studiot

How many times do you have to be told by several differe nt members in several different ways. There is no end to the line. There is no last number, symbol, hobgoblin or anything else. (what comes after X ?) Infinity is not only not a number it is not a member of the natural numbers, N, the rational numbers, Q or the Real numbers R. There is no point at infinity in any of these systems. All of these statements are equivalent and mean that to 'reach infinity' you have to step outside the existing system and establish a new one. If you do this you have to prove that all the existing rules and relationships hold good in your new system, you can't just assume them and carry on as though they apply. You have definitely not done any of this spadework, yet you complain bitterly that we are hiding something from you. Every time I ask some more searching questions about your system dodge the issue and offer this reply Your flawed line is not the 'heart of the issue' - We need to go into all this because the examples demonstrate where the flaws lie. When you have accepted those the process of what to do about them can commence. The simplest answer is that you don't have to worry about the end, because the process does not end. I'm sorry but if you want more you will have to do more maths. There is no other way. For instance in my three examples, the first two will never produce a resultant length, but the last one does so a study of this difference is highly relevant.

Most definitely, +1 So why do you keep trying to do it ? (I didn't say -1 here because I don't give negative points.) Let us look more closely at the famous painted line of yours. How wide is it ? Why ever should you think this ? There are whole infinities of points that to use that ugly phrase, "that cannot be reached " For instance every point in the plane alongside your famous painted line. How long is this painted line ? Consider three cases ? You paint the first metre segment and chalk up n = 1 You paint the second metre segment and chalk up n = 2 and so on. A German Mathmatician would say USW = und so weiter BUT You paint the first 1/1 metre segment and chalk up n = 1 You paint the second 1/2 metre segment and chalk up n = 2 You paint the third 1/3 metre ssegment and chalk up n = 3 USW OR You paint the first 1/1 metre segment and chalk up n = 1 You paint the second 1/4 metre segment and chalk up n = 2 You paint the third 1/9 metre segment and chalk up n = 3 Which strip is the longest ? Can you say anything else about the lengths of these strips ? You can say they all have exactly the same count of segments.

As far as I know the only way to use MS (Word) formula writer is to make picture of the formula and paste that. If there is a way to post a formula, wriitten in world I'm all ears. 🙂

[math]\sqrt[3]{{7 + \sqrt {50} }} + \sqrt[3]{{7 - \sqrt {50} }}[/math] [math]\sqrt[3]{{7 + \sqrt {50} }} + \sqrt[3]{{7 - \sqrt {50} }}[/math] If you select the markup and choose the size dropdown menu you get the second large line with size = 36

Thank You. I can't therefore see the problem. The blue dots constitute part of the plane. The white areas constitute the rest of the plane and to satisfy you first condition (non intersection with the blue dots) the shape must lie entirely on the white area. Any circle or disc, centered at the intersection of the diagonals between adjacent four blue dots of radius less than half the interdiagonal length will lie entirely on the white, between the four dots. However you have also stated a second condition that the area has to be less than 1 area unit. Since the larger circles and discs have a area greater than 1, we must slim down the acceptable range of radii to also being less than 1/square root (PI).

Yes this is one place you are most definitely going astray. Please note this does not mean I consider you to be in any way an idiot. An infinite line does not have a length, any more than N has a greatest or last (END) element. In order for a line to have a length it must have two ends. A beginning and a termination. The natural number line has a beginning but no end (termination) The integer line, the rational number line and the real number line have neither beginning nor end (termination).

The rules are not clear to me. Do you mean that the shape must not cross any of the gridlines that the blue points form the intersection points of ? Or do you mean not intersect any of the points themselves ? Incidentally the solution to the problem of not crossing any gridlines is explored quite deeply in structural chemistry and mineralogy

When you say 'increase by 1' what exactly do you mean ? Are you counting or performing arithmetic ? These two processes are very different things, obeying different rules. In order to appreciate this here is a Wikipedia list of what you need to study, which demonstrates why I said it would take so long. I note you didn't reply to that post. List of statements independent of ZFC - Wikipedia It shouldn't be an issue once you realise this as things change once you include infinity. It is an axiom of (finite) arithmetic that there is a unique number Y which has the property that when added to any number , X, the result is x X + Y = X This unique number is of course more familiarly known as zero. However if you introduce infinity you also have X + ∞ = X Contradicting the axiom of 'additive identity' as this axiom is called. +1 to wtf and pzkpfw

