studiot

Hello, Sensei. You misunderstood my comment about simple and compound. Perhaps it's a language thing. I am using the terms in a specific technical way, in this case referring to the number of independent quantities that have to be measure to complete the measurement. For instance for the position of the ship two quantities have to be measured, latitude and longitude. For the position of the mountain summit, three viz lat, long and elevation. These are compund measurements. Simple measurments measure only one thing and can be directly measured, for instance area and volume can both be directly measured as can the number of apples in the bag or its weight. Hope this clarifies my point. I didn't say they were measurements of physical objects. Do you consider physicality in some way necessary to measurements?

In my experience, "what exactly is.......?" questions end up going round in circles more often than not. Because you have to start somewhere (definitions as Klaynos mentions) in order to have some discussion material to work on. 'exactly' starts by implying a measurement. 'is' opens up the whole existance/reality debate. 'what' presupposes existence, since if there is no existence how can the existence of the 'what' be discussed? I'm not trying to be akward, only a three word +1 sentence and already we see how difficult general philisophy is. Perhaps the best we can do in these circumstances is to introduce, consider and discuss, specific aspects. One issue is what to include and what to exclude in what we regard as 'measurement'. ydoaPs provides a concrete example, but what if we are measurand is abstract, say the position of a mountain or a ship? We cannot measure the position without performing (or having some machine perform for us) calculations. So are calculations measurements? Is the calculation 3 x 4 = 12 in some way a measurement of 12? This brings in another point. In simple arithmetic we confidently expect that 3 x 4 = 4 x 3. That is order does not matter when making our measurement that involves more than one thing. Can measurements be broken down into simple and compound? But wait, the whole basis of modern quantum theory can be traced to the pure mathematical notion that order does matter and the Heisenberg Uncertainty Principle and other uncertainty principles can be derived from the fact that in some mathematical calculations [math]A*B \ne B*A[/math] Klaynos also mentioned accuracy. Modern measurement theory does not divorce accuracy from the measurement. So the question and answer is interesting. How much does that bag of apples you are carrying weigh, Bill? Ten tons. Well is that inaccurate estimate a 'measurement'? If not, how near does the estimate have to be to be considered a measurement? And what is the difference between a measurement and an estimate? Which brings us back to my earlier point. Do we consider 3x4=12 a measurement? If you say no, what if I asked for the area of a room, you had measured as 3m x 4m? ydoaPs you have introduced a very tricky subject rather neatly. +1

Your musings about matrices are pretty close to the mark. Dual spaces turn up in all sorts of places and functional analysis is of great importance in applied maths. Keep up the good work.

Why would that be? Please let's not go too far down this side road, about sets. The thread is about properties of members of sets.

Yes sort of. You have the right idea but be careful with the teminology. This only comes with practise. Is R2 just a vector space? for instance? Wouldn't that statement imply that all transformations in R2 are linear, Are they?

Does this not prove there is only one empty set?

I don't know if your reading since August will have taken you this far but I would say that your article is referring to linear functionals. Extract from Wikipedia. Functional analysis is important, and was all the rage 50 years ago. But to appreciate it you need to know what functionals and the dual space are. Indeed you need to know what a space is. So a set is simply a collection of memebrs. A space is when we impose a 'structure' to that set. Roughly a structure means rules and relationships between members. Sometimes we include more than one set in the space. This is the case with vector spaces. A vector space has two sets, the set of vectors and a second set of coefficients. A requirement of the second set is that it forms a 'Field'. As you so rightly observe certain specific number sets such as the reals form a field. A functional is a function that associates each vector in the set of vectors with an element from the field set in the space. A common example is the definite integral. The set of all functionals is called the 'dual space' of the vector space.

Somewhere, buried in set theory, you will find a theorem that the empty set is a subset of every set. In other words there is one and only one empty set. Please remember ( I admit to difficulties here as well), this thread is about properties of the members, not the set.

Yes, you have to be at least a little bit mad to be one.

It has no members, so how many properties do its members have? In a way this a quite a good question since there are zero members on the membership list, there are zero member properties. I don't know if this is a valid answer though, according to ZF axioms. Thank you , ajb, for introducing those. How would the null set play with axiom3, the exclusion or russian doll axiom? That would suggest that no (zero) properties can be associated with the null set. Axiokm3 would allow (demand?) the formation of a subset of {3, 26, goats} which contains only numbers

The discrepancy discovery is not yours to share. That honour belongs to messrs Hall and Rossi.

