Genady

Notice that the axiom of choice has been used in the proof above.

I've started studying measure theory by this book: Measure, Integration & Real Analysis (Graduate Texts in Mathematics) (It is free on Kindle, btw.) It pretty much begins with proving that it is impossible to simply extend the concept of length from a real interval to an arbitrary set of real numbers. The proof is a bit formal, and I want to make it more intuitive while still rigorous. Would like to hear if the following description is good and if it can be improved. So, we want to have a real-valued, non-negative function 'length' defined for any set of real numbers with the following desired properties: a) it gives length of b-a for an interval [a, b], b) if set A is a union of disjoint sets A1, A2, etc., then length(A) = length(A1) + length(A2) + ..., c) length of a set does not change if the set shifts as a whole along the real line right or left. Let's assume that such function exists. Consider interval [0, 1]. It can be partitioned so that each number x belongs to a subset consisting of all numbers whose distance from x is a rational number. IOW, if y-x is rational then y and x are in the same subset. Let set V to contain exactly one element from each subset of the partition above. If we shift V by a rational distance r, we get a set Vr which contains different elements from the subsets. Since all elements in each subset are separated from each other by rational distances between 0 and 1, the union of all sets Vr which are shifted by all rational distances r between -1 and 1, covers the entire interval [0, 1]. So, the sum of lengths of all sets Vr is greater than or equal to the length of the interval [0, 1], i.e. greater than or equal to 1. Since every Vr is just a shifted V, they all have the same length, which is therefore greater than 0. OTOH, all Vr are shifted from the original V by maximum 1 to the right or to the left. Thus, their union is covered by the interval [-1, 2] and therefore the length of the union is less than or equal to 3. By taking enough of the shifted sets Vr we can get the sum of their lengths to be greater than 3. Then we get disjoint sets with the sum of their lengths being greater than the length of their union. This contradicts the property b) above. Thus, a function with the properties a), b), and c) does not exist. Is it clear enough? How can this explanation be improved?

Here is a nice demonstration of the reversibility described above:

Viruses (not cells, of course) do. I don't think that all of cell components being present in a mix at the same time is good. They rather need to come in a right order.

Like in nuclear transfer?

As I said above, what you think is mathematics in fact is accounting.

Dream on.

The morning sun here, too.

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Math definitely is NOT a series of small, logical steps, and it is nothing like solving a puzzle. Contrary to this advice, math problems are usually solved by embedding in ADDED constructions.

I see the problem now. I was talking about mathematics while you are talking about accounting.

Thank you, Paul. This was helpful.

The OP question should be changed to "Is it in the interest of SSA not to delay processing of applications for benefits?"

The update on the issue: none. More than three months later, my application is still pending. All attempts to get a hint from SSA on why it takes so long, failed. SSA in the US tells me to contact their office in DR. A phone message in their office in DR plays for 9-10 minutes and then disconnects. Awaiting their email response, for two weeks now...

It was a rhetorical question. The point of that question was that spacetime is often curved in vacuum without mass or energy being anywhere in the vicinity.

Where are these mass and energy when the spacetime is curved in vacuum?

You can find an overview here: Chemical Kinetics: Rate of reaction, types of reaction on basis of their rate. Notice this:

It says nothing about a process that led to its appearance.

Because I think it is a large-scale process.

You have answered your own question.

We have a direct experimental observation of the two states, without life and with life. If the process of changing the former into the latter is such that it takes, say, at least a million years, we will never directly observe it.

13.8 billion years ago, there was no life in the universe. 13.8 billion years later, there is life. A state without life has evolved into a state with life. This is abiogenesis.

You can take a snapshot of an accelerometer that you hold in your hand. It will show if you accelerate or not.

BTW, the bubbles may be also misleading. When diving Molokini Crater we got caught in a down current once. We held to a rock, but our bubbles were pulled down. The visibility was perfect thought, so we did not have a problem to see where the surface is.

Don't hold your breath. Chances are nobody will respond because, according to the forum Guidelines,