R4: the OOP claims that Banach-Tarski implies that 1+1=3. They just repeatedly restate Banach-Tarski, ignoring that I understand what it is, and also ignoring that that's not how addition works.
Ah, but you are assuming the transitive property of equality works on god, but as trinitarian Christians keep insisting to me you can have 3 things all be the same thing, but also not the same as each other, and they get real mad if you ask if they mean some other relationship than actual equality and start spewing the names of ancient people and their supposed heresies.
Not a day goes by without someone refuting the most basic linguistics concepts with the worst sources you've ever seen.
I honestly think it's worse than mathematics, because most people at least have some background in basic arithmetic, and maybe algebra. Nobody knows anything about linguistics, so people just make stuff up 😔
I feel like that's true for the opposite reason? Most people have very little math at all, and tend to be aware of that at least a bit, and also have very little linguistics, but they speak a language, so how hard can linguistics be?
The thing about language is that even though people use their native language proficiently, they have no memory of how they learned it. They have no awareness of the subconscious rules they use that linguists are most interested in. In fact, the only "rules" they are aware of are the ones they were taught in school, and almost by definition, if something needs to be taught, it isn't what linguists would call a "rule."
That's why people consciously understand the grammar and syntax of foreign languages they barely speak better than of their own native language. And it's also why they have such weird notions of what linguists must do.
This is super evident whenever I hear English speakers who are learning another language talking about how stupid gendered articles are. They complain about how contrived and illogical it is. But if you point out how ridiculous the a/an "rules" are, they rarely seem to realize its the same thing.
It might not be so everywhere, but for most people, a/an is fully regular. "A" precedes a vowel, and "an" precedes a consonant. I know some people do use "an" before certain words starting with h, but that's not really the norm. There are also two forms of the word "the," with /θij/ preceding a vowel and /θə/ preceding a consonant.
But it's regular in any case. Gender agreement is certainly regular. It's just not always obvious which gender a given noun has, so it is an extra thing you have to remember (sort of like remembering irregular plurals).
That one at least is useful for philosophical arguments that we could have started with different structures for our math and it may have ended up looking drastically different and our intuitions very different, leading us to contknuously re-examine the foundations we have and their limits and utility.
OOP is essentially doing an internal critique and failing hard.
I like this one cause on top of being wrong, they keep repeating they "don't know how to explain something so basic/low level". It's supposed to be a way to diminish/shame op, but if you truly understand something, you actually ARE able to explain it. It just shows that they only have a superficial understanding of it (which makes sense as they claim something false)
The Banach-Tarski paradox concerns finitely-additive measures rather than the usual countably-additive ones. If you only require your measure to be finitely-additive, locally-finite, and translation-invariant, then every subset of R is measurable. Same with R2, actually. With R3, SO(3) is a nonamenable group, so no such measure exists.
My intuition could be wrong here because I'm not analyst... is this related at all to SO(3) being a simple Lie group? I'm more familiar with representation theory than analysis in general.
It’s because SO(3) contains a copy of F_2, the free group on two generators.
Interestingly, SO(3) is amenable as a topological group (as it’s compact, so has finite Haar measure), so the Banach-Tarski decomposition is necessarily discontinuous.
I could be wrong, but I think every set is measurable in ZF? (EDIT: said that totally wrong as pointed out in the comment below). Or at least it's consistent with ZF that every set is measurable.
But yeah, if every set is measurable in ZFC then 1+1=3.
"Every set is measurable in ZF" would be wild because then ZFC would have a contradiction (since it has both ZF and not measurable sets). But yeah, it's consistent with ZF.
More there are models of ZF where choice fails and all subsets of the reals are lebesgue measurable, you cannot prove from ZF that all sets are measurable as there are models where they are and models where there are non-measurable ones (e.g. all models of ZFC).
Edit: for some reason your edit didn't show up on my end until I posted.
Consistent with ZF+DC+”there is an inaccessible cardinal”. That’s of course still consistent with ZF, but the inaccessible cardinal is important to the argument as shown by Shelah.
That's true if you add DC, or at least that Lebesgue measure is sigma-additive (athough I'm not sure that's enough). Otherwise the innaccessible is not needed. But I guess people wouldn't truly speak of Lebesgue measure when its definition doesn't yield a sigma additive measure.
Sorry, non-set-theorist here: if ZF+all measurable is equiconsistent with ZF+inaccessible which is equiconsistent with ZF, how can the cardinal be important?
Define = to be the equivalence relation on the naturals as follows: we say n=m if you can decompose n balls into a finite union of disjoint subsets that can be put together to yield m balls. The quotient of the naturals by this equivalence relation collapses to the zero-ring and we recover the correct formula of 1+1=3. QED (/s if it wasn't obvious)
More like 1=2, but you can derive anything from that. The whole reason it’s a paradox is that we end up increasing the measure of an object while doing rotations and messing about with null sets, both of which don’t change measure. We do depart from the mesurable sets, though.
