The problem with this definition is that the set of all sets that do not contain themselves as subsets is shown, by Russell's Paradox, to be logically contradictory. Your definition requests that we posit an object which is logically contradictory, and then remove
X
X from it. This is akin to requesting the reader to take the smallest prime number with exactly three divisors, subtract it from itself, and then insist that the answer is 0. — Tommy

My argument is to define X without X as a subset of itself regardless of whether it can or can't be such. In this way the paradox is avoided by defining a set that contains 'All sets...' but not X. X is then included in X₂ and the paradox is avoided. The same argument is then applied to X₂, X₃... The result is that the 'Set of all...' is really an infinity of sets each containing the other.

You should consider checking out NBG class-set theory which is an alternative formulation of set theory. — Tommy

My idea is that it can be framed in terms of set theory alone without the invention of classes. My definition is that if X as a subset of itself is excluded this leaves X\X' where X' is the subset.

But 0.999... = 9 * (0.111...) = 9 * (1/9) = 1 — fdrake

This still begs the question about the difference between a limit and an actual infinite sum. Your reasoning shows that you don't run out of natural numbers 'until' infinity.

You're going in circles. 1 is one of the possible things that sum can be. Pause for a second and think about this; otherwise this could continue forever. — InPitzotl

The reason the unit circle is a standard in geometry is because what applies to the unit applies to a circle of radius 10 or 100 or 1000. It is the internal geometric logic of the thing that matters, not the arbitrary measurement. It is like the difference between inches and centimeters. 1 inch = 2.5 cm. This has more to do with convention than anything fundamental.

If you do. It can also be nothing (0). It can also not be anything. — InPitzotl

But if you draw the x - axis and mark off one unit, there you have it. The sum of dimensionless points add up to a unit width.

Yes, and 0+0+0... can be equal to 1. And 50. And a billion. And negative 7. — InPitzotl

But you still have 0 + 0 + ... = something

No, it's not arbitrary. It's just infinitely non-specific. — InPitzotl

Say 0 + 0 + 0 + ... = 50 units. Simplify-

(0 + 0 + 0 + ...)/50 = 1

But 0/50 = 0. It's just a matter of reducing to the lowest terms and the same logic obtains.

That doesn’t sum to 1, that sums to 1/9. — Pfhorrest

That was a typo, I meant to say 9/10 + 9/100 + ...

If we see 0 repeated an infinite number of times in a sum, we tend to say that the result is undefined. — InPitzotl

Yes, because it can't be defined in terms of calculus but the question remains, what is it?

But you could do the same thing with a segment of length 2, 50, 0, and -7. — InPitzotl

Yes, but that is arbitrary as the unit can be taken as any width, as in geometry - the unit radius can be 1 inch or 1 light year.

Sorry, what is c here and how does that relate to 0.999...? — InPitzotl

It's the same idea 0.999.. = 9(1/10 + 1/100 + ...)

What seems to be happening here is that 1/x = 0 at infinity.

Then you have the absurd(???) conclusion that

1/infinity = 0

1 = 0 x infinity

So 0 + 0 + ... = 1 after an infinity of terms.

There was an Indian mathematician in the Middle Ages who asserted this (I forget his name)

But most mathematicians probably would not accept this.

But there is also a geometric way to "prove" this.

Take the x,y axis and mark off the unit length from 0 to 1.

This unit represents an infinite string of points all lined up in a straight line.

What is the width of each point? Zero. They are dimensionless.

But they add up to 1 unit width. How do da?

zero width + zero width + ... = extension???

Every time a mathematician draws a graph on the x,y axis they are implicitly accepting that 0 + 0 + ... = 1 because they are working under the assumption that an infinity of dimensionless points add up to extension; the unit. Go figure...

When you extend your inequality to infinity, x isn't finite, and you can't say 0 < 1/x < 1 for an infinite x. You never have an infinite number of a finite 1/x where 0 < 1/x < 1. — InPitzotl

Ok, but isn't this what happens with 1/2^c in the sum 1/2 + 1/4...? If we're talking about an infinite sum the same applies: by that way of looking at it you would not have 0 < 1/2^c < 1

Or in terms of 0.999..you can't, by this criterion, say 0 < 1/10^c < 1

But the assertion being made is that 1/10 + 1/100+... can be taken to an infinity of terms and summed to 1. If we take a finite number of terms it won't be 1. That's what I mean by an actual or literal sum. You have to go the whole way.

Such as the vicissitudes of these things. Ramanujan summed the natural numbers and got -1/12.

Calculus is a way of reasoning but there are other ways of reasoning that arrive at different results. Calculus is fine as long as we are talking about finite sums tending towards a limit. But when you go to infinity things get sticky. This is one of the reasons calculus was formulated in terms of finite sums going to the limit, so that the paradoxes at infinity would not interfere.

(a) a value x such that 0 < 1/x < 1, (b) an infinite number of those values. You can't have (b) with any finite number. You can't say (a) "at infinity". Since you need both, and never have both, you cannot apply Theorem 1. — InPitzotl

Ok, but can't this be also said for 0.999...? Adding terms and then saying 'at infinity'. You can't have (b) at any finite number of the terms 9/10, 9/100...but ya gotta say 'at infinity' sometime if you assert that the literal infinite sum is 1.

