Proofs That 2=1

[PirateMonk]

Traditional (Divide by Zero)
1. a=b
2. a2=ab
3. 2a2=a2+ab
4. 2a2-2ab=a2-ab
5. 2(a2-ab)=(a2-ab)
6. 2=1

Abusing Electromagnetic Equations
The force exerted on a wire in a uniform magnetic field is equal to the product of the field strength, the current in the wire, and the length of the wire (F=BIl). Imagine two parallel wires with the same current in them, on of length l, one on length 2l. Because the current is the same, each wire creates a magnetic field at the other one of the same strength, B. By Newton's Third Law, the forces each exerts on the other are equal: 2BIl=BIl. Dividing through by the variables, this yields 2=1.

Liar's Paradox
Let us assume that 2=1 is false. We can construct the sentence 'This sentence is false or 2=1.' Given our assumption, the sentence is true if the first clause is true and false if it is false, which is paradoxical. This contradiction arises entirely because of our initial assumption, and is easily avoided by instead assuming its negation. Since any assumption which leads to a contradiction is flawed, we are forced to discard it. Therefore, 2=1.

Are there any other good ones?

[Qaera]

.99 repeating equals 1.
6x9=42

~ ♅

[Serpentine]

I prefer my version:
http://fc09.deviantart.net/fs30/f/2008/107/d/f/It__s_true__by_Serpentine16.jpg

[Don Julio Anejo]

I prefer my version:
http://fc09.deviantart.net/fs30/f/2008/107/d/f/It__s_true__by_Serpentine16.jpg
2.4 in A1 and B1 and set the whole thing to round to the nearest number? :smallamused:

[Keld Denar]

e^(i pi) = cos(pi) + isin(pi) = -1 + 0i = -1

e^(3i pi) = cos(3 pi) + isin(3 pi) = -1 + 0i = -1

Since -1 = -1

then e^(i pi) = e^(3i pi)

ln(e^(i pi)) = ln(e^(3i pi))

i pi = 3i pi

1 = 3

Eulered!

[ZombyWoof]

Excel's powers cannot be quantified! (http://www.youtube.com/watch?v=tSGraO5mHcs)

[Elder Tsofu]

Playing with the viewing settings in Excel? Surely you can do better than that Serpentine? :smallwink:

[Lady Moreta]

I prefer my version:

Teehee... Peregrine has that on a t-shirt (http://www.thinkgeek.com/tshirts-apparel/unisex/generic/60f5/) - it confuses small children (and me)

[Mono Vertigo]

I DO NOT BELIEVE YOU.
Y'all are trying to make me question reality and my sanity! Well, YOU WON'T SUCCEED. :smallfurious::smallbiggrin:
[/has a very bad relationship with maths]
No, seriously, I don't hate any of you. It's the maths that hate me. MATHS. WITH YOUR ILLOGICAL... LOGIC... LOGIC.

[The_Final_Stand]

.99 repeating equals 1. This is true.
X = 0.999...
10X = 9.999...
9X = 9
X = 1
1 = 0.999...


6x9=42. In base 13.



Answers in bold.

[Manga Shoggoth]

2 = 1, because I have a degree and I said so*.





* Argument from Authority

[Douglas]

Excel's powers cannot be quantified! (http://www.youtube.com/watch?v=tSGraO5mHcs)
Bah, on seeing the title of that vid I was expecting it to be the Excel 97 easter egg (http://www.youtube.com/watch?v=c6nY0QkG9nQ).

[Renegade Paladin]

Traditional (Divide by Zero)
1. a=b
2. a2=ab
3. 2a2=a2+ab
4. 2a2-2ab=a2-ab
5. 2(a2-ab)=(a2-ab)
6. 2=1
http://img.photobucket.com/albums/v350/RenegadePaladin/DivideByZero.jpg

:smalltongue:

[tensai_oni]

Abusing Electromagnetic Equations


Wires do not generate simple, uniform magnetic fields. This proof seems true only because your model is extremely simplified.



Liar's Paradox


'This sentence is false or 2=1.' has a truth value of 0. Neither the sentence is false nor 2=1.

But it doesn't mean the sentence is true. Its truth value is unknown. So there is no paradox, only rejection of the law of excluded middle.

[PirateMonk]

Wires do not generate simple, uniform magnetic fields. This proof seems true only because your model is extremely simplified.



'This sentence is false or 2=1.' has a truth value of 0. Neither the sentence is false nor 2=1.

But it doesn't mean the sentence is true. Its truth value is unknown. So there is no paradox, only rejection of the law of excluded middle.

Well yes, obviously they don't actually work.

If both clauses of the disjunction are false, as you claim, then the sentence overall is false. I'm not sure rejecting the law of the excluded middle is sufficient.

[Charity]

Traditional (Divide by Zero)
1. a=b
2. a2=ab
3. 2a2=a2+ab
4. 2a2-2ab=a2-ab
5. 2(a2-ab)=(a2-ab)
6. 2=1


There is a slightly tidier version (still relying on dividing by zero)

An incorrect (?) algebraic proof:
Prove that 2 = 1.
Proof:
Let a = b ..........................multiply by b
ab = b2 ..........................subtract a2
ab – a2 = b2 – a2 ..............factorise
a(b – a) = (b + a)(b – a) ......divide by (b – a)
a = b + a .............................replace b with a
a = a + a ............................collect like terms
a = 2a .................................divide by a
Therefore 1 = 2

[Blue Ghost]

Self-referential sentences cannot have meaningful truth values, I believe. Or something like that.

[Admiral Squish]

Here's the question. if its POSSIBLE to create a proof that tells us that 2=1, doesn't that sort of call the validity of all mathematics into question?

[ForzaFiori]

Here's the question. if its POSSIBLE to create a proof that tells us that 2=1, doesn't that sort of call the validity of all mathematics into question?

The only way for the proof to work is if A and B both equal 0, or something like that, which means that they're dividing by 0 in the last step, which is impossible. So while it LOOKS like a proof, it's actually breaking the laws of math.

At least, I'm pretty sure that's what my Calc teacher told us. it was something weird like that.

[Whiffet]

The only way for the proof to work is if A and B both equal 0, or something like that, which means that they're dividing by 0 in the last step, which is impossible. So while it LOOKS like a proof, it's actually breaking the laws of math.

At least, I'm pretty sure that's what my Calc teacher told us. it was something weird like that.

It's not so much A and B both equal 0 as it is that A and B equal each other. So if you subtract one from the other, you get 0. The traditional "proofs" have a point where you divide by the quantity A minus B or B minus A.

[Deathslayer7]

The only way for the proof to work is if A and B both equal 0, or something like that, which means that they're dividing by 0 in the last step, which is impossible. So while it LOOKS like a proof, it's actually breaking the laws of math.

At least, I'm pretty sure that's what my Calc teacher told us. it was something weird like that.

you are dividing by zero, but you can use any number not just zero.

