## Who invented Infinity=infinity+1?

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mollwollfumble
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### Who invented Infinity=infinity+1?

Looking on the web, I find that John Wallace invented the symbol we use for infinity, in order to describe the infinitesimal, in his: "Arithmetica infinitorum, sive nova methodus inquirendi in curvilineorum quadraturam aliaque difficiliora matheseos problemata" (1655).

But who invented the statement "infinity = infinity + 1"? And under what circumstances?
Was it Cantor in 1895? Or earlier than that? I think it must be earlier.

I need to know, because I'm working with the surreal numbers.

Qaanol
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### Re: Who invented Infinity=infinity+1?

mollwollfumble wrote:Looking on the web, I find that John Wallace invented the symbol we use for infinity, in order to describe the infinitesimal, in his: "Arithmetica infinitorum, sive nova methodus inquirendi in curvilineorum quadraturam aliaque difficiliora matheseos problemata" (1655).

But who invented the statement "infinity = infinity + 1"? And under what circumstances?
Was it Cantor in 1895? Or earlier than that? I think it must be earlier.

I need to know, because I'm working with the surreal numbers.

Try reading the Wikipedia page on infinity as a starting point. If that doesn’t satisfy your needs with, for example, its statement that “The Isha Upanishad of the Yajurveda (c. 4th to 3rd century BC) states that "if you remove a part from infinity or add a part to infinity, still what remains is infinity".” then try following its citation links. That’s a good way to begin researching a topic, or at least discovering keywords to search. If you still need more, hit up JSTOR.

Also, the statement “∞=∞+1” is pretty much a standard definition of infinity. Namely, the cardinality of a set is infinite if the set can be put into bijective correspondence with one of its proper subsets.
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mollwollfumble
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### Re: Who invented Infinity=infinity+1?

Ta, but that doesn't answer the question yet.

> Also, the statement “∞=∞+1” is pretty much a standard definition of infinity.

It's an awful definition of infinity. I can easily find you five better ones, like infinity is the surreal number {Z|} for Z as the integers, or infinity is S(N) where N are the natural numbers and S is the ZFC successor function, or infinity is the first non-finite von Neumann ordinal, or infinity is the sequence s_n=n, or infinity from Nonstandard Analysis, or infinity is the number of natural numbers, or is {N}, etc. In all of those infinity is not equal to infinity plus one.

skeptical scientist
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### Re: Who invented Infinity=infinity+1?

mollwollfumble wrote:...infinity is the surreal number {Z|} for Z as the integers, or infinity is S(N) where N are the natural numbers and S is the ZFC successor function, or infinity is the first non-finite von Neumann ordinal, or infinity is the sequence s_n=n, or infinity from Nonstandard Analysis, or infinity is the number of natural numbers, or is {N}, etc. In all of those infinity is not equal to infinity plus one.

Surely number of natural numbers is equal to itself plus one, under any reasonable definitions.
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mr-mitch
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### Re: Who invented Infinity=infinity+1?

Do you mean none? My definition of infinity is that for all other numbers x, infinity >= x.
This implies infinity = infinity + 1, because of the antisymmetry of >=.

Somewhat unrelated definition, that a number n is sufficiently large under some precision e > 0 if 1/n < e or m is insignificant compared to n if m/n < e.

These definitions are likely to have many problems, though.

Giallo
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### Re: Who invented Infinity=infinity+1?

mr-mitch wrote:Do you mean none? My definition of infinity is that for all other numbers x, infinity >= x.
This implies infinity = infinity + 1, because of the antisymmetry of >=.

Somewhat unrelated definition, that a number n is sufficiently large under some precision e > 0 if 1/n < e or m is insignificant compared to n if m/n < e.

These definitions are likely to have many problems, though.

What about the fact that the logarithmic spiral [imath]r(\phi) = a^{\phi}e^{i\phi}[/imath] is infinite?
What about the concept of infinity in the complex numbers (the [imath]\geq[/imath] is not defined there...)?

I think that the standard definition:
Qaanol wrote:The cardinality of a set is infinite if the set can be put into bijective correspondence with one of its proper subsets.

is the best one possible.
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Qaanol
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### Re: Who invented Infinity=infinity+1?

mollwollfumble wrote:Ta, but that doesn't answer the question yet.

I wasn’t trying to answer the question, I was trying to teach you how to fish for the answer on your own. And in doing so, I provided evidence that the statement in question was over 2,000 years older than your first guess.
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gmalivuk
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### Re: Who invented Infinity=infinity+1?

