞/2 a quantity equal to the total quantity of positive even numbers?

[Anonymous No.16115618]

>>16113638
Wot if we had a philosophy called EXTREME Ninetyninsm that aid you can’t have a hundred of something. Wot would be the implications?

[Anonymous No.16115667]

>>16115592
If ∞ is the quantity of natural numbers, as calculated by summing the function f(n) = 1 over all natural numbers, then yes.
If you're talking about sets and their cardinality, as measured by the existence of bijective correspondences between them, then no.

[Poster of the 10th Reply No.16115675]

>>16115667
>If ∞ is the quantity of natural numbers, as calculated by summing the function f(n) = 1 over all natural numbers, then yes

Is that quantity equal to the area of an acute triangle with area ∞?

[OP No.16115679]

>>16115675
Whoops left my name on from elsewhere. That was the OP posting that

[Anonymous No.16115681]

>>16115675
By definition of the area of an acute triangle with area "∞", yes.

[Anonymous No.16115683]

To cut the chase, non-Archimedean fields are a well defined mathematical object within which you can do normal algebra with infinite quantities.

[OP No.16115699]

>>16115681
So let's define ∞/2 as A, the quantity of natural numbers as B, and ∞ as C
You said "If ∞ is the quantity of natural numbers, as calculated by summing the function f(n) = 1 over all natural numbers, then yes"

So A=B

Then said "By definition of the area of an acute triangle with area "∞", yes."
So C=B

So if A=B, and C=B, then A=C

So, ∞/2 - the area of the right triangle formed by having the acute triangle - is equal to the area of the original acute triangle!

Your intuition is quite right that this is nonsensical, hence how it can be used to prove contradictions as is done in the OP.

[Anonymous No.16115724]

>>16115699
>So let's define ∞/2 as A, the quantity of natural numbers as B, and ∞ as C
Ok.

>You said "If ∞ is the quantity of natural numbers, as calculated by summing the function f(n) = 1 over all natural numbers, then yes"
Yes.

>So A=B
Doesn't follow.

>Then said "By definition of the area of an acute triangle with area "∞", yes."
Yes.

>So if A=B, and C=B, then A=C
Yes. (but A=B is false)

>So, ∞/2 - the area of the right triangle formed by having the acute triangle - is equal to the area of the original acute triangle!
Doesn't follow.

[OP No.16115736]

>>16115724
>Doesn't follow.
You said yourself that it does! Remember? You said the two quantities were equal.

[Anonymous No.16115741]

>>16115736
>You said yourself that it does! Remember? You said the two quantities were equal.
I never said this.
I said that if ∞ is the quantity of natural numbers, as calculated by summing the function f(n) = 1 over all natural numbers, then ∞/2 is the total quantity of positive even numbers, which is correct, and doesn't imply that ∞/2 equals ∞.
The quantity of positive even numbers is assumed to be calculated by summing the function f(2n) = 1, f(2n+1) = 0 over all natural numbers.

[Anonymous No.16115749]

>>16113638
[math]\infty[/math] is a limit, not a literal algebraic number. in fact, unless otherwise specified, such as with an explicit extension, [math]\infty \notin \mathbb{R}[/math]
however:
[math]\infty \in \bar{\mathbb{R}}[/math], where [math]\bar{\mathbb{R}}=\mathbb{R}\cup\{-\infty,\infty\}[/math] ([math]\bar{\mathbb{R}}[/math] is what we call the extended reals).

what all of that means is that your sum over [math]\mathbb{N}[/math] in the denominator never gets to [math]\infty[/math], it only approaches it. that is why the sums can become arbitrarily long, but the summands never reach zero. if you instead summed over [math]\mathbb{N}\cup\{\infty\}[/math], the summands for the final term DO reach zero (unless you're working in the surreal or hyperreal number systems or something like them... which i seriously doubt any finitist would be able to stomach), making your entire argument moot, because even the infinite summation of that final term never equals anything BUT zero.

[Anonymous No.16115753]

>>16115749
huh, that's weird. [math]\mathbb{N}\cup\{\infty\}[/math] should show up. is there a cap on inline TEX formatting?

[Anonymous No.16115757]

>>16115753
...guess not. what am i doing wrong there? that's basically how i formatted the definition of the extended reals... which worked just fine.

[OP No.16115764]

>>16115741
To be frank I simply don't see how this in some way advances your point that (∞/2)≠∞

You're describing two countable infinities, both equal to ℵ0, then saying "and they aren't the same size". Can you explain your point in some more detail?

