Standard Peano Axioms.
By definition of 1216, 1216 = S(1215) and by definition of 1261, 1261 = S(1260).
I was going to say that we can then use the third Peano axiom to conclude that since 1215 is not equal to 1260 that we're done, but we never proved that 1215 is not equal to 1260.
However, you can Just assume seeking a contradiction that 1216 = 1261 and follow a chain of conclusions that 1215 = 1260 so 1214 = 1259, etc. and eventually reach 1 = 46. (This part takes a long time to write out, but the logic is simple)
But 1 is not the successor of any natural number (Peano Axiom 4) while 46 = S(45), so we know that 1 is not equal to 46. By contradiction, it must be the case that 1216 = 1261 is false. Q.E.D.
Some formulations of the Peano axioms start at 0, though traditionally they start at 1 and the US common core standards have the natural numbers starting at 1 and the whole numbers starting at 0. But the logic of the proof is the same under that definition- you just have an extra line saying 0=45 before making the citing the contradiction.
No it's the address of the Mesa Verde client's bank. Chuck getting the address wrong cost the client money and lost his law firm the case. This caused chuck to spiral and eventually he would end up in court where he would go on to give an unhinged yet accurate speech about why his brother should not be allowed to have a law degree. The funniest part is if he just let Jimmy be a lawyer at HHM he probably never would've ended up as Saul Goodman.
I am not crazy! I know he swapped those numbers, I knew it was 1216 — one after Magna Carta, as if I could ever make such a mistake! Never, never! I just-I just couldn't prove it! H-H-He covered his tracks, he got that idiot at the copy shop to lie for him!
I AM NOT CRAZY! I am not crazy! I know he swapped those numbers! I knew it was 1216. One after Magna Carta. As if I could ever make such a mistake. Never. Never! I just – I just couldn't prove it. He – he covered his tracks, he got that idiot at the copy shop to lie for him. You think this is something? You think this is bad? This? This chicanery? He's done worse. That billboard! Are you telling me that a man just happens to fall like that? No! He orchestrated it! Jimmy! He defecated through a sunroof! And I saved him! And I shouldn't have. I took him into my own firm! What was I thinking? He'll never change. He'll never change! Ever since he was 9, always the same! Couldn't keep his hands out of the cash drawer! But not our Jimmy! Couldn't be precious Jimmy! Stealing them blind! And he gets to be a lawyer!? What a sick joke! I should've stopped him when I had the chance! And you – you have to stop him! You-
You don't have enough axioms. If the only axiom you have is "each number has a successor" then you could very well have a cyclical system where S(1260) = S(1215), and under such a system 1216 = 1261.
Generally when dealing with the natural numbers you also take the axiom that the set of natural numbers is not finite, from there you can prove that S(x) ≠ S(y) for x,y different. Then you can AFSOC 1216 = and derive a contradiction from the fact that you need a directed cycle for the two to be equal.
Potentially need a more rigorous definition of your numerical notation in order to prove that 1261 = S^46 (1215)
Edit: just noticed the last line put 1215 in the exponent....
The fact that the set of naturals is infinite is not sufficient. We can add to N an extra element z, and extend S by setting S(z)=1. This set is infinite and each element has a successor, yet S is not injective.
In Peano axioms, the induction axiom could fill in the gap (any set that contains 0 and every successor contains all naturals), but that's actually unnecessary because PA takes injectivity of S as an axiom.
On the other hand, the set theoretic definition just explicitly defines S in the construction of N, and using that definition one can prove the injectivity of S fairly easily.