Qaanol wrote:I provided evidence that the statement in question was over 2,000 years older than your first guess.
I'm not sure why that should be terribly surprising, either. Just because the now-standard symbol for (a point at) infinity was invented in 1655 doesn't mean the concept itself only showed up around the same time.

After all, it's not like the symbols for most of our other numbers are very old, either.
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MartianInvader
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### Re: Who invented Infinity=infinity+1?

mollwollfumble wrote:Ta, but that doesn't answer the question yet.

> Also, the statement “∞=∞+1” is pretty much a standard definition of infinity.

It's an awful definition of infinity. I can easily find you five better ones, like infinity is the surreal number {Z|} for Z as the integers, or infinity is S(N) where N are the natural numbers and S is the ZFC successor function, or infinity is the first non-finite von Neumann ordinal, or infinity is the sequence s_n=n, or infinity from Nonstandard Analysis, or infinity is the number of natural numbers, or is {N}, etc. In all of those infinity is not equal to infinity plus one.

All of these definitions seem to imply that the number of real numbers is not infinity. That would be a shame.
Let's have a fervent argument, mostly over semantics, where we all claim the burden of proof is on the other side!

lalop
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### Re: Who invented Infinity=infinity+1?

Why is everyone here talking as if infinity was a number?

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### Re: Who invented Infinity=infinity+1?

MartianInvader wrote:
mollwollfumble wrote:Ta, but that doesn't answer the question yet.

> Also, the statement “∞=∞+1” is pretty much a standard definition of infinity.

It's an awful definition of infinity. I can easily find you five better ones, like infinity is the surreal number {Z|} for Z as the integers, or infinity is S(N) where N are the natural numbers and S is the ZFC successor function, or infinity is the first non-finite von Neumann ordinal, or infinity is the sequence s_n=n, or infinity from Nonstandard Analysis, or infinity is the number of natural numbers, or is {N}, etc. In all of those infinity is not equal to infinity plus one.

All of these definitions seem to imply that the number of real numbers is not infinity. That would be a shame.

It isn't? It's 2Z.
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Qaanol
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### Re: Who invented Infinity=infinity+1?

lalop wrote:Why is everyone here talking as if infinity was a number?

Because infinity and 5 are effectively indistinguishable.
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MartianInvader
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### Re: Who invented Infinity=infinity+1?

Robert'); DROP TABLE *; wrote:
MartianInvader wrote:
mollwollfumble wrote:Ta, but that doesn't answer the question yet.

> Also, the statement “∞=∞+1” is pretty much a standard definition of infinity.

It's an awful definition of infinity. I can easily find you five better ones, like infinity is the surreal number {Z|} for Z as the integers, or infinity is S(N) where N are the natural numbers and S is the ZFC successor function, or infinity is the first non-finite von Neumann ordinal, or infinity is the sequence s_n=n, or infinity from Nonstandard Analysis, or infinity is the number of natural numbers, or is {N}, etc. In all of those infinity is not equal to infinity plus one.

All of these definitions seem to imply that the number of real numbers is not infinity. That would be a shame.

It isn't? It's 2Z.

Indeed it is. But 2Z is not S(N), nor is it the first non-finite von Neumann ordinal, or the number of natural numbers, etc. so by those definitions it's not infinity. This is, of course, the problem with treating infinity like a number instead of an adjective.
Let's have a fervent argument, mostly over semantics, where we all claim the burden of proof is on the other side!

lightvector
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### Re: Who invented Infinity=infinity+1?

Defining "infinity" to be any one thing is silly. There are a lot of concepts that all fall under the umbrella of "infinity" and which one is most useful depends on what properties you are interested in. No single definition captures all of these properties. For example,

The concept of infinity in the context of size or the number of discrete things one has leads to the definition of infinite cardinality and infinite sets.

The concept of infinity as an element that comes after infinitely many things in sequence leads to the definition of the ordinal numbers.

The concept of infinity as an arithmetic quantity that can be added and multiplied leads to various things like the hyperreal and the surreal numbers.

The concept of infinity as a topological limit in different ways such as as "infinitely positive/negative" or "infinitely far away" leads to the extended real numbers, the projective real line, and the Riemann sphere.

Yesila
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### Re: Who invented Infinity=infinity+1?

Qaanol wrote:
lalop wrote:Why is everyone here talking as if infinity was a number?

Because infinity and 5 are effectively indistinguishable.

And here I always thought it was infinity and 8 that were hard to tell apart.

tomtom2357
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### Re: Who invented Infinity=infinity+1?