EX: say a=b=5

when you divide by (a-b) you are actually dividing by (5-5=0) zero. Therefore the proof is not possible.

[PirateMonk]

Self-referential sentences cannot have meaningful truth values, I believe. Or something like that.

'This sentence is six words long.'

Even if you only mean sentences which reference their own truth value, that may not help. As I noted above, denying that 'This sentence is false' is false makes it false. Just as calling it true makes it false, denying that it has a meaningful truth value gives it a meaningful truth value. I suppose you could refuse to get pulled back into it, which would leave you in the odd position of asserting that ''This sentence is false' is false' is true, but 'This sentence is false' is not.

[Tirian]

Here's the question. if its POSSIBLE to create a proof that tells us that 2=1, doesn't that sort of call the validity of all mathematics into question?

Of course not. The conclusion that 2=1 calls all of arithmetic into question, because then any two numbers would be equal.

[Heliomance]

Here's the question. if its POSSIBLE to create a proof that tells us that 2=1, doesn't that sort of call the validity of all mathematics into question?

It's been said already, but not explicitly. The "proof" involves a step which is not mathematically valid (dividing by zero). It hides this well, to maintain its believability, but the fact that that step is in there at all automatically invalidates the entire proof.


An incorrect (?) algebraic proof:
Prove that 2 = 1.
Proof:
Let a = b ..........................multiply by b
ab = b2 ..........................subtract a2
ab – a2 = b2 – a2 ..............factorise
a(b – a) = (b + a)(b – a) ......divide by (b – a)
a = b + a

From here, you could also go subtract a from both sides
b=0

Congratulations, you have just proved that any number is equal to zero. (Except, y'know, not.)

[WalkingTarget]

'This sentence is six words long.'

Even if you only mean sentences which reference their own truth value, that may not help. As I noted above, denying that 'This sentence is false' is false makes it false. Just as calling it true makes it false, denying that it has a meaningful truth value gives it a meaningful truth value. I suppose you could refuse to get pulled back into it, which would leave you in the odd position of asserting that ''This sentence is false' is false' is true, but 'This sentence is false' is not.

This is, essentially, Gödel's Incompleteness Theorem in plain English instead of fancy math terms.

Kurt Gödel proved that in any non-trivial mathematical system (one that can perform the basic arithmetical operations) there will always be true statements in the system that cannot be derived from the axioms of that system.

Basically, the ability to make self-referential statements (like "This sentence is false.") causes mathematical-style logic to break down.

[Holocron Coder]

It's been said already, but not explicitly. The "proof" involves a step which is not mathematically valid (dividing by zero). It hides this well, to maintain its believability, but the fact that that step is in there at all automatically invalidates the entire proof.



From here, you could also go subtract a from both sides
b=0

Congratulations, you have just proved that any number is equal to zero. (Except, y'know, not.)

Actually, it is entirely possible to divide by zero. We disregard this in simple mathematics as it has a useless result: infinity.

So, the proof is actually the one that proves that 2 infinities is equal to 1.

And yes, I'm not actually making that up :smallbiggrin: it's true.

[Bryn]

Infinity is not really a number (I guess this use of 'infinity' is a way to talk about limits or something like that?), and dividing by 0 doesn't give you infinity. You can see this if you take 1/x and let x go from, say, -1 to 0. As x approaches 0, 1/x goes to -infinity, not +infinity. 1/0 is undefined, not infinity.

[Dr.Epic]

http://img.photobucket.com/albums/v350/RenegadePaladin/DivideByZero.jpg

:smalltongue:

That happened to me once.

[PirateMonk]

This is, essentially, Gödel's Incompleteness Theorem in plain English instead of fancy math terms.

Kurt Gödel proved that in any non-trivial mathematical system (one that can perform the basic arithmetical operations) there will always be true statements in the system that cannot be derived from the axioms of that system.

Basically, the ability to make self-referential statements (like "This sentence is false.") causes mathematical-style logic to break down.

Gödel's Incompleteness Theorem just deals with the distinction between truth and theoremhood in a formal system, not with the Liar's Paradox directly.
''''''This sentence is false' is false' is false' is false' is true' is true' is true...
The proof of Tarski's Theorem (http://en.wikipedia.org/wiki/Tarski%27s_truth_theorem), on the other hand, does seem to deal directly with the LP.

[DraPrime]

.99 repeating equals 1.

Actually that one makes a lot of sense. If you subtract .9 repeating from one, you'll have 0.01 with infinite zeroes. Infinite zeroes will make this infinitely small, which means that it is nothing, or at least so some math teacher told me. So it's really just two different ways of writing the same number, kind of like how 4/2 is also 2.

[Manga Shoggoth]

Infinity is not really a number (I guess this use of 'infinity' is a way to talk about limits or something like that?), and dividing by 0 doesn't give you infinity. You can see this if you take 1/x and let x go from, say, -1 to 0. As x approaches 0, 1/x goes to -infinity, not +infinity. 1/0 is undefined, not infinity.

Nope.

1/0 = Infinity; usually treated as a number in mathematics and physics (although you are correct in your comment about limits).

0/0 is undefined because it expands out as 0 * 1/0:


0 * x = 0
x/0 = infinity
Therefore the value of 0/0 cannot be determined, since it cannot be both 0 and infinite.

In fact 0/0 is referred to as the first indeterminate form. There are several others.

[AsteriskAmp]

Actually that one makes a lot of sense. If you subtract .9 repeating from one, you'll have 0.01 with infinite zeroes. Infinite zeroes will make this infinitely small, which means that it is nothing, or at least so some math teacher told me. So it's really just two different ways of writing the same number, kind of like how 4/2 is also 2.

Actually we assume that happens because otherwise things stop working, it's based on the fact our system isn't perfect, the conversion system for periodic numbers is flawed, which gives us that. We assume that yes, 0.periodic 9=1, but it is not, we just work with it as that because we need it and it's a damn good one, just not perfect.

[Lord Vampyre]

Actually we assume that happens because otherwise things stop working, it's based on the fact our system isn't perfect, the conversion system for periodic numbers is flawed, which gives us that. We assume that yes, 0.periodic 9=1, but it is not, we just work with it as that because we need it and it's a damn good one, just not perfect.

Actually .9 repeating is equal to exactly 1.

Consider:

1/9 = .1 repeating now if you multiply both sides by 9

1 = .9 repeating

This is far from a formal proof on the subject, but should be sufficient to illustrate the point.


Nope.

1/0 = Infinity; usually treated as a number in mathematics and physics (although you are correct in your comment about limits).

0/0 is undefined because it expands out as 0 * 1/0:

0 * x = 0
x/0 = infinity
Therefore the value of 0/0 cannot be determined, since it cannot be both 0 and infinite.

In fact 0/0 is referred to as the first indeterminate form. There are several others.