[OP No.16115773]

>>16115749
I think you're right on the money!
>is a limit, not a literal algebraic number
That's basically what finitism argues - you can have potential infinities ("the denominator never gets to ∞
, it only approaches it", as you put it) but never actual infinities (i.e. what you get when you DO "reach" it, which is where the logical contradictions start)

[Anonymous No.16115774]

>>16115764
I'm using three basic rules of summation that you can prove by yourself.
>inb4 I get fucked in the ass by latex

[math]
\sum_n^N f(n) = \sum_n^{N/2} f(2n) + \sum_n^{N/2} f(2n+1)
[/math]
[math]
\sum_n^N 1 = N
[/math]
[math]
\sum_n^N 0 = 0
[/math]

then for the function f(n) = 1 summed up N = ∞, it is self evident that the summation results in ∞, whatever it is.

For the function f(2n) = 1, f(2n+1) = 0 we get:

[math]
\sum_n^\infty f(n)
=
\sum_n^{\infty/2} f(2n) + \sum_n^{\infty/2} f(2n+1)
=
\sum_n^{\infty/2} 1 + \sum_n^{\infty/2} 0
= \infty/2
[/math]

this calculation can be performed in any suitable non-Archimedean field, which is as easy to provide as adjoining an element called ∞ to the field of real numbers and positing ∞ > n for any natural number.

>ℵ0
You're doing algebra in OP, and I'm doing algebra. Stop talking about set theory as if it means anything in this context.

[Anonymous No.16115790]

>>16113638
Treat infinity as 0. Multiplying and dividing doesn't do anything here.

[OP No.16115820]

>>16115774
When you say "For the function f(2n) = 1, f(2n+1) = 0 we get: ∑∞nf(n)=∑∞/2nf(2n)+∑∞/2n", there seems to be some confusion, or perhaps assuming a thing to prove it. It does not sum to infinity divided by two, it simply sums to infinity because it is infinite. The operation you're performing on the set of natural numbers (separating into even and odd) doesn't change the fact that both subsets are infinite.

It's like saying if we have a group of 100 balls, half of which are yellow and half of which are red, if we split them into yellow and red then the two groups are not 50 each but instead (100/2). The two quantities are synonymous.

[Anonymous No.16115826]

>>16115820
>It's like saying if we have a group of 100 balls, half of which are yellow and half of which are red, if we split them into yellow and red then the two groups are not 50 each but instead (100/2). The two quantities are synonymous.
Yes, 100/2 is indeed synonymous with (equal to) 50. You seem very confused.

[Anonymous No.16115831]

>>16115820
>The operation you're performing on the set of natural numbers (separating into even and odd) doesn't change the fact that both subsets are infinite.
I'm not performing any operation on the "set" of natural numbers, by the way. I'm just doing algebra.

[OP No.16115843]

>>16115826
>Yes, 100/2 is indeed synonymous with (equal to) 50.
M-hm, and ∞/2 is, similarly, equal to ∞.

>>16115831
Your math doesn't make sense. Again it's like insisting 50 and 100/2 are different amounts if you split the two groups of 100 balls by color. That doesn't matter if we're just looking at the amount - and math is the study of the behavior of amounts, you know. The history of how things got into a group has no bearing on the group's size

[Anonymous No.16115845]

>>16115843
>∞/2 is, similarly, equal to ∞.
∞/2 is equal to ∞/2, in the same way x/2 is equal to x/2.

>Again it's like insisting 50 and 100/2 are different amounts
50 and 100/2 aren't different amounts, but 100/2 and 100 are. Again, you're very confused.

[Anonymous No.16115860]

>>16115773
i explicitly showed what happens when you add the limits as members - your second step stops working completely. saying it equals anything but 0 is just wrong. there's no contradiction at all.

[OP No.16115878]

>>16115845
>100/2 aren't different amounts, but 100/2 and 100 are
M-hm. The whole issue with infinity is that is behaves in massively different ways from finite numbers.

To go back to the triangle example, say our original triangle was itself the product of bisecting a triangle with area ∞. Would that mean that its area wasn't really ∞/2 as well that whole time? No of course not. Historical information doesn't effect the amount.

[Anonymous No.16115891]

>>16115878
>M-hm. The whole issue with infinity is that is behaves in massively different ways from finite numbers.
If you work in an appropriate mathematical structure that doesn't have to be the case.

>To go back to the triangle example, say our original triangle was itself the product of bisecting a triangle with area ∞. Would that mean that its area wasn't really ∞/2 as well that whole time? No of course not. Historical information doesn't effect the amount.
You're contradicting yourself here. If the bigger triangle has area ∞, then the original triangle has area ∞/2, not ∞, and the triangles bisected from it have area ∞/4. But the original triangle has area ∞, so the bigger triangle has to have area 2∞. Put any number in place of ∞ and recognize that what you're saying is absolute lunacy.