There's nothing wrong with using an addition symbol to represent disjoint union, and it behaves fairly similarly to addition of naturals. In fact, the whole idea behind decategorification is that addition of naturals is nothing more than to isomorphism classes of sets. Addition really is just disjoint union of sets.
So I don't think you rebutted the badmather very effectively. I also think appealing to your credentials is a low form of argumentation.
The real mistake the badmather made wasn't using + to represent disjoint union. It was using 1 to represent his infinite set. Like I was saying, people use addition to represent disjoin union of sets all the time. This is the basis of cardinal arithmetic. In cardinal arithmetic, as well as basic thought experiments like the Hilbert hotel, statements like aleph0 + aleph0 = aleph0 are easily checked.
And that's the type of weirdness that is involved in Banach-Tarski, decomposing an infinite set into two constituent parts and showing that each part is isomorphic to the whole. It's more complex than a simple cardinality argument, because of the stuff about nonmeasurable sets and nonamenable groups, and how the resulting reconstituted sets are not just same cardinality as the starting sphere, but in fact they are the same measure, and can be constructed with only finitely many rigid moves.
So TLDR, his mistake wasn't using + for disjoint union. That's a standard thing. His mistake was using 1, a symbol for a finite set, for a counterintuitive equation that relies in a fundamental way on the summands being infinite.
R4: the OOP claims that Banach-Tarski implies that 1+1=3. They just repeatedly restate Banach-Tarski, ignoring that I understand what it is, and also ignoring that that's not how addition works.
claim: 1+1 = 3
proof: geometric set theory.
Q.E.D
claim: Taylor Swift is God
proof: u/Taytay_Is_God on Reddit says so
Q.E.D.
Well, guess I'm no longer an atheist, but that still won't make me listen to her music.
You have been banned from r/infiniteones
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Corollary:
Taylor Swift is God.
You are an atheist.
From 2, God does not exist.
From 3 and 1, Taylor Swift does not exist.
QED
Ah, but you are assuming the transitive property of equality works on god, but as trinitarian Christians keep insisting to me you can have 3 things all be the same thing, but also not the same as each other, and they get real mad if you ask if they mean some other relationship than actual equality and start spewing the names of ancient people and their supposed heresies.
God is the Father but also the Mother. Can't get much more trans than that.
I can’t find any counter examples so this proof is very convincing
Alternate proof : it's in the name.
Did you get bored with SPP?
1 father + 1 mother = 3 total family members.
Sex proves that 1+1=3
E = mc2 + Sex
Can’t prove me wrong
By the famous LinkedIn theorem, E=mc2 + AI
This therefore implies AI = Sex
Idk I don’t buy it, did you check your work?
Wait, are you saying that my AI girlfriend is NOT real???
This shows that at rest, sex = 0. You have to give what you take.
It's true because I get 0 sex
Damn, I've been missing a step in all of my calculations since high school 😔
what
wait a minute...
Clearly, sex = sqrt((mc2)2 + p2 c2) - mc2
They are one person
They are too alone
They are three together
They are for each other
Holy shit... Banach-Tarski in popular culture this whole time.
*sigh*. I sympathize. Trying to respond to things on math subreddits can quickly lead to insanity.
This guy can count sets, but he doesn't know that he can't measure them.
1∞ + 1∞ = 3∞
Bazinga, I can't make this easier, pleb.
1*0 + 1*0 = 3*0 Depending how you interpret 'sizeless set'. Infinite points of zero size or something else.
I mean if you are willing to abuse notation and definitions then proving 1+1=3 is easy.
Take 1 pile of stuff. Split it into thirds. You now have 3 piles of stuff. Checkmate, mathematics!
That didn’t even require addition! You’ve just proven 1=3.
Ah ha! I'm even greater than I thought I was.
/s I originally had a different example and rewrote it for brevity. Woops.
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You may have a "PhD," but they googled something for 2 minutes.
This is reddit on literally every single topic known to man.
Not a day goes by without someone refuting the most basic linguistics concepts with the worst sources you've ever seen.
I honestly think it's worse than mathematics, because most people at least have some background in basic arithmetic, and maybe algebra. Nobody knows anything about linguistics, so people just make stuff up 😔
I feel like that's true for the opposite reason? Most people have very little math at all, and tend to be aware of that at least a bit, and also have very little linguistics, but they speak a language, so how hard can linguistics be?
An F in English? Bobby, you speak English.
Ugh I remember in 6th grade my mom said exactly that but said 92 instead of F and my name isn’t Bobby.