Actually, this is incorrect.

1/x goes to infinity as x goes from 1 to 0, however 1/x goes to negative infinity as x goes from -1 to 0. In mathematics you can not actually divide by zero as the number will be undefined. However, you can divide by a number that is approaching zero, thus giving you positive or negative infinity.

[AsteriskAmp]

Actually .9 repeating is equal to exactly 1.

Consider:

1/9 = .1 repeating now if you multiply both sides by 9

1 = .9 repeating

This is far from a formal proof on the subject, but should be sufficient to illustrate the point.
Again, conversion system that isn't perfect. It's DAMN GOOD, it's very near it, but it's not perfect, so we assume the error to be correct because the difference is theoretical (in practice you'll never get to the infinite decimal were the difference is) and because otherwise we are screwed.

[averagejoe]

Actually, it is entirely possible to divide by zero. We disregard this in simple mathematics as it has a useless result: infinity.

So, the proof is actually the one that proves that 2 infinities is equal to 1.

And yes, I'm not actually making that up :smallbiggrin: it's true.


Infinity is not really a number (I guess this use of 'infinity' is a way to talk about limits or something like that?), and dividing by 0 doesn't give you infinity. You can see this if you take 1/x and let x go from, say, -1 to 0. As x approaches 0, 1/x goes to -infinity, not +infinity. 1/0 is undefined, not infinity.


Nope.

1/0 = Infinity; usually treated as a number in mathematics and physics (although you are correct in your comment about limits).

0/0 is undefined because it expands out as 0 * 1/0:


0 * x = 0
x/0 = infinity
Therefore the value of 0/0 cannot be determined, since it cannot be both 0 and infinite.

In fact 0/0 is referred to as the first indeterminate form. There are several others.

None of these is correct. 1/0 (or similar) has meaning sometimes, but it depends on the formulation you're using. In pedestrian math it has no well-defined solution.

To understand the problems with 1/0, one first has to examine the motivation for division, arguably the most abstract of the arithmetical operations. In the traditional sense, saying x/y (or, x divided by y) means, "I have x things, and I want to divide them into y equal parts. How many will be in each divided part?" Looking at this, 1/0 is plainly nonsense (or any other number over zero). "I have one item, and I want it evenly divided into zero parts. How much will each part get?" One seems like the obvious answer to this, but then you've divided it into one part with a whole item in it. One might imagine semi-physical or philosophical scenarios in which you look at how much you can divide something until each piece is no longer a section of the original whole, but an entirely new thing, but that has no mathematical significance, and probably isn't practical anyways.

In algebra, you get equations like 2a=4. In other words, there's some numerical value attached to something, and you know that twice that numerical value equals four. The act of dividing each side by two is a shorthand way of saying, "I know two quantities are equal. Therefore, I also know, that if I divide each of these quantities by two, they will continue to be equal." Again, since a/0 has no arithmetic meaning, you cannot guarantee equality if you divide two equal numbers by zero. It's a nonsense thing to say, like, "b used in a Shakespearean sonnet = 2 used in a Shakespearean sonnet." Another way to look at a/0 in algebra is to say, "What number multiplied by 0 equals a?" Such a number does not exist.

In calculus you can make 1/0 more meaningful by putting the division in terms of limits, but even then it isn't well defined. lim(x->0)1/x can be either positive or negative infinity (though lim(x->0)1/x^2 is infinity. Again, it depends on how you define things.)

There's more to it than that, but to get into it more one has to talk about infinity and higher mathematical systems. In some formulations there is even a so-called "point at infinity." Things like infinity or 1/0 can have meaning, it just depends on how you're formulating things. It's a mistake many people make, I think. They look at math as some abstract thing, and forget that there are real motivations for performing certain operations or obtaining certain values, and that the answer to any mathematical question depends at least somewhat on these things.


Actually we assume that happens because otherwise things stop working, it's based on the fact our system isn't perfect, the conversion system for periodic numbers is flawed, which gives us that. We assume that yes, 0.periodic 9=1, but it is not, we just work with it as that because we need it and it's a damn good one, just not perfect.

This is also pretty much completely false (for the real numbers). It's not an unreasonable point of view, but it ultimately comes from people's intuitions that have to do with reality, whereas the mathematical world has different rules than what we're used to. The reason people tend to think this is that 0.9999etc.=1 is a kind of uncomfortable result, and it is a really uncomfortable result. Yet there it is.

If it's easier, just think of it as two different ways of writing the same number. There are trivially infinite ways of writing any rational number (ex. 1=1.0=1.00, etc.), and no one has any problems with this. Similarly, ending any number in repeating nines is just another way of writing that number with the last digit before the nines being bumped up by one.

However you look at it, to me, the idea that the two are not equal is even more uncomfortable. For them to be unequal would require the existence of real numbers that are infinitesimally small; real numbers so small that no other numbers are smaller than them. It's sheer insanity!

The equality of those values is well-proven using many different, rigorous processes. Our intuition tells us it cannot be so, and yet it is!

[Keld Denar]

Well, in limit calculus, you learn that different values of infinity are more or less the same.

1*infinity = 3*infinity

approximately. The only time its different is when you have differing exponents of infinity.

(infinity)^2 > (infinity)^1

Otherwise, the difference between x and 2x is aproximately 0 as x goes to infinity, while x^2 is much larger than x as x goes to infinity.

[averagejoe]

Well, in limit calculus, you learn that different values of infinity are more or less the same.

1*infinity = 3*infinity

approximately. The only time its different is when you have differing exponents of infinity.

(infinity)^2 > (infinity)^1

Otherwise, the difference between x and 2x is aproximately 0 as x goes to infinity, while x^2 is much larger than x as x goes to infinity.

Intuitively this seems wrong to me. Where does this result come from?

[Bryn]

In calculus you can make 1/0 more meaningful by putting the division in terms of limits, but even then it isn't well defined. lim(x->0)1/x can be either positive or negative infinity (though lim(x->0)1/x^2 is infinity. Again, it depends on how you define things.)
This is what I wished to say*, but failed to. Sorry.

[Capt Spanner]

Infinity is hard.

I like the idea that from 2 = 1 you can prove anything.

For example:

The set of people containing me and Roland contains 2 people.

However, since 2 = 1, that set contains 1 person.

Hence Roland and I are the same person.

Hence I am Roland.

[Mono Vertigo]

I am he as you are he as you are me and we are all together.

I'm getting my mathematically-challenged coat.

[AsteriskAmp]

This is also pretty much completely false (for the real numbers). It's not an unreasonable point of view, but it ultimately comes from people's intuitions that have to do with reality, whereas the mathematical world has different rules than what we're used to. The reason people tend to think this is that 0.9999etc.=1 is a kind of uncomfortable result, and it is a really uncomfortable result. Yet there it is.