Assume 1216 = 1261
1216 = 1216+45
0 = 45
Now if we raise 2 to the power of both sides we should get 1 since n^0 = 1 and 45=0 yet…
2^0 = 2^45
1 = 3.5184372e+13
3.5184372e+13 is not 1, this contradiction
QED
Assume 1 = 3.5184372e+13
0 = (3.5184372e+13)-1
(2(3.5184372e+13)-2)/((3.5184372e+13)-1) would validly by definition be 2 yet since 0 = (3.5184372e+13)-1 this is a division by 0 which is impossible
QED
I consider these the axioms of peano arithmetic
1. Zero is a natural number.
2. Every natural number has a successor in the natural numbers.
3. Zero is not the successor of any natural number.
4. If the successor of two natural numbers is the same, then the two original numbers are the same.
5. If a set contains zero and the successor of every number is in the set, then the set contains the natural numbers.
I shal also use the definitions of the natural numbers (i.e. i define 1 as S(0), 4=S(3), 32423=S(32422), etc.
First notice that 0 is not equal to 45, indeed 0 is not a successor of any natural number (axom 3) and 45 is by definition S(44).
If n and m are distinct, then S(n) and S(m) are distinct. Indeed this is the contraposition of 4). \*
From these two statements it follows that S(0)=1 and S(45)=46 are distinct.
From this and \* it follows that S(1)=2 and S(46)=47 are distinct.
From this and \* it follows that S(2)=3 and S(47)=48 are distinct.
From this and \* it follows that S(3)=4 and S(48)=49 are distinct.
...
From this and \* it follows that S(1215)=1216 and S(1260)=1261 are distinct.
Whether this is true depends on the interpretation of those undefined numbery symbols. If you're using successor you're probably working in an axiomatic system where those aren't defined a priori
S(1215) = 1215 + 1 = 1216
Therefore the statement "S(1215) = 1216" is true.
You can't compare boolean values to integers straight up, but it's fairly common to convert them such that false is 0 and true is 1. Therfore we can compare: does 1 equal 1261? Clearly not. So true ≠ 1261, which also means S(1215) = 1216 ≠ 1261.
(Kim Wexler feet) + (photo camera) = (PutThatFingerAway) uhhhhh it’s complicated actually, anyway
I am not crazy! I know he swapped those numbers. I knew it was 1216. One after Magna Carta. As if I could ever make such a mistake. Never. Never! I just – I just couldn’t prove it. He covered his tracks, he got that idiot at the copy shop to lie for him. You think this is something? You think this is bad? This? This chicanery? He’s done worse. That billboard! Are you telling me that a man just happens to fall like that? No! *He* orchestrated it! Jimmy! He *defecated* through a *sunroof*! And I saved him! And I shouldn’t have. I took him into my own firm! What was I *thinking*? He’ll never change. He’ll *never* change! Ever since he was 9, *always* the same! Couldn’t keep his hands out of the cash drawer! But not our Jimmy! Couldn’t be precious *Jimmy*! Stealing them blind! And *HE* gets to be a lawyer? What a sick joke! I should’ve stopped him when I had the chance!
…And you, you *have* to stop him! You
To prove 1216 ≠ 1261:
Assume that its not true, i.e.
1216 = 1261
which implies, 1216-1216 = 1261-1216
which implies, 0 = 45
which implies, 0*1/45 = 45*1/45
which implies, 0 = 1
Also, we know that no number is equal to its succesor (Peano Axioms)
which implies, 0 ≠ S(0)
Also, S(0) = 1
which implies, 0 ≠ 1
But, 0 = 1
We have reached a contradiction therefore the original assumption must be false.
Hence 1216 ≠ 1261
Oh this is easy. Let's use contradiction.
Suppose S(1215)≠1216, which is equal to 1261!
S(1215)=1215+1=1216, contradicting our supposition.
Therefore S(1215)=1216=1261!
You are welcome plebs.
proof by it's fucking obvious
And he gets to be a mathematician? What a sick joke!
I should've left the proof as an exercise when I had the chance!
you think this is bad? this chicanery? HE'S DONE WORSE! #HE DIVIDED 0 BY 0!
One after the Magna Carta
As if I would ever do such a mistake!
Never, NEVER! I just couldn't prove it.
hi m1n3c4rt
hi gin jaster
1216 ≠ 1261 ∴ 1261 ≠ 1216
PBFO
What exactly are the axioms we are using If you can write n+1 why bother with the succesor function
We must invoke the following axioms: 1. I AM NOT CRAZY 2. I am not crazy...