I really started by thinking about the two ways of defining a set. 1) List all the members 2) Specify a common property possessed by all members. Now every member of every set has one property. The property of being on the list of members of that set. Your {my brother} example identifies one aspect. It does not include all men, or even maybe all your brothers. So there are potential male members that are excluded. But let us move on to numbers to make this point clearer. Suppose we make our set The set of all numbers greater than zero. 1) The membership list property includes all the positive numbers 2) Every number on the list has a second property that has to be stated, that of exceeding one. We also know that there are more numbers that are not on the membership list. So we have at least two properties, and we don't need to mention any more, even though we could easily propose many. But what about the set {3, 26, goats} Do we need a second property?

At least one. I was thinking that either we have to do away with the ability of specifying set membership by property or accept two as a minimum number of properties.

I'd be more interested in how he explains his handle.

What is the minimum number of properties posessed by members of a set?

Perhaps you would like to rephrase this?

Would you like to explain this. It makes no sense to me. Nor does showing pictorial sequences of allegedly convergent or divergent boundaries all the same distance apart. I would suggest you need to conduct some old fashioned geological investigation before offering some of the above statements of geological history. You should check that the actual rocks found in the field conform to your theory. Are they of the correct Age? Are they of the correct type - sedimentary or igneous? What is the orientation of their bedding? Are they the right way up or is there an inversion unconformity? For instance a recent poster failed to do this with another theory of the Himalaya http://www.scienceforums.net/topic/91603-evolution-of-himalayas-and-tibet-and-the-great-volcano/ And failed to realise that the region comprises two distinct blocks one sedimentary and one igneous. Please make sure you do not fall into the same trap. Incidentally no one here is trying to cleverly trap you. But we will rigorously test your statements for logical consistency, both with known observed facts like the known geological maps and sections and other measurements, and internally with themselves. Nice to see you back.

Brett, please note that Strange offered you a formula involving delta t or time differences. Your formula involves t or time alone. That implies some sort of absolute time or time synchronisation between two reference frames, which you cannot have.

Yes, Mr Zeeman induces them.

Sounds good, but In even modest sized outfits the telephone and broadband is no longer powered by the telco line, it is powered from the same mains as the server. So unless you also have a UPS on you telecoms equipment your server will be unable to send an Email.

It is a question of patiently putting in the correct masses and friction coefficients at the appropriate points. I don't get exactly the same results, but it would depend upon the value adopted for g. Here are my calculations 1) [math]acceleration = \frac{{Tension}}{{totalmass}} = \frac{{248}}{{92 + 17}} = 2.275m/{s^2}[/math] 2) [math]Force = mass*acceleration = 17x2.275N[/math] 3) [math]Max{F_{top}} = mass*\max acceleration[/math] [math]0.8*17*g = 17*{a_{\max }}[/math] [math]{a_{\max }} = 0.8g = 0.8*9.81 = 7.848m/{s^2}[/math] [math]Max{F_{total}} = Mas{s_{total}}*{a_{\max }} = 109 *7.848 = 855.432N[/math] 4) [math]{F_{topsliding}} = Mas{s_{top}}*acceleratio{n_{top}}[/math] [math]0.62*17*g = 17*{a_{topsliding}}[/math] [math]{a_{topsliding}} = 0.62g = 0.62*9.81 = 6.0822[/math] 5) [math]acceleration = \frac{{netforce}}{{mas{s_{botom}}}} = \frac{{Tensionpull - frictionfromtop}}{{92}} = \frac{{1223 - 0.62*17*9.81}}{{92}}12.17m/{s^2}[/math]

As regards the decay, the is a difference between the rate of decay and the decay itself. The rate of decay tends towards the exponential curve as the number of trials tends to infinity. The decay itself is random. This mean that if you lined up 1,000,000 U238 atoms and i pointed to number 1093738 in the row it's decay or not would occur in a totally random fashion. That particular atom could be stable for longer than the age of the universe or it could decay in the next second, and you have no way of predetermining which will occur. That is also why we can construct a clock from radioactive decay, as I said earlier and why this process can also be used to show the need for a time coordinate as well as measure it. Length is not involved, only number.

Fully random? Does your claim have a meaning other than being just plain wrong.

No it is a much much much wider subject than that. Most mathematics is linear, and the first approach to any non linear maths is to try to linearise it (=find a linear approximation). You really need to find out and understand what linear maths is. I do not mean study all its ins and outs, that would take years, just find out enough to recognise what is linear and what is not and to appreciate the principle consequences of that distinction. The following polynomical falls into the ambit of linear maths y = ax6 + bx5 + cx4 + dx3 + ex2 + fx because it is a linear combination of basis polynomials x6, x5, x4, x3, x2 x. this polynomial does not (is non linear) y = x + xy Note the word is basis not base. Detail is important in mathematics.

'close to' is introducing the mathematica (topological)l concepts of connectedness, neighbourhoods, compact sets, all of which I think are violated by the big bang itself.