My middle school had
A: 97 and up
A-: 93 to 96
B+: 91 to 92
B: 88 to 90
B-: 85 to 87
C+: 82 to 84
C: 78 to 81 And I don’t remember what was lower than that. It’s so many years later but hot dang it was traumatic.
Oh absolutely, but if linguistics was available outside of higher education I think it would prevent a lot of that sentiment.
"I speak language therefore I know linguistics"
The thing about language is that even though people use their native language proficiently, they have no memory of how they learned it. They have no awareness of the subconscious rules they use that linguists are most interested in. In fact, the only "rules" they are aware of are the ones they were taught in school, and almost by definition, if something needs to be taught, it isn't what linguists would call a "rule."
That's why people consciously understand the grammar and syntax of foreign languages they barely speak better than of their own native language. And it's also why they have such weird notions of what linguists must do.
This is super evident whenever I hear English speakers who are learning another language talking about how stupid gendered articles are. They complain about how contrived and illogical it is. But if you point out how ridiculous the a/an "rules" are, they rarely seem to realize its the same thing.
It might not be so everywhere, but for most people, a/an is fully regular. "A" precedes a vowel, and "an" precedes a consonant. I know some people do use "an" before certain words starting with h, but that's not really the norm. There are also two forms of the word "the," with /θij/ preceding a vowel and /θə/ preceding a consonant.
But it's regular in any case. Gender agreement is certainly regular. It's just not always obvious which gender a given noun has, so it is an extra thing you have to remember (sort of like remembering irregular plurals).
Exactly the opposite. "A" comes before a consonant, and "an" comes before a vowel.
Lol whoops. I made an mistake. A error.
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Axiom of choice states you can choose 1+1 to equal whatever you want
Which actually proves that p = np if we set p = 1+1.
Where "1" is an element of set {"n", "p"}
Oh yeah? Well I choose to reject the axiom of choice. Checkmate.
But only as a treat
Rule 1 of posting incorrect math on Reddit: a professor with a goofy name will correct you.
False.
true
proof by construction
construction? construction of what?
a refrigerator, perchance?
March 3rd 2032
False
Proof by contradiction
True
This is just the "1 mom+1 dad becomes mom, dad and baby so 1+1=3" argument.
That one at least is useful for philosophical arguments that we could have started with different structures for our math and it may have ended up looking drastically different and our intuitions very different, leading us to contknuously re-examine the foundations we have and their limits and utility.
OOP is essentially doing an internal critique and failing hard.
Dude just do geometric set theory to it. How fucking hard is that you idiot!
I like this one cause on top of being wrong, they keep repeating they "don't know how to explain something so basic/low level". It's supposed to be a way to diminish/shame op, but if you truly understand something, you actually ARE able to explain it. It just shows that they only have a superficial understanding of it (which makes sense as they claim something false)
Also, I think "1+1=3 is not true" is much more low-level than the Banach-Tarski paradox.
Yeah claiming banach tarski is low level is wild
Proof by contradiction.
Everybody knows 1+1 = 2
It takes some ungodly number of pages to show that 1+1 = 2
I was more willing to read about Banach-Tarski than 1+1, and I almost understood it.
Therefore Banach-Tarski is lower level than 1+1 = 2
QED
Where is the contradiction?, you ask.
I contradicted you, didn't I?
Proving 1+1=3 by Fermat’s Little Theorem
QED
I misread that as Fermat's Last Theorem and got so confused at first
Honestly, sets are an area where 1+1=1 is more justified.
|{x}| = 1, |{y}| = 1, so surely |{x} ∪ {y}| = 2, right?
Well . . . not if x = y.
So if we use + for ordinary unions instead of disjoint unions, then 1 + 1 can be either 1 or 2.
okay but have you considered banach tarski paradox? checkmate matheist.
Well, "If every set is measureble, then 1+1=3." is a true statement.
The Banach-Tarski paradox concerns finitely-additive measures rather than the usual countably-additive ones. If you only require your measure to be finitely-additive, locally-finite, and translation-invariant, then every subset of R is measurable. Same with R2, actually. With R3, SO(3) is a nonamenable group, so no such measure exists.
My intuition could be wrong here because I'm not analyst... is this related at all to SO(3) being a simple Lie group? I'm more familiar with representation theory than analysis in general.
It’s because SO(3) contains a copy of F_2, the free group on two generators.
Interestingly, SO(3) is amenable as a topological group (as it’s compact, so has finite Haar measure), so the Banach-Tarski decomposition is necessarily discontinuous.
Oh, thanks! It took me a while to process that but I think that makes sense.
Don't feel bad. I am an analyst and I also have no idea wtf is going on here.
I could be wrong, but I think every set is measurable in ZF?(EDIT: said that totally wrong as pointed out in the comment below). Or at least it's consistent with ZF that every set is measurable.But yeah, if every set is measurable in ZFC then 1+1=3.