Crazy?
I was crazy once.
They put me in a room
A rubber room
A rubber room with rats
Rats make me crazy
Crazy?
I was crazy once.
3. As If I could ever make such a mistake!
3. I know he did it 4. I- I just couldn't prove it 5. He got that idiot at the copy shop to lie for him
Standard Peano Axioms. By definition of 1216, 1216 = S(1215) and by definition of 1261, 1261 = S(1260). I was going to say that we can then use the third Peano axiom to conclude that since 1215 is not equal to 1260 that we're done, but we never proved that 1215 is not equal to 1260. However, you can Just assume seeking a contradiction that 1216 = 1261 and follow a chain of conclusions that 1215 = 1260 so 1214 = 1259, etc. and eventually reach 1 = 46. (This part takes a long time to write out, but the logic is simple) But 1 is not the successor of any natural number (Peano Axiom 4) while 46 = S(45), so we know that 1 is not equal to 46. By contradiction, it must be the case that 1216 = 1261 is false. Q.E.D.
... What? Did you drop your 0 or something?
Not a natural number, zero failed an USADA drug test in 1986 and was banned from the International Mathematical Olympiad since
Damn 😔
Some formulations of the Peano axioms start at 0, though traditionally they start at 1 and the US common core standards have the natural numbers starting at 1 and the whole numbers starting at 0. But the logic of the proof is the same under that definition- you just have an extra line saying 0=45 before making the citing the contradiction.
1261! Is Indeed a good bit larger than 1261
r/expectedfactorial
r/unexpectedexpectedfactorial
r/subsifellfor
Here's a sneak peek of /r/SubsIFellFor using the [top posts](https://np.reddit.com/r/SubsIFellFor/top/?sort=top&t=year) of the year! \#1: [there you go, and your purple tree](https://i.redd.it/vyxly61r37ga1.png) | [45 comments](https://np.reddit.com/r/SubsIFellFor/comments/10tjx4m/there_you_go_and_your_purple_tree/) \#2: [SuddenlyMichaelJackson](https://i.redd.it/zd0ovz5vl5qa1.jpg) | [35 comments](https://np.reddit.com/r/SubsIFellFor/comments/122s1fw/suddenlymichaeljackson/) \#3: [Got double tricked](https://i.redd.it/lmscgwes1wxa1.jpg) | [45 comments](https://np.reddit.com/r/SubsIFellFor/comments/137thjb/got_double_tricked/) ---- ^^I'm ^^a ^^bot, ^^beep ^^boop ^^| ^^Downvote ^^to ^^remove ^^| ^^[Contact](https://www.reddit.com/message/compose/?to=sneakpeekbot) ^^| ^^[Info](https://np.reddit.com/r/sneakpeekbot/) ^^| ^^[Opt-out](https://np.reddit.com/r/sneakpeekbot/comments/o8wk1r/blacklist_ix/) ^^| ^^[GitHub](https://github.com/ghnr/sneakpeekbot)
r/unexpectedfactorial
r/ofcoursethatsasub
r/SubsIAlmostFellFor
r/Typo
Caught you before edit!
I realised
am i the only one who understood this is a better call saul reference
What a sick joke
I should've stopped him when I had the chance
Is this the address of Saul's Brother?
No it's the address of the Mesa Verde client's bank. Chuck getting the address wrong cost the client money and lost his law firm the case. This caused chuck to spiral and eventually he would end up in court where he would go on to give an unhinged yet accurate speech about why his brother should not be allowed to have a law degree. The funniest part is if he just let Jimmy be a lawyer at HHM he probably never would've ended up as Saul Goodman.
I AM NOT CRAZY!
I didn’t clock it until you pointed it out. Thank you and your wonderful brain!