I think I got it (incidentally, c=k here, right?) — InPitzotl

Yes, you got it. The point I'm making is that 1/k is positive and > 0. Even as you go to infinity 1/x can't be zero. So, you are summing an infinity of positive terms > 0 which is infinity, right? As the number of terms taken increases 1/x decreases but never becomes zero.

1/2+1/4+1/8+...+1/2^1023+1/2^1024 < 1024+1024+...+1024 — InPitzotl

I'm not talking about the 1024th term. I'm saying-

1/2+1/4+1/8+...+ 1/2^c to k terms - whatever the value of k

>1/2^c + 1/2^c + 1/2^c to k terms. There are the same number of terms in each series.

Contemplate this; I have three dozen pies. I divide them amongst the nine of us. — Banno

Personally I think mathematics is not really about numbers. Mathematics is more about harmonies and proportion. Numbers are 'markers' in the symphony of proportion and relation. The real music of mathematics is beyond numbers. Just a thought...

So that it works at term 10 is irrelevant, because the inequality fails at term 1024 and for all terms after it. You can't go from 10 into infinity without going passing 1024. — InPitzotl

I'm not up to speed on binary. I don't think you understand what I'm saying.
Are all of the terms in 1/2, 1/4, 1/8...positive and > 0? Yes.
For all c 1/2^c is positive and > 0.

Let c run to infinity and sum. Now you have an infinite sum of positive quantities > 0 and that's infinite.

There's no point in saying calculus says otherwise because calculus does not deal explicitly in infinite sums. You need the God calculator for that.

There is no infinite sum of equals on the left side. — InPitzotl

1/2 + 1/4 + 1/8+....+1/1024 > 1/1024 + 1/1024 + 1/1024+....+1/1024

LHS has the same number of terms as RHS

Now let the number of terms run to infinity and the sum on RHS is infinite at infinity.

The next inequality would be-

1/2 + 1/4 + 1/8+....+1/2048 > 1/2048 + 1/2048 + 1/2048+....+1/2048

in fact, they're smaller than 1/2^1024. — InPitzotl

It doesn't matter. An infinite sum of equal infinitesimals must be infinite.

After the 1024th term, we're adding numbers on the left much smaller than 1/1024; in fact, they're smaller than 1/2^1024. And on the right, for each of these, we're just adding 1/1024. — InPitzotl

What is $1/2^c$ added to itself infinitely?

$1/2^c$ added $2^c$ times is $1$

Now add another $2^c$ terms$= 1$

1 + 1 + ... = $\infty$

Your left sum is 0.111....11 with 1024 1 bits in binary. Your right sum is 1. Is 0.111...11 with 1024 1-bits greater than 1? — InPitzotl

The sum of terms 1/x is infinity if 1/x > 0.

...and you'll find the inequality always breaks down for some number of terms, and all terms after that. In fact, you can cheat... whatever integral x you specify, it will break down at the xth term. — InPitzotl

x is a power of 2.

1/2 + 1/4 + 1/8+....+1/1024 >

1/1024 + 1/1024 + 1/1024+....+1/1024 for the same number of terms.

Now go to infinity with x. You are still adding an infinity of positive terms.

This is false soon as the number of terms is greater than or equal to x, after which point the bottom sum is greater than 1 and the top sum is still less than 1. — Pfhorrest

x is not static. I'm saying if there are the same number of terms in each. Now increase x indefinitely with the same number of terms top and bottom.

1/2 + 1/4 + 1/8+....1/x >

1/x + 1/x + 1/x+...+1/x

Now let x go to infinity-

$\sum_{i=1}^{\infty} 1/x = \infty$

The point of a limit is that the sum never exceeds it. No matter how many terms you add to that convergent series, it will never exceed 1. Why then would you think it could ever add to to infinity? If it could, that would make it a divergent series, one with no limit, by definition. — Pfhorrest

I don't insist it doesn't sum to 1. You may well be right. I'm saying we don't know because we can never have an actual infinite sum. An infinity of positive quantities are being summed and any positive quantity summed infinitely, is infinity.

What is the 'last' term in the sequence 1/2, 1/4, 1/8...? No need to answer, it is a rhetorical question. But surely all terms - the whole infinity of them - are positive and > 0. Right? Now sum an infinity of positive quantities...

One of those series diverges; it does not have a limit. The other converges: it has a limit. The second one never gets anywhere close to infinity no matter how long you run it. It would only ever even get up to 1 if you ran it forever, with your “God-calculator”. — Pfhorrest

But that begs the question: you say it don't sum to infinity because it don't sum to infinity. That is the very thing that is being questioned. I know the limit is 1. But that limit is defined by finite arithmetic. I am asking what really happens at infinity. An infinity of positive quantities are being summed and any positive quantity summed infinitely, is infinity.