If it's easier, just think of it as two different ways of writing the same number. There are trivially infinite ways of writing any rational number (ex. 1=1.0=1.00, etc.), and no one has any problems with this. Similarly, ending any number in repeating nines is just another way of writing that number with the last digit before the nines being bumped up by one.

However you look at it, to me, the idea that the two are not equal is even more uncomfortable. For them to be unequal would require the existence of real numbers that are infinitesimally small; real numbers so small that no other numbers are smaller than them. It's sheer insanity!

The equality of those values is well-proven using many different, rigorous processes. Our intuition tells us it cannot be so, and yet it is!
They are not equal, they are equivalent and as such we treat them as equals because the system says so, but there is still a minuscule difference so small that it stops making sense and isn't allowed in the system, so we treat and consider them, and heck, say what they represent is equal. The idea makes no sense, it's the system the beats the sense into it since well, it works in base of axioms which make it work.

If you however work with the hyperreals things change abruptly since that REALLY infinitesimal quantity can be represented. But yes, different number system.

[averagejoe]

This is what I wished to say*, but failed to. Sorry.

Yeah, I figured, I just wasn't quite sure of your meaning. Also:


Infinity is hard.

Edit:


They are not equal, they are equivalent and as such we treat them as equals because the system says so, but there is still a minuscule difference so small that it stops making sense and isn't allowed in the system, so we treat and consider them, and heck, say what they represent is equal. The idea makes no sense, it's the system the beats the sense into it since well, it works in base of axioms which make it work.

If you however work with the hyperreals things change abruptly since that REALLY infinitesimal quantity can be represented. But yes, different number system.

Based on what? What is your reasoning for this?

[AsteriskAmp]

Based on what? What is your reasoning for this?

Let there be a recurring decimal 0.2999999999999999999.......
When we try to convert this into a fraction, this is what happens.
x = 0.2999999999999999999............
10x = 2.999999999999999999999......
100x = 29.9999999999999999999999......
90x = 27
x = 27/90 = 3/10, which is 0.3

If we did the math on any non-repeating to infinity (truncated) version, we would notice there was a one at the end, however this would mean there is something akin to 26.(9)1 [Where ( ) is infinite]

They work in theory, and yes, we say 1=0.9999999999999...
It misses by an infinitesimal amount, however we work around that through limits and... voila, the infinitesimal amount doesn't really matter since it is so small it's importance is non-existant.

Infinity Abuse

To put it in a different yet mathematical context.

Take a point out of a sphere, points are adimensional yet spheres are built out of points, so we could say the sphere has lost a point yet no part of itself, now take another point, it still holds true, repeat ad infinitum, you now have two spheres with the same measurements. If we had infinitesimal quantities we could break the system this way, so to prevent abuse we say 1=0.9999... so that that REALLY small number doesn't work toward breaking our system.

Math makes weird sense when you consider what numbers are made of, 1 is equal to 1/2+1/2, which in turn and in turn, and if you go on you'll reach the point where you realise one is made of nothing actually.

[averagejoe]

Let there be a recurring decimal 0.2999999999999999999.......
When we try to convert this into a fraction, this is what happens.
x = 0.2999999999999999999............
10x = 2.999999999999999999999......
100x = 29.9999999999999999999999......
90x = 27
x = 27/90 = 3/10, which is 0.3

If we did the math on any non-repeating to infinity (truncated) version, we would notice there was a one at the end, however this would mean there is something akin to 26.(9)1 [Where ( ) is infinite]

:smallconfused: But if there's a one at the end, it isn't infinite. It terminates.

Also, 0.29 repeating is equal to 0.3. I basically claimed this in my prior post. I honestly don't know what you're trying to say about any of this.


They work in theory, and yes, we say 1=0.9999999999999...
It misses by an infinitesimal amount, however we work around that through limits and... voila, the infinitesimal amount doesn't really matter since it is so small it's importance is non-existant.

It really doesn't have anything to do with limits, though. I mean, I'm sure there are proofs that use limits, but there are many that don't that are equally useful.


Infinity Abuse

To put it in a different yet mathematical context.

Take a point out of a sphere, points are adimensional yet spheres are built out of points, so we could say the sphere has lost a point yet no part of itself, now take another point, it still holds true, repeat ad infinitum, you now have two spheres with the same measurements. If we had infinitesimal quantities we could break the system this way, so to prevent abuse we say 1=0.9999... so that that REALLY small number doesn't work toward breaking our system.

This is untrue, and I'm not sure what it has to do with the aforementioned equality. The "hole," in such a sphere is still mathematically significant. No mathematician would call both the same sphere (though they would have many similar properties.)

Edit: I apologize, I just really don't know what you're trying to say.

[AsteriskAmp]

:smallconfused: But if there's a one at the end, it isn't infinite. It terminates.

Also, 0.29 repeating is equal to 0.3. I basically claimed this in my prior post. I honestly don't know what you're trying to say about any of this.

I'm describing why it works in theory but it isn't part of reality, we have to assume they are the same and they are equal even though they aren't in reality, they are equivalent. Math infinite works... in math, reality has no infinite that we know off.


It really doesn't have anything to do with limits, though. I mean, I'm sure there are proofs that use limits, but there are many that don't that are equally useful.
Limits are what make infinite make sense, so to talk about an infinite string we have to talk about where the string is going, infinites contrary to what intuition tell us do end, they end up at a limit.


This is untrue, and I'm not sure what it has to do with the aforementioned equality. The "hole," in such a sphere is still mathematically significant. No mathematician would call both the same sphere (though they would have many similar properties.)

Edit: I apologize, I just really don't know what you're trying to say.
Nope, the point is adimensional, so there is no hole, there are infinite points, and they are the only thing making the sphere up. And infinite -infinite =infinite, so you could go on taking points and build another sphere of infinite points from the first without there being any difference at all in the first one.

Banach-Tarski mathematically illustrates the issue with math in itself, it is internally consistent, the moment it goes out of it's theoretical realm however it is not assured (though it will be very near perfect) it will work.

What I mean is that we assume, work with, assign the value of, use to refer as the same, treat as equal 1 with 0.99999... to prevent this sort of breakage, where we could use infinitesimals.

For example, in the Hyperreals, where we do have infinitesimals we have

0.999.... = 1 - 1/10H
...(____)
.......H

[PirateMonk]

Infinity is hard.

I like the idea that from 2 = 1 you can prove anything.

For example:

The set of people containing me and Roland contains 2 people.

However, since 2 = 1, that set contains 1 person.

Hence Roland and I are the same person.

Hence I am Roland.

More directly, it's relatively trivial to prove ~(2=1) even in highly rigorous formal systems. In propositional logic, (P/\~P)->Q for any Q (the Principle of Explosion (http://en.wikipedia.org/wiki/Principle_of_explosion)) requires only a slightly longer proof.


Infinity Abuse

To put it in a different yet mathematical context.