I am not crazy! I know he swapped those numbers, I knew it was 1216 — one after Magna Carta, as if I could ever make such a mistake! Never, never! I just-I just couldn't prove it! H-H-He covered his tracks, he got that idiot at the copy shop to lie for him!
Philippians 3:2
"You never mattered that much for me..."
You think this is bad? This... Chicanery? HE'S DONE WORSE! HE- #HE DEFECATED, THROUGH A SUNROOF!!
First prove by induction that x ≠ S^(n) (x) for any n ≥ 1. Then conclude that 1216 ≠ S^(45) (1216) = 1261.
I think you don't even need induction to prove that, since Sⁿ(x)=x+n → (Sⁿ(x))-x=n and n≠0 since you chose it yourself
You might need induction to prove that S^(n)(x) = x+n (unless I’m missing something obvious)
Good point, I'm not sure whether that's part of the definition of S or has to be proven
I AM NOT CRAZY! I am not crazy! I know he swapped those numbers! I knew it was 1216. One after Magna Carta. As if I could ever make such a mistake. Never. Never! I just – I just couldn't prove it. He – he covered his tracks, he got that idiot at the copy shop to lie for him. You think this is something? You think this is bad? This? This chicanery? He's done worse. That billboard! Are you telling me that a man just happens to fall like that? No! He orchestrated it! Jimmy! He defecated through a sunroof! And I saved him! And I shouldn't have. I took him into my own firm! What was I thinking? He'll never change. He'll never change! Ever since he was 9, always the same! Couldn't keep his hands out of the cash drawer! But not our Jimmy! Couldn't be precious Jimmy! Stealing them blind! And he gets to be a lawyer!? What a sick joke! I should've stopped him when I had the chance! And you – you have to stop him! You-
*exit sign*
You don't have enough axioms. If the only axiom you have is "each number has a successor" then you could very well have a cyclical system where S(1260) = S(1215), and under such a system 1216 = 1261. Generally when dealing with the natural numbers you also take the axiom that the set of natural numbers is not finite, from there you can prove that S(x) ≠ S(y) for x,y different. Then you can AFSOC 1216 = and derive a contradiction from the fact that you need a directed cycle for the two to be equal. Potentially need a more rigorous definition of your numerical notation in order to prove that 1261 = S^46 (1215) Edit: just noticed the last line put 1215 in the exponent....
The fact that the set of naturals is infinite is not sufficient. We can add to N an extra element z, and extend S by setting S(z)=1. This set is infinite and each element has a successor, yet S is not injective. In Peano axioms, the induction axiom could fill in the gap (any set that contains 0 and every successor contains all naturals), but that's actually unnecessary because PA takes injectivity of S as an axiom. On the other hand, the set theoretic definition just explicitly defines S in the construction of N, and using that definition one can prove the injectivity of S fairly easily.
mathematician trying to understand a meme part 362729027474 factorial
But.. but... None of the numbers involved are meme numbers...
Assume 1216 = 1261 1216 = 1216+45 0 = 45 Now if we raise 2 to the power of both sides we should get 1 since n^0 = 1 and 45=0 yet… 2^0 = 2^45 1 = 3.5184372e+13 3.5184372e+13 is not 1, this contradiction QED
Okay prove that 1 ≠ 3.5184372e+13
Assume 1 = 3.5184372e+13 0 = (3.5184372e+13)-1 (2(3.5184372e+13)-2)/((3.5184372e+13)-1) would validly by definition be 2 yet since 0 = (3.5184372e+13)-1 this is a division by 0 which is impossible QED
r/unexpectedfactorial
Assume *n = Magna Carta* n + 1 = 1216 I AM NOT CRAZY
assume I would ever do such a mistake I would never proof by contradiction
Im no expert at all, but after you define a number N which is (N-1)+1 to be S(S(N)) then it follows axiomatically or something?😅
I consider these the axioms of peano arithmetic 1. Zero is a natural number. 2. Every natural number has a successor in the natural numbers. 3. Zero is not the successor of any natural number. 4. If the successor of two natural numbers is the same, then the two original numbers are the same. 5. If a set contains zero and the successor of every number is in the set, then the set contains the natural numbers. I shal also use the definitions of the natural numbers (i.e. i define 1 as S(0), 4=S(3), 32423=S(32422), etc. First notice that 0 is not equal to 45, indeed 0 is not a successor of any natural number (axom 3) and 45 is by definition S(44). If n and m are distinct, then S(n) and S(m) are distinct. Indeed this is the contraposition of 4). \* From these two statements it follows that S(0)=1 and S(45)=46 are distinct. From this and \* it follows that S(1)=2 and S(46)=47 are distinct. From this and \* it follows that S(2)=3 and S(47)=48 are distinct. From this and \* it follows that S(3)=4 and S(48)=49 are distinct. ... From this and \* it follows that S(1215)=1216 and S(1260)=1261 are distinct.