There is a system in which infinity does not equal infinity+1, this is the ordinals!
I have discovered a truly marvelous proof of this, which this margin is too narrow to contain.

Proginoskes
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### Re: Who invented Infinity=infinity+1?

tomtom2357 wrote:There is a system in which infinity does not equal infinity+1, this is the ordinals!

And the surreals. In fact, (one) infinity is [imath]\omega = \{Z\, | ~\}[/imath], but [imath]\omega + 1 = \{\omega | ~\}[/imath].

snow5379
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### Re: Who invented Infinity=infinity+1?

For every number between 0 and 2 there's a number between 0 and 1 that's half of it. Therefore there are at least as many numbers between 0 and 1 as 0 and 2. In other words 1 x infinity = 2 x infinity. Basically... infinity plus, minus, times, or divided by anything is either infinity or undefined.

Qaanol
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### Re: Who invented Infinity=infinity+1?

snow5379 wrote:For every number between 0 and 2 there's a number between 0 and 1 that's half of it. Therefore there are at least as many numbers between 0 and 1 as 0 and 2. In other words 1 x infinity = 2 x infinity. Basically... infinity plus, minus, times, or divided by anything is either infinity or undefined.

I’d be pretty comfortable with |ℤ|÷|ℝ| = 0, actually.
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fishfry
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### Re: Who invented Infinity=infinity+1?

Qaanol wrote:
snow5379 wrote:For every number between 0 and 2 there's a number between 0 and 1 that's half of it. Therefore there are at least as many numbers between 0 and 1 as 0 and 2. In other words 1 x infinity = 2 x infinity. Basically... infinity plus, minus, times, or divided by anything is either infinity or undefined.

I’d be pretty comfortable with |ℤ|÷|ℝ| = 0, actually.

Hi there,

Would you be comfy with |R| * 0 = |N| ?

If so, you'll soon be in trouble; and if not, what do you mean by the division sign?

Qaanol
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### Re: Who invented Infinity=infinity+1?

fishfry wrote:Would you be comfy with |R| * 0 = |N| ?

Nope, |ℝ|*0 will remain undefined.

fishfry wrote:If so, you'll soon be in trouble; and if not, what do you mean by the division sign?

An operator in its own right that happens to coincide with the functional inverse of multiplication when restricted to finite values.
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fishfry
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### Re: Who invented Infinity=infinity+1?

Qaanol wrote:
fishfry wrote:Would you be comfy with |R| * 0 = |N| ?

Nope, |ℝ|*0 will remain undefined.

fishfry wrote:If so, you'll soon be in trouble; and if not, what do you mean by the division sign?

An operator in its own right that happens to coincide with the functional inverse of multiplication when restricted to finite values.

I see that you are saying that if A and B are cardinals with A < B (meaning that there exists an injective function from A into B, but no injective function from B into A) that A / B = 0. There's no problem with that, but you are only using '/' as a notation for A < B.

Are there any other pairs of cardinals for which your division operation would be defined? For example do you define a value for |N| / |N|, or 2|N| / |N|, or 2|N| / 2, or any other pair of cardinals?

Or is your use of the division operator simply a shorthand or alternate notation for A < B?

tomtom2357
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### Re: Who invented Infinity=infinity+1?

Guys, in my mind there are three way to define infinity, as the limit of the natural numbers (which isn't very good as you bump into some problems there), as an ordinal, or as a cardinal.In the first way, infinity+1 is undefined, in the second way, infinity/=infinity+1, but in the third way, Infinity=Infinity+1, so decide what way you think of infinity (I personally like the ordinals) and that will decide the problem for you! I have discovered a truly marvelous proof of this, which this margin is too narrow to contain.

gfauxpas
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### Re: Who invented Infinity=infinity+1?

Isn't infinity shorthand for "for every M > 0 there is some [imath]\delta >0[/imath] etc, or "for every M there is an N etc., or for every [imath]\epsilon[/imath] there is some M etc.? Or is that only in very specific contexts? I haven't learned much set theory, but it sounds like from this thread that infinity has very specific meanings in specific contexts.

Another definition I came across is "something contains an infinite number of elements iff there is some bijection from it to an infinite set". Is that what you guys mean by "cardinals"?

gmalivuk
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### Re: Who invented Infinity=infinity+1?

gfauxpas wrote:Or is that only in very specific contexts?
Yeah, that's just what it means when you're talking about limits in a set that doesn't contain infinity, such as the reals or integers.
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