Take a point out of a sphere, points are adimensional yet spheres are built out of points, so we could say the sphere has lost a point yet no part of itself, now take another point, it still holds true, repeat ad infinitum, you now have two spheres with the same measurements. If we had infinitesimal quantities we could break the system this way, so to prevent abuse we say 1=0.9999... so that that REALLY small number doesn't work toward breaking our system.

If you use a solid sphere, you only need five pieces (http://en.wikipedia.org/wiki/Banach-Tarski_Theorem). (Ninja'd).


Math makes weird sense when you consider what numbers are made of, 1 is equal to 1/2+1/2, which in turn and in turn, and if you go on you'll reach the point where you realise one is made of nothing actually.

An infinite number of infinitely small things is not "nothing".

[AsteriskAmp]

If you use a solid sphere, you only need five pieces (http://en.wikipedia.org/wiki/Banach-Tarski_Theorem).
I did mention Banach-Tarski



An infinite number of infinitely small things is not "nothing".
If we defined the small as.

f(x)=1/10x as x approaches positive infinity y approaches 0.
If we used the idea of infinite divisibility, we could say that in fact an infinite number of infinitely small will reach zero as long as "small" reaches infinity.

[averagejoe]

I'm describing why it works in theory but it isn't part of reality, we have to assume they are the same and they are equal even though they aren't in reality, they are equivalent. Math infinite works... in math, reality has no infinite that we know off.

:smallconfused: Yeah, but this is a discussion about math. Though math corresponds with reality with surprising regularity, mathematics has never had any claim at all on modeling reality. Your argument strikes me as philosophical, since we have no ability to measure, "A 0.999..." Indeed, I'm not sure where one would start, since mathematically it's the same thing as measuring a 1. Speculating on immeasurable and unknowable parts of reality will never get you a better argument than, "Because I think so."


Limits are what make infinite make sense, so to talk about an infinite string we have to talk about where the string is going, infinites contrary to what intuition tell us do end, they end up at a limit.

The concept of infinity predates the concept of limits. All you need for infinity is the natural numbers.


What I mean is that we assume, work with, assign the value of, use to refer as the same, treat as equal 1 with 0.99999... to prevent this sort of breakage

That's a strong claim. I'm afraid I must ask for a source on that.

[PirateMonk]

I did mention Banach-Tarski

I was repeatedly interrupted while writing that post and forgot to preview. That was entirely in response to the quoted section, which sounded like something slightly different.


If we defined the small as.

f(x)=1/10x as x approaches positive infinity y approaches 0.
If we used the idea of infinite divisibility, we could say that in fact an infinite number of infinitely small will reach zero as long as "small" reaches infinity.

If you do it like that, then the number of "pieces" of one will be 10x for the same x, so it cancels out to 1 as common sense would dictate even as it approaches infinity.

[Force]

Wellll...

http://img.photobucket.com/albums/v216/FaxCelestis/FaxCelestisAvatar.png

+

http://upload.wikimedia.org/wikipedia/commons/thumb/6/66/Venus_symbol.svg/50px-Venus_symbol.svg.png

=

http://a3.sphotos.ak.fbcdn.net/hphotos-ak-snc6/253801_180196862038878_100001456953766_491910_4687 112_n.jpg

My apologies to the innocent named in this post. :smallcool:

[PirateMonk]

Nope, the point is adimensional, so there is no hole, there are infinite points, and they are the only thing making the sphere up.

Wrong. In a three-dimensional Cartesish coordinate system, a sphere centered on the origin can be specified with the equation x2+y2+z2=r2 (or x2+y2+z2<r2 for a solid sphere). If you remove a point, there will be some coordinate triplet which satisfies the equation but still is not in the sphere.

[Heliomance]

Nope, the point is adimensional, so there is no hole, there are infinite points, and they are the only thing making the sphere up. And infinite -infinite =infinite, so you could go on taking points and build another sphere of infinite points from the first without there being any difference at all in the first one.

What PirateMonk said. S2 is not mathematically equivalent to S2\{(0,0,1)}, for example.

[AsteriskAmp]

:smallconfused: Yeah, but this is a discussion about math. Though math corresponds with reality with surprising regularity, mathematics has never had any claim at all on modeling reality. Your argument strikes me as philosophical, since we have no ability to measure, "A 0.999..." Indeed, I'm not sure where one would start, since mathematically it's the same thing as measuring a 1. Speculating on immeasurable and unknowable parts of reality will never get you a better argument than, "Because I think so."
The issue is that mathematically 1=0.9999..., geometric progressions and limits confirm it, however, equal here works because of the definitions and the way the system was made. We assume it works and use it that way because it works so well we don't have issue with that almost nothing, it is equivalent, all else is because of the way defined the system in which it happens, other number systems have different ways of working around the issue. We assume it because our system has no way around it.


The concept of infinity predates the concept of limits. All you need for infinity is the natural numbers.

Yes, but the ability to work with infinity requires Limits. Also, infinity changes according to the definition. For example Greek infinity doesn't end, ours does, at a limit.


That's a strong claim. I'm afraid I must ask for a source on that.
Disregard, error based on extremely simplified model. But assertion was made with the fact that infinitesimals don't exist in our system.


If you do it like that, then the number of "pieces" of one will be 10x for the same x, so it cancels out to 1 as common sense would dictate even as it approaches infinity.
You can't divide infinity by infinity, it leaves a big pile of undefined. However, the result will be undefined eitherway. (I didn't check the second part somehow, damn multiplication by zero), also, it's extremely simplified and fails, analytical geometry disproves it. Either way Banach Tarski still holds in demonstrating the issue.

[Heliomance]

The easiest way to divide infinity by infinity - or limits approaching infinity by other limits approaching infinity, anyway - is using l'Hopital's Rule.

[Orzel]

I can prove 1+2=1

*gives the next poster 2 cookies*

1 playgrounder + 2 cookies = ?

[enderlord99]

Yum!

*dunks in milk*

[PirateMonk]

The issue is that mathematically 1=0.9999..., geometric progressions and limits confirm it, however, equal here works because of the definitions and the way the system was made. We assume it works and use it that way because it works so well we don't have issue with that almost nothing, it is equivalent, all else is because of the way defined the system in which it happens, other number systems have different ways of working around the issue. We assume it because our system has no way around it.

No, we prove it fairly rigorously.


Yes, but the ability to work with infinity requires Limits. Also, infinity changes according to the definition. For example Greek infinity doesn't end, ours does, at a limit.

I'm pretty sure limits involving infinity just refer to what happens when x grows without limit, not that infinity is somehow limited.


You can't divide infinity by infinity, it leaves a big pile of undefined. However, the result will be undefined eitherway. (I didn't check the second part somehow, damn multiplication by zero), also, it's extremely simplified and fails, analytical geometry disproves it. Either way Banach Tarski still holds in demonstrating the issue.