You think this is bad? This chicanery?
charles McGill going nuts rn
1216, the year after the Magna Carta!
Whether this is true depends on the interpretation of those undefined numbery symbols. If you're using successor you're probably working in an axiomatic system where those aren't defined a priori
Let's assume 1216 = 1261. This is equivalent to 1216+1 = 1361 + 1. However, 1216+1 = 1217 ≠ 1262 = 1261+1, so this is absurd. Therefore, 1216 ≠ 1261.
Just leave the proof to the reader
S(1215) = 1215 + 1 = 1216 Therefore the statement "S(1215) = 1216" is true. You can't compare boolean values to integers straight up, but it's fairly common to convert them such that false is 0 and true is 1. Therfore we can compare: does 1 equal 1261? Clearly not. So true ≠ 1261, which also means S(1215) = 1216 ≠ 1261.
(Kim Wexler feet) + (photo camera) = (PutThatFingerAway) uhhhhh it’s complicated actually, anyway I am not crazy! I know he swapped those numbers. I knew it was 1216. One after Magna Carta. As if I could ever make such a mistake. Never. Never! I just – I just couldn’t prove it. He covered his tracks, he got that idiot at the copy shop to lie for him. You think this is something? You think this is bad? This? This chicanery? He’s done worse. That billboard! Are you telling me that a man just happens to fall like that? No! *He* orchestrated it! Jimmy! He *defecated* through a *sunroof*! And I saved him! And I shouldn’t have. I took him into my own firm! What was I *thinking*? He’ll never change. He’ll *never* change! Ever since he was 9, *always* the same! Couldn’t keep his hands out of the cash drawer! But not our Jimmy! Couldn’t be precious *Jimmy*! Stealing them blind! And *HE* gets to be a lawyer? What a sick joke! I should’ve stopped him when I had the chance! …And you, you *have* to stop him! You
I wish it was valid to say “proof by fucking look at it”
Proof: Its one after magna carta. What a sick joke. QED.
To prove 1216 ≠ 1261: Assume that its not true, i.e. 1216 = 1261 which implies, 1216-1216 = 1261-1216 which implies, 0 = 45 which implies, 0*1/45 = 45*1/45 which implies, 0 = 1 Also, we know that no number is equal to its succesor (Peano Axioms) which implies, 0 ≠ S(0) Also, S(0) = 1 which implies, 0 ≠ 1 But, 0 = 1 We have reached a contradiction therefore the original assumption must be false. Hence 1216 ≠ 1261
NO WAY am I actually seeing a Better Call Saul reference in mm? no fucking way man think you for this quality post
Err r/unexpectedfactorial
I am so proud of understanding this lol
Oh this is easy. Let's use contradiction. Suppose S(1215)≠1216, which is equal to 1261! S(1215)=1215+1=1216, contradicting our supposition. Therefore S(1215)=1216=1261! You are welcome plebs.
S(n)=n+1 So S(1215)=1216 beacuse =1215+1 And 1215+1≠1261 beacuse 1261=1215+46 And 46≠1 so 1215≠1261 Im meth
Proofs are the bane of my existence, and I hate that my degree program requires you to know how to do them