This becomes false as soon as k > 2. — Pfhorrest

I know, I made a mistake. Let me rethink how to formulate it...

rethinking...

you cannot talk about this reified "actual sum" unless you can talk about it, and I'm not sure you've convinced me there's a thing to talk about. — InPitzotl

What I'm saying is very simple. Suppose you had a kind of God calculator that would print out the actual addition of 9/10 + 9/100... to an infinity of terms, what would that be, 1 or infinity? That's what I mean by the actual sum.

To show how quirky infinite sums are consider the following (this is not meant to answer anything, it is just to illustrate how strange things become at infinity)

Theorem 1
Define 1/x such that 0 < 1/x < 1. If 1/x is summed to itself infinitely often, the sum is infinity. From this we conclude that any positive quantity added infinitely sums to infinity

Now sum 1/2 + 1/4 + 1/8...in view of the above theorem. No term in this series is zero, they are all positive quantities. So we are summing an infinity of positive quantities, some of the bigger than others...

Again, I'm not trying to answer anything here but it is worth contemplating.

Apparently not... see the underlined as evidence for your continued confusion of the same point. The sum is by definition the same as the limit. — InPitzotl

I missed this post. Yes, the definition of the sum is the same as the limit. But I am talking about an actual sum. An explicit infinite sum that you can write down. This of course is not possible because it requires an infinity of calculations. That's the point I'm making. I understand sums and limits but that is not really how the question can be answered. It can only be answered by an explicit infinite sum. What makes me suspicious is the paradoxes that exist at infinity.

1. Is there a theory of intelligence that explains these statistics?

No.

2. Are the statistics a reflection of systemic bias against women? — TheMadFool

No.

You have to account for social factors. Only in affluent societies do women have the means to peruse science etc. Women don't go into scientific areas as much as men, even when they have the means to. Men who are breadwinners will sometimes have jobs that could lead them a nobel prize. Women often choose types of work, like medical care, that won't lead to a nobel prize. There are dozens of reasons why.

A ninth of that particular pie is a particular quantity. A ninth, or 1/9, is not a particular quantity. Are you capable of understanding this? — Metaphysician Undercover

1/9 is a proportional relationship. In geometry if the side of the square is $1$ the diagonal is $\sqrt{2}$ and the proportion is $1:\sqrt{2}$

Hmm, have you ever asked yourself a simple question "why"? Definitely we can, and we do, for centuries (since Leonard Euler's time) — Andrey Stolyarov

No, we know what a limit is. But as I keep saying, a limit is not a literal infinite sum. Weierstrass did not formulate calculus in terms of literal infinities. He formulated it in terms of finite sums converging to a limit. I'm not arguing that limits are not as they are defined. I'm saying a literal infinite sum is qualitatively different to a finite sum. The definition of a limit only makes sense in terms of finite sums converging to a limit. This is how the limit was rigorously defined by Weierstrass, Cauchy...That definition says that the finite sum can get ever closer to the limit. But you are jumping from the finite to the infinite and saying finite calculations still apply to infinite sums.

The limit of 1/2 + 1/4 + 1/8... is 1. But what is the infinite sum, as a literal infinity? We cannot assume it is 1 just because finite arithmetic points in that direction.

The infinite sum itself has been defined to be the limit... by mathematicians — InPitzotl

I'm aware of that. But it has not been explicitly defined. It has been defined in terms of limits which are limits of finite sums.

Mathematicians regularly compute what they mean by it. — InPitzotl

They compute limits which are not the same as sums. A limit is what a finite sum converges to.

Actual mathematicians do. — Pfhorrest

See next answer.

An infinite sum is defined because the mathematics community defined it; same as "twelve" is defined because English speakers defined it. — InPitzotl

That still begs the question what is an infinite sum if nobody has ever computed it? You can't jump from the finite to the infinite and expect finite rules to apply. And it is questionable that it has been defined. All that has been rigorously defined is a limit.

The infinite sequence of additions implied by a series cannot be effectively carried on (at least in a finite amount of time). However, if the set to which the terms and their finite sums belong has a notion of limit, it is sometimes possible to assign a value to a series, called the sum of the series. This value is the limit as n tends to infinity (if the limit exists) of the finite sums of the n first terms of the series, which are called the nth partial sums of the serie — Wikipedia

Emphasis mine.

They're talking about infinite series, and saying that the limit is the sum of that series. I didn't quote the whole article, just the relevant part. Click the link and read for yourself. — Pfhorrest

It hardly matters. An infinite sum is undefined because nobody has ever computed it. The problem is that while it is ok to apply the logic to finite quantities, nobody knows what an infinite sum is.

When this limit exists, one says that the series is convergent or summable, or that the sequence is summable. In this case, the limit is called the sum of the series. — Wikipedia

They don't say the infinite sum, ie the sum of all terms.

When this limit exists, one says that the series is convergent or summable, or that the sequence is summable. In this case, the limit is called the sum of the series. — Wikipedia

You need to provide the link. As a working definition that may well be useful but there are still problems with infinite sums.

A limit is by definition something that will not be exceeded. — Pfhorrest

By a finite number of terms.

What do you think "the limit is 2" means, if not "this will never add up to more than 2, no matter how many terms you add"? — Pfhorrest

Yes, you are correct but what is an infinite sum of terms? What happens when you sum an infinity of terms? Calculus does not account for this.