You can, however, divide a limit as x goes to infinity by itself. You were just talking about splitting 1 into smaller and smaller parts, so this is the proper way to look at it.


I can prove 1+2=1

*gives the next poster 2 cookies*

1 playgrounder + 2 cookies = ?

That just proves that p+2c=1, which has infinitely many solutions.

[RebelRogue]

araveugnitsuga is talking about non-standard analysis/hyperreals here. Hyperreal numbers include infinitesimals.

Edit: seems like a missed a cookie :smallfrown:

[Heliomance]

Actually, I think what Orzel was going for is p + 2c = p, as the cookies will vanish.

[AsteriskAmp]

No, we prove it fairly rigorously.

Through our numbering systems (and some others).
There are others systems which address the issue of the .999... in different ways.


I'm pretty sure limits involving infinity just refer to what happens when x grows without limit, not that infinity is somehow limited.

Infinites do end all the time. And they end in a limit.


You can, however, divide a limit as x goes to infinity by itself. You were just talking about splitting 1 into smaller and smaller parts, so this is the proper way to look at it.
I meant infinite successive divisions.

[PirateMonk]

Through our numbering systems (and some others).
There are others systems which address the issue of the .999... in different ways.

This is true, but it does not imply that the standard approach is in any way a shaky assumption used to patch a hole in our system, which is what you seemed to be claiming.


Infinites do end all the time. And they end in a limit.

What?


I meant infinite successive divisions.

The proper way to analyze this situation is with a limit as x approaches infinity.

[AsteriskAmp]

This is true, but it does not imply that the standard approach is in any way a shaky assumption used to patch a hole in our system, which is what you seemed to be claiming.

There is a hole in the way it works, we have to assume it because we don't have infinitesimals, if we did, then we could work another way around.


What?

Some infinites end in limits, not all are endless, all are infinite however.

[Orzel]

But playgrounders are cookies right?

*sharp teeth*

[tonberrian]

On .9 repeating:

I did a proof of this in math class. Basically, it is provable that between any two different real numbers, there exists a real number between them. If you take .9 repeating and 1, you can't choose a real number between them. Ergo, .9 repeating = 1, because they cannot be two different real numbers.

[PirateMonk]

There is a hole in the way it works, we have to assume it because we don't have infinitesimals, if we did, then we could work another way around.

Again, prove, not assume.


Some infinites end in limits, not all are endless, all are infinite however.

I'm not sure I know enough math to respond to this properly, so I'll leave it to someone else.


But playgrounders are cookies right?

*sharp teeth*

Then you get 3c=c, so c=0. Congratulations, you just proved cookies don't exist. I hope you're happy.

[The Extinguisher]

Some infinites end in limits, not all are endless, all are infinite however.

Please expand, because I have never, ever heard of this before. It seems fairly ridiculous, given what a "limit" actually means in mathematics. Saying something ends in a limit is about as nonsensical as saying a sequence ends in a derivative.

[Peregrine]

e^(i pi) = cos(pi) + isin(pi) = -1 + 0i = -1

e^(3i pi) = cos(3 pi) + isin(3 pi) = -1 + 0i = -1

Since -1 = -1

then e^(i pi) = e^(3i pi)

ln(e^(i pi)) = ln(e^(3i pi))

i pi = 3i pi

1 = 3

Eulered!

Umm. Wow. I've never seen this before. :smalleek: It's like saying that since cos π = cos 3π, π = 3π and 1 = 3... except there's no obviously wrong step.

That is, I know where it goes wrong (where you introduce the logarithms), but I don't know why. Apparently logarithms don't work normally when you have complex exponents...?


On .9 repeating:

I did a proof of this in math class. Basically, it is provable that between any two different real numbers, there exists a real number between them. If you take .9 repeating and 1, you can't choose a real number between them. Ergo, .9 repeating = 1, because they cannot be two different real numbers.

I like that one. :smallsmile: One of many excellent proofs. The one I find easiest to remember is:
0.9... = 0.1... × 9
0.1... = 1/9
∴ 0.9... = 1/9 × 9 = 9/9 = 1


There are many reasons why people don't believe that this can be so; averagejoe hit my personal reason dead on, but I think it's worth repeating.

Aside from trivial cases (like adding 0s at either end), we're never taught in school that one number can have two distinct decimal representations.

I think I had this epiphany when learning about balanced ternary (http://en.wikipedia.org/wiki/Balanced_ternary), which is a number system that has quite obvious distinct representations of each number.

So, 0.9... = 1. There is no difference. There is no ...1 "at infinity" (i.e. the difference is not 0.00...001). Even if...


Some infinites end in limits, not all are endless, all are infinite however.

Let's assume this is true. But this is an infinity that is endless. There is no last decimal place for a 1 to exist in. It's zeroes all the way down.

[Heliomance]

Umm. Wow. I've never seen this before. :smalleek: It's like saying that since cos π = cos 3π, π = 3π and 1 = 3... except there's no obviously wrong step.

That is, I know where it goes wrong (where you introduce the logarithms), but I don't know why. Apparently logarithms don't work normally when you have complex exponents...?


Yep. IIRC, that should technically be log rather than ln. The standard definition of logarithms doesn't work when using complex numbers, so we have to define a new one which is equivalent under most, but not all, circumstances. This does mean however that it is no longer the inverse function of exponentiation.

[Bryn]

Umm. Wow. I've never seen this before. :smalleek: It's like saying that since cos π = cos 3π, π = 3π and 1 = 3... except there's no obviously wrong step.

That is, I know where it goes wrong (where you introduce the logarithms), but I don't know why. Apparently logarithms don't work normally when you have complex exponents...?
Yeah (http://en.wikipedia.org/wiki/Logarithm#Complex_logarithm). When going from
ln(e^(i pi)) = ln(e^(3i pi))
it should have been
i(pi+2n*pi)=i(3pi+2m*pi) (where n and m are integers)


On 0.999..., Wikipedia actually has a really nicely written article, particularly from this bit (http://en.wikipedia.org/wiki/0.999...#Discussion) onwards.

[Tirian]

That is, I know where it goes wrong (where you introduce the logarithms), but I don't know why. Apparently logarithms don't work normally when you have complex exponents...?

The logarithm is the inverse of the exponential function. And that's all well and good over the reals, where the exponential is one-to-one. But when a function isn't one-to-one, then you can't go from f(x)=f(y) to x=y without a lot more care.

A simpler example is with the square root function and negative numbers. We know that (-4)^2 = 4^2, but it would be a mistake to conclude that -4=4.

[Peregrine]

Thanks -- it seems so obvious in hindsight. When exponentiation becomes periodic, of course the logarithm must be multi-valued. I practically wrote that in my own post (with my analogy to cosine) without even realising the implications of what I'd said!