And look, I understand that you watched a video or two on cardinal arithmetic on Numberphile or whatever so you feel like you have a pretty good grasp about this whole infinity business, but do you realize that even in your favorite popmath set theory there's more to it than just cardinal arithmetic? In ordinal arithmetic addition isn't even commutative, yet I don't see you rave about that.
Now go ahead and find me a set with cardinality ½, since you said that 1 = ½+½.

[OP No.16115942]

>>16115891
>If you work in an appropriate mathematical structure that doesn't have to be the case.
To be blunt, you seem to be making random things and rules up rather than using any recognized mathematical structure. Can you find me anyone else who agrees with this "half of infinity isn't infinite" approach?

I'm very curious if even one other person says that we can distinguish between ∞/2 and ∞ the way you're saying. Is there?

[Anonymous No.16115949]

>>16115942
https://en.wikipedia.org/wiki/Non-Archimedean_ordered_field
https://en.wikipedia.org/wiki/Levi-Civita_field

[OP No.16115977]

>>16115949
In other words you're not even talking about the same thing as I am, just a construct in some other system that happens to use the same symbol. Like saying "nuh uh 11+1 is 100" while talking about binary.
But even within that framework, is there any person who says what you're saying? Can you show any articles or papers or anything saying this?

[Anonymous No.16115992]

>>16115977
You're the one mixing algebra with set theory. I'm responding to faulty algebra with correct algebra. Mathematics might not be for you.

[OP No.16116003]

>>16115992
That's like looking at a chess puzzle and solving it using the "correct" rules of checkers, moving the pieces like checkers instead of chess pieces. "We get different results if we use math that has different rules" is again about as significant as saying "100 is REALLY 11+1" and then linking to the Wikipedia article on binary.

If what you're saying is "correct algebra", why can't you point to anybody who actually agrees with it? I could give you legions upon legions saying that "half" an infinite is itself infinite.

[Anonymous No.16116019]

>>16116003
The internet was a better place when your kind didn't know how to access it.

[Anonymous No.16116169]

>>16113638
>If we accept actual infinities, then eventually we arrive at:
>1=[(1/∞)+(1/∞)+(1/∞)...]

>Allowing for actually infinites eventually we arrive at
>2={[(1/∞)+(1/∞)+(1/∞)...]+[(1/∞)+(1/∞)+(1/∞)...]}

>However, if you double infinity or if you half it, it doesn't become bigger or smaller. If you cut an infinity in half the two halves are both the same quantity as the whole was before. If you double an infinity it's the same quantity as it was before.

each of these is unsupported

[OP No.16116218]

>>16116019
It seems like an important concern. Sometimes if you have what seems like a very great but unique mathematical finding, it CAN be a great discovery, but often it can also be the result of a mistake you've made, or mistaken assumptions you didn't realize you were holding. One way to check this is to see the result others get. Has anyone gotten your same result?

[Anonymous No.16116682]

>>16114924
this. wantonly using infinity in formulae without a rigorous foundation is indicative of debilitating inbreeding in one's immediate ancestry.

[Anonymous No.16116702]

>>16113638
>if you double infinity or if you half it, it doesn't become bigger or smaller
Please prove this in the context on which you are using it

[Anonymous No.16116916]

>>16116169
>each of these is unsupported
Allowing for actual infinities, what's the most something could be subdivided? An actually infinite number of times, right?

And half or double an actual infinity isn't actually a different amount (in a sense, it also IS in a sense, but the whole point is that it brings contradictions). Like the famous example of there being as many even numbers as there are natural numbers if you look at them in "total".

[Anonymous No.16116974]

>>16116916
> An actually infinite number of times, right?
no. you should define the notion of "subdivided an infinite NUMBER of times" rigorously. and while you are at it, do away with the puerile notion that any sort of infinity is a number.

[OP No.16117002]

>>16116974
>no.
Then how many?

>do away with the puerile notion that any sort of infinity is a number.
I actually disagree with that on a grammatical level: numbers are amounts and when people talk about infinity they're talking about an amount. I 100% absolutely don't believe anything ever can be that amount but grammatically it's hard to deny that it is one.

[Anonymous No.16117069]

Infinities aren’t real in nature. There is no point in discussing them other than as a thought experiment?
>There are infinity natural numbers!!!!
No. There are as many as you can bother to calculate, nothing more.