Yeah, there is a model of ZF in which every set of reals is Lebesgue measurable.
https://www.jstor.org/stable/1970696
Ah, published in Annals. Nice!
"Every set is measurable in ZF" would be wild because then ZFC would have a contradiction (since it has both ZF and not measurable sets). But yeah, it's consistent with ZF.
Ah right lol, I phrased that pretty badly
More there are models of ZF where choice fails and all subsets of the reals are lebesgue measurable, you cannot prove from ZF that all sets are measurable as there are models where they are and models where there are non-measurable ones (e.g. all models of ZFC).
Edit: for some reason your edit didn't show up on my end until I posted.
Consistent with ZF+DC+”there is an inaccessible cardinal”. That’s of course still consistent with ZF, but the inaccessible cardinal is important to the argument as shown by Shelah.
That's true if you add DC, or at least that Lebesgue measure is sigma-additive (athough I'm not sure that's enough). Otherwise the innaccessible is not needed. But I guess people wouldn't truly speak of Lebesgue measure when its definition doesn't yield a sigma additive measure.
Yes, my mistake. Of course I meant to put DC in there. Will edit in a moment.
Sorry, non-set-theorist here: if ZF+all measurable is equiconsistent with ZF+inaccessible which is equiconsistent with ZF, how can the cardinal be important?
Here’s a link to the paper where Shelah proves it.
Define = to be the equivalence relation on the naturals as follows: we say n=m if you can decompose n balls into a finite union of disjoint subsets that can be put together to yield m balls. The quotient of the naturals by this equivalence relation collapses to the zero-ring and we recover the correct formula of 1+1=3. QED (/s if it wasn't obvious)
The Beatles proved that 1+1+1=3
Dunno whether they proved that, or posited it as an axiom…I only remember that line and not those around it.
I see this same sort of attitude all the time when arguing with flat-earthers. Guess it's not limited to those morons, sadly.
You have to accept the fact that this is what normal people think winning an argument looks like
"i dont know how to explain this" thats because you dont know how it works.
More like 1=2, but you can derive anything from that. The whole reason it’s a paradox is that we end up increasing the measure of an object while doing rotations and messing about with null sets, both of which don’t change measure. We do depart from the mesurable sets, though.
I want to argue against him, but it’s posted in /r/truths.
They are absolutely now trolling in this thread. Claiming they didn't say what they said right there in the screenshot.
yeah I know lol
If I take red paint and yellow paint, I can make red paint, yellow paint, and orange paint. Therefore, 1 + 1 = 3.
BREAKING: taytayisgod outside of infinitenines
There's nothing wrong with using an addition symbol to represent disjoint union, and it behaves fairly similarly to addition of naturals. In fact, the whole idea behind decategorification is that addition of naturals is nothing more than to isomorphism classes of sets. Addition really is just disjoint union of sets.
So I don't think you rebutted the badmather very effectively. I also think appealing to your credentials is a low form of argumentation.
The real mistake the badmather made wasn't using + to represent disjoint union. It was using 1 to represent his infinite set. Like I was saying, people use addition to represent disjoin union of sets all the time. This is the basis of cardinal arithmetic. In cardinal arithmetic, as well as basic thought experiments like the Hilbert hotel, statements like aleph0 + aleph0 = aleph0 are easily checked.
And that's the type of weirdness that is involved in Banach-Tarski, decomposing an infinite set into two constituent parts and showing that each part is isomorphic to the whole. It's more complex than a simple cardinality argument, because of the stuff about nonmeasurable sets and nonamenable groups, and how the resulting reconstituted sets are not just same cardinality as the starting sphere, but in fact they are the same measure, and can be constructed with only finitely many rigid moves.
So TLDR, his mistake wasn't using + for disjoint union. That's a standard thing. His mistake was using 1, a symbol for a finite set, for a counterintuitive equation that relies in a fundamental way on the summands being infinite.
Someone should slap that dude with a hardcover Federer.
Only if you allow (1/0)+(1/0)=(3/0)
I love u/Taytay_is_God
oh uh thanks...
btw, what's an anagram of Banach-Tarski?
Banach-Tarski Banach-Tarski
i think its funny how clear it is that they have no idea what theyre saying and are just stating something they saw in a tiktok
As someone who hasn't learned Banach-Tarski, is this essentially saying 1×∞+1×∞=3×∞, therefore 1+1=3?
except that the 1 is actually represents an infinite set
heh, I recognize you from my "Taylored series" joke.
I'm not knowledgeable about BT though, so that's all I can say.
u/Epicnessofcows lol
If 1 = 3/2, problem solved!
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There is finite model theory, if you are interested in restricting yourself in a similar way.
But "refusing to do anything" it a bit tricky, since you wouldn't be able to quantify over functions, open sets, sequences, etc.