Did I mention that I've just gotten my first job as a maths teacher...? :smallredface:

[RebelRogue]

Yep. IIRC, that should technically be log rather than ln.
Depends on who you ask! Some (mostly 'pure') mathematicians use log to mean natural logarithm. Just about everybody else use 'log' to denote a base 10 logarithm and use ln for natural ones.


Did I mention that I've just gotten my first job as a maths teacher...? :smallredface:
I was in the same boat as you half a year ago. So far, I've found it to be a lot of fun :smallsmile:

[erikun]

Well, in limit calculus, you learn that different values of infinity are more or less the same.

1*infinity = 3*infinity

approximately. The only time its different is when you have differing exponents of infinity.

(infinity)^2 > (infinity)^1

Otherwise, the difference between x and 2x is aproximately 0 as x goes to infinity, while x^2 is much larger than x as x goes to infinity.
Not quite. What taking the limit of a value tells you is the relative scale of the two values, not the equality of them. That is, 1*infinity and 3*infinity increase at the same rate, not that they are the same number.

In fact, take this example problem:
f(x) = 3x / (x + 1)

We cannot simplify the problem any further, but we want to know that happens to this forumla as we get increasingly large x values. Thus, we take the formula to infinity1 to see what happens.
f(x) lim(x->inf) = 3*infinity / (infinity + 1)

Clearly, the +1 is insignificant when dealing with infinity, so we end up with 3*infinity/1*infinity. This gives us a final value of 3/1, which is indeed what the function 3x/(x+1) converges upon with increasingly large values of x.

1Technically, we take the derivatives and mathematically solve from that. The derivative of 3x is 3, while the derivative of x+1 is 1. We then solve f(x)' = 3/1, which gives us our answer.
My display here is mainly a rough visualization of how the process works.


As for something like (infinity)^2 vs. (infinity)^1, we are comparing the rate that the two functions increase. It is not that (infinity)^2 is larger than (infinity)^1, but that it increases at a greater rate. [Note, though, that a x^2 function will overtake a x^1 function at any suitably large value of x, regardless of how large a constant we add to x^1.]

To use our example from before, take the function:
f(x) = 3x2 / (x + 1)

As you can see, the upper value 3x2 is going to increase dramatically faster than the lower value (x+1). If we were to look at the "final" value, it would be 3*infinity^2 / (1*infinity + 1). While they both have the same result (infinity), the numerator increases at a much higher rate, infinity*infinity / infinity, and so the entire equation continues to expand to infinity (increasingly large values of y) as we attain increasingly large values of x.


And I think that's enough of a math lesson for today.

[Science Officer]

-snip-


Thanks, I was just going to jump in to say something like this.
Actually, I'm far too lazy.

General: While there are larger and smaller infinities (http://en.wikipedia.org/wiki/Aleph_number) it's not quite like 3*infinity or infinity^2.

As for 0.999... equaling one, it does (http://en.wikipedia.org/wiki/0.999...). Not sort of, not equivalent to, it does.


And to confuse the whole issue of infinity and dividing by zero, there's this fellow (http://www.bbc.co.uk/berkshire/content/articles/2006/12/12/nullity_061212_feature.shtml).

[Talya]

Actually that one makes a lot of sense. If you subtract .9 repeating from one, you'll have 0.01 with infinite zeroes. Infinite zeroes will make this infinitely small, which means that it is nothing, or at least so some math teacher told me. So it's really just two different ways of writing the same number, kind of like how 4/2 is also 2.

http://en.wikipedia.org/wiki/0.999...

[The_Final_Stand]

For further reading: http://qntm.org/pointnine

Similarly, look at the common counter arguments, and the part just above that.

On another note: Troll pi! (http://qntm.org/trollpi)

[Tengu_temp]

"I still don't believe it and I'm entitled to my own opinion."

In regular science, we have theories. A theory is proposed in order to explain observations, and can be overturned in light of new, inexplicable observations. Multiple theories and opinions may compete with one another. There are fashions. There is room for debate.

In mathematics, we have theorems instead of theories. A theorem is the result of a mathematical proof. A theorem is a fact. A theorem cannot be overturned and is not a matter of opinion. Once proven, a theorem stands for eternity. Mathematics is not ideological.

Thanks to the many proofs above, "point nine recurring equals one" is just such a theorem. So, your opinion is wrong. And sorry, but no: you're not entitled to be wrong in mathematics. That's not how it works.

I love this quote. It's made of truth.

[Mono Vertigo]

Also, because no science thread is complete without XKCD: http://xkcd.com/263/

[Maralais]

I am he as you are he as you are me and we are all together.

I'm getting my mathematically-challenged coat.

Goo goo go-joo. Had the same idea when I saw that :smallbiggrin:

[PirateMonk]

You're not entitled to be wrong, but you are entitled to create your own system with whatever axioms you want. It will just make it hard to talk to other people about math.

[Peregrine]

You're not entitled to be wrong, but you are entitled to create your own system with whatever axioms you want. It will just make it hard to talk to other people about math.

As long as it's consistent. That's something I came to realise while learning how to teach concepts like complex numbers. You can assume anything you like, make up anything you like, and as long as you get consistent results at the end, voila, you've got workable mathematics.

And then if it's consistent and useful, people might actually take notice. :smallwink:

Take Science Officer's link to that "nullity" article, for instance...

And to confuse the whole issue of infinity and dividing by zero, there's this fellow (http://www.bbc.co.uk/berkshire/content/articles/2006/12/12/nullity_061212_feature.shtml).

Dr Anderson is entitled to say that "0/0 = nullity" if he likes, but I wonder whether it really gives consistent results, and I seriously doubt that it gives useful results. (I can also find one absolutely false statement in his answers. You can't represent his "nullity" in binary. You can encode it in binary, choosing a binary number to symbolise it, but without that encoding your binary number will actually represent an ordinary, integer value.)

[Tirian]

Dr Anderson is entitled to say that "0/0 = nullity" if he likes [...]

He actually isn't. Division is defined to be the solution to the associated multiplication problem. When I want to know what 12/3 is, I ask whether there is a number x such that 3x = 12. There is, in fact, a unique solution to that problem, so we say that 12/3 = 4.

Division by zero gets borked in two different ways, depending on what you're dividing by 0. If you want to calculate 1/0, for instance, you get stuck because there is no real number x such that 0x = 1. Since that number doesn't exist, you can imagine a totally new thing that you define to be the solution to the problem and call it infinity if that's what makes you happy and do whatever you want with it as long as you recognize that it isn't a real number.

0/0 is a different concept, though. It's not that there is no real number such that 0x = 0, the problem is now that every real number satisfies that equation. Which one IS 0/0? Not only can't you choose one of the real numbers fairly, but you can't even create a new concept to be THE solution to the problem like you could with 1/0.

Anderson's notion that you can create an overarching solution to the problem is almost exactly the same as the way that modern computers do math, which is to create a garbage value called NaN ("not a number") for unsolvable problems and any calculation involving NaN. NaN is practical and realistic in ways that Anderson's theory isn't, though, so there is no reason to claim that he solved this problem first or that his solution is preferable in any way.