[Anonymous No.16117079]

>>16117002
>then how many?
that question makes no sense until you define what 'subdivided an infinite number of times' means. is it possible you never encountered mathematical definitions?
>numbers are amounts
that's very far from mathematical rigour. natural numbers are an (equivalence class of) initial object in a category. rationals are an equivalence class of pairs of integers wrt a specific equivalence relation. reals are isomorphic to Dedekind cuts of rationals. surreal numbers are a kind of generalization of Dedekind cuts etc.
there is no universal notion of infinity. you introduce it for every case separately, with its own axioms. for instance if you want to build a monoid with the minimum-of-two-numbers operation, you introduce a neutral element such that for any x, min(x, neutral-element) is x. then you have some right to denote this neutral element with ∞. or if you want to formalize the (somewhat primitive) intuition of 'adding up an infinite number of values', then you end up with the notion of the limit of a series.
in your case you haven't come up with jack shit.

[Anonymous No.16117081]

>>16117069
> Infinities aren’t real in nature.
this is such a 'so fucking what?' statement ...

[OP No.16117097]

>>16117079
>that question makes no sense until you define what 'subdivided an infinite number of times' means
Unless you're genuinely having difficulty understanding what's being said you're on a road that we can already prove leads nowhere.
Gödel's incompleteness theorems prove that definitions can get more and more and more rigorous: endlessly, forever.

>natural numbers are an (equivalence class of) initial object in a category. rationals are an equivalence class of pairs of integers wrt a specific equivalence relation. reals are isomorphic to Dedekind cuts of rationals. surreal numbers are a kind of generalization of Dedekind cuts

Well clearly not, since people were talking about many of these first concepts before the latter ones came into being.

This is one thing I really like about finitism, and which in a way makes me feel attracted to ultrafinitism as well: to keep mathematics tied to reality.

>etc.

That "etc" doesn't stop anywhere. There's no point at which you have a complete, rigorous definition of _anything_ using axioms like this.

[OP No.16117100]

>>16116702
You have an army that has ∞ soldiers. Your country is very progressive so it's half men and half women. They all get married and have two kids. All the kids are very patriotic and join the army too. How many soldiers have you got now? ∞.

[Anonymous No.16117133]

>>16117100
What is the minimum average birthrate to sustain such a population?

[OP No.16117162]

>>16117133
depends how good of a general you are

[Anonymous No.16117223]

>>16113638
Interesting so the universe is finite I have a feeling there is more to the universe than what we can see

[OP No.16117304]

>>16117223
For sure! That's one big reason this is so important. It has extremely substantial impacts on how we view our world

[Anonymous No.16117361]

>>16117097
oh you are just a schizo. as you were, then. I won't waste more time with someone who does not understand the need for definitions.

[OP No.16117382]

>>16117361
There's a big difference between the need for definitions and something I've noticed some people on the internet do, where they insist that you can't do mathematics without more and more and more and more specific definitions. Even objecting to 1+1=2 with that sort of reasoning.

But that's a loser's game. We can prove that cycle never stops. Eventually we have to agree "OK I understand what you're talking about" since a definition in an axiomatic system like this can never _truly_ be complete.

So if you have something you would like to see specifically clarified, we can do that. But if you say "natural numbers are an (equivalence class of) initial object in a category. rationals are an equivalence class of pairs of integers wrt a specific equivalence relation. reals are isomorphic to Dedekind cuts of rationals. surreal numbers are a kind of generalization of Dedekind cuts, now keep going with this until you get to the end" then you don't understand what you're asking since there is no end to that process of more and more and more rigorous definitions.

[Anonymous No.16117406]

>>16113638
>we arrive at
no we don't, infinity is not a number even if it has a "number-like" symbol

[Anonymous No.16117408]

>>16115273
>See now how the infinitist argues!
If you really cared, you could look up the Cauchy's construction of R and verify for yourself how the continuous R's built up from discrete N using Peano's axioms. But you don't care, so you deserve to be called a mouth-breathing retard, as they anon correctly pointed out.

[Anonymous No.16117409]

there is a limited number of possible humans. high, but limited.

[Anonymous No.16117411]

>>16113638
>If we accept actual infinities, then eventually we arrive at:
>1=[(1/∞)+(1/∞)+(1/∞)...]
bullshit, wrong from there on out, that does not at all follow from the first statement, nor doesn anybody non-retarded claim it does.

Mongoloid.

[OP No.16117451]

>>16117411
What's the most you can divide 1 by?

[Anonymous No.16117863]

>>16117097
>This is one thing I really like about finitism, and which in a way makes me feel attracted to ultrafinitism as well: to keep mathematics tied to reality.
>>>/x/

[OP No.16118481]

>>16117863
Quite the opposite. Think about it: if we were discussing, say, chemistry, would you keep asking "what is nitrogen?". Of course not, nitrogen is a well-defined concept and the nuances like the various isotopes are well known to anyone with even a slightly more than cursory interest.

Just as chemistry is the study of chemicals and their behavior, mathematics is the study of amounts and their behavior.