[Heliomance]

Depends on who you ask! Some (mostly 'pure') mathematicians use log to mean natural logarithm. Just about everybody else use 'log' to denote a base 10 logarithm and use ln for natural ones.


Until you start playing with complex exponentials and logarithms, at which point I seem to remember it becomes 'log' again.

[Bryn]

The nearest book to me (Mathematical Methods for Physics and Engineering by Riley, Hobson and Bence) uses Ln (as opposed to ln) for the multi-valued complex logarithm. I think it's quite arbitrary what you call it.

[Peregrine]

He actually isn't. Division is defined to be the solution to the associated multiplication problem.

Let me rephrase. He's entitled to say "0/0 = nullity", if and only if he can provide a definition of nullity, and division, and possibly zero, that provides internally consistent results. (Division doesn't have to be the inverse of multiplication, though that's the most widely used, and useful, definition.)

The definitions probably won't be acceptable if they're wildly different to the customary meanings, of course. And the whole thing would be ignored anyway if it didn't provide some sort of useful advantage over existing solutions like NaN.

I do agree with you, though; I doubt that Dr Anderson did anything of the sort. He specifically refutes the idea that his "nullity" is an indeterminate value like NaN; he says it's a number, one of his "trans-real" numbers, for which he claims consistent results -- but I don't see how 0/0 can be consistent in a system that includes the reals. He also claims that the indeterminacy of NaN (and relatedly, the fact that NaN != NaN) is not "natural" to programmers. As a programmer (teaching came as a career switch), I say he's bonkers.

I once got bored and came up with an algebra that allowed division by zero and made infinity a number. Actually it had three symbols -- zero, infinity, and U -- or six if you count the negative of each (yes, it needed signed zero). Unlike Dr Anderson's "trans-reals", it didn't include the reals; it couldn't. It mapped to them, though. U (for "unknown") mapped to any positive real number, so converting back to the reals gave indeterminate results (which Anderson says he managed to avoid).

PS I haven't had any luck finding details of the trans-reals on Dr Anderson's website (http://www.bookofparagon.com/). I'm getting lost in the maze of claims to have unravelled feelings, free will, and no doubt Life, the Universe and Everything.

PPS Found it! This "exact arithmetic" PDF (http://www.bookofparagon.com/Mathematics/SPIE.2002.Exact.pdf) contains Anderson's "trans-rationals" rather than the "trans-reals", but it nicely demonstrates how he's treating 0/0. It is, for all intents and purposes, the same as NaN. As far as I can see, all addition, multiplication, or inversion involving 0/0 gives 0/0 as the answer. The only difference is that he axiomatically states that 0/0 = 0/0.

[Vacant]

lern2poststructuralism:

1. Arbitrary signifiers are arbitrary.
2. Define "1" and "2" to equal the same thing.
3. 1=2, both may also equal 3.

It works to make any two things equal, by the way.

[PirateMonk]

PS I haven't had any luck finding details of the trans-reals on Dr Anderson's website (http://www.bookofparagon.com/). I'm getting lost in the maze of claims to have unravelled feelings, free will, and no doubt Life, the Universe and Everything.

There's a PDF in the Books section which looks promising.

Edit: Imagine two objects, one going at 2 m/s, the other at 1 m/s. In zero seconds, each object travels zero meters, so, by Anderson's methods, the instantaneous velocity of each is nullity. Since nullity is equal to itself, 2=1.

Does any of this actually contradict his axioms?

[enderlord99]

2(Aleph-null)=1(Aleph-null)

divide by (Aleph-null)

2=1

[Vacant]

I love this quote. It's made of truth.

No, it is made of words, so it is made of metaphors. See my earlier proof on 1=2.

[Bryn]

lern2poststructuralism:

1. Arbitrary signifiers are arbitrary.
2. Define "1" and "2" to equal the same thing.
3. 1=2, both may also equal 3.

It works to make any two things equal, by the way.
Couldn't we, though, reasonably assume that people talking about the statement '1=2' are almost certainly assuming commonly used, standard definitions of '1' and '2' (and '=')? It seems very unlikely that anyone wants a proof of any statement which takes the form '1=2'.

(I think that where people are talking about '1=2' in this context, they probably mean '1', '2' and '=' as defined in the Peano axioms (http://en.wikipedia.org/wiki/Peano_axioms), where 1≡S(0) and 2≡S(S(0)) )

I fear I've made an easily countered claim, as I am aware I know very little about post-structuralisim.

[tyckspoon]

I fear I've made an easily countered claim, as I am aware I know very little about post-structuralisim.

I suspect you've just missed the joke. Post-structuralism is primarily an art/literature criticism philosophy, which basically has jack-all to do with math at any point; you can't counter a claim based on it with anything related to actual math.

[Vacant]

Yeah, basically what tyckspoon said. I mean, sometimes it has to do with math, but only as an absurd joke or as a metaphor.

[Capt Spanner]

2(Aleph-null)=1(Aleph-null)

divide by (Aleph-null)

2=1

Afraid you can't divide by Aleph-Null, since it's not a "number" as such, and thus division is undefined.

[enderlord99]

Afraid you can't divide by Aleph-Null, since it's not a "number" as such, and thus division is undefined.

That's correct. However, all "proofs" that 2=1 are inaccurate; mine's just more obvious.

[RebelRogue]

Afraid you can't divide by Aleph-Null, since it's not a "number" as such, and thus division is undefined.
Division is defined for cardinals, actually. However, it is not always unique. http://en.wikipedia.org/wiki/Cardinal_number#Division

[The Extinguisher]

That's correct. However, all "proofs" that 2=1 are inaccurate; mine's just more obvious.

That's not nessecarily true, but any proof that does show 2=1 and is accurate is derived from axioms that aren't useful or the normally excepted ones.

[RebelRogue]

That's not nessecarily true, but any proof that does show 2=1 and is accurate is derived from axioms that aren't useful or the normally excepted ones.
Alternatively, you have proven, that ZF(C) is in fact not consistent after all.

[Capt Spanner]

That's not nessecarily true, but any proof that does show 2=1 and is accurate is derived from axioms that aren't useful or the normally excepted ones.

It shows that your axioms are internally inconsistant.

I think I've demonstrated earlier that if 2 = 1 I am Roland St. Jude. Since I can't wield any mod powers, I therefore conclude that 2 =/= 1.

[PirateMonk]

It shows that your axioms are internally inconsistant.

I think I've demonstrated earlier that if 2 = 1 I am Roland St. Jude. Since I can't wield any mod powers, I therefore conclude that 2 =/= 1.

That relies on certain assumptions about what '2', '=', and '1' mean. As long as ~(2=1) is not a theorem of the system, it is perfectly consistent (although probably not useful).