As I see your argument in plain English it is this:- Taking you list of sets from post#1 Set the indexing line counter aside for the moment as it is not really needed. Consider the set which contains every set on your list. If such a set exists, call it W . The listing of W then appears as in your list of sets without the indexing. IF you go on long enough why do you not arrive at the set {1,2,3,4...., (n-1), n (n++1)...}, why of course is N ? Of course N is also the indexing set we have ignored up to now. Note also that all the sets up to N are finite, but N itself is transfinite. I mentioned Russel's Paradox which queries the existence of W. This was one of the earliest expositions of many paradoxes that appeared around the 1890s to do with the size of sets. Hints of these difficulties go right back to the Ancient Greeks and Zeno in particular, although they did not have more modern set theory to place the questions in. A proper course of study into the whys and wherefores of these matters takes more than a year so most folks don't attempt it but look for a quick fix explanation. My offering to you is to consider the Greek approach, where they realised that there is more than one infinity. They distinguished two types of infinity viz potential and actual infinity. They believed that there are no instances of actual infinity, which we observe as for instance, the count of numbers between 1 and 2. But their potential infinity does not exist' either for a different reason. It does not exist because no finite process can ever get there. In other words the process does not terminate or goes on forever. Which is what I am suggesting is the reason why your list will never arrive at N. Cantor's approach considering magnitudes has run into difficulties why has yet to be fully resolved. There are at least three different mathematical/logical schemes to try to achieve this. After your year and more of study you would find that none are totally satisfactory as they all wrestle with the idea that some sets are just too big to be contained in other sets. Hopefully you can now sleep happy at night.

Thanks for looking into it and the knowledgeable answer. +1

Not quite. n must be the value of a function that takes on natural numbers ( or the positive integers if you prefer) as its values. That is the only way n can be a different natural numbers in different lines in your OP list.th You are correct about (x+1) also called successor property of the natural numbers. It is this property that invokes infinity and leads to the countable infinity property of the natural numbers. Infinity has many properties, but the one we want in this case is that it cannot be reached. The ancient Greeks called this the potential infinity because it is never actually realised. This is imprecise and inaccurate. Please try to be correct. The list does not have a numerical value or increase by one. Nor does a set have a numerical value. I know what you mean (and your meaning is OK, but it isn't what your words are saying) Until you can state it correctly yourself I doubt if anyone can help you understand.

This rather hinges on what property of infinity is being invoked. What sort of mathematical object do you think n is ? Surely you can't have two different definitions of the same thing ?

Let's revisit your opening post from a new point of view. The sets {1} {1, 2} {1, 3} etc Are all finite sets. This means that if I write out their contents as a series, The partial sums are all finite. Do you know what a partial sum is ? For the three example sets they are 1 1, 3 1,3, 6 The last sum always arrives at a finite number. In other words the series is always convergent or unconditionally convergent. Now look at what happens with infinite series. The 1, 3, 6 pattern goes on forever, getting larger and large at every partial sum. That is the infinite series is divergent. You can add or subtract or do other more complicated arithmetic with any of these finite series, by replacing the series with its final partial sum. So {1+2} + {1+2+3} = 3 + 6 = 9 but what happens if you try to perform these tasks with an infinite series ? {1+2} + {1 +2 + 3+ 4...} = {1+2} + N = 3 + ??? This is the problem lying at the base of simple set theory Note some infinite series are convergent for example the series 1/n2. So as soon as you try to introduce N into your list of sets, you loose all the set operations -Union, sum, difference etc.

Some sets are open, some are closed But some are neither open nor closed and yet others are both open and closed. That much has been known for a long time. cl stands for clopen , and is the subject of study in current maths. As such terminology varies a good deaql. R stands for real You will also references to lambda or theta and supercontinuous functions. So the article I linked to has some of these, rather formally, There is plenty more especially if you look for supercontinuity.

I was originally not going to bother to follow your link as it should be unnecessary to leave the forum to find out what it is all about. However as some members are obviously thinking about it and perhaps benefitting I decided to look. As a result I have some suggestions. Firstly I see that you must have put in a great deal of effort to to complete this table. But I would suggest some improvements are necessary as you have to know quite a bit of chemistry to be able to use it successfully The title is Periodic Table Reactions which is not really quite accurate. It would be better titled Periodic Table Bonding. This is because there is no guarantee that if you placed some of these elements in contact they would actually react. But very often your table would indeed describe the bonds if they dis nindeed react. Secondly I can find any simple covalent bonds ? Are you shure that the Carbon - hydrogen bond in say methane is polar-covalent or dative which is another word for the charge separation effect ? Then of course your tabular entries for molecules such as oxygen., hydrogen, nitrogen are blank, whereas they should be covalent. How would you classify ozone ? You also have blanks where it should read 'metallic bonding'. As regards Science Forums, perhaps if you pasted a screenshot of just the top left corner, and satated that the full table is available at xxx that would cover forum rules. Ask a moderator, some have chemistry backgrounds.

The explanation it a bit tougher I'm afraid. a0c62f4626de2e369d6d43aab0099057c283.pdf (semanticscholar.org)

Oh interesting! I will have to read up on this. +1 Taking a tiredness break and then thinking out it a bit more is exactly the right way to go. +1 I have thrown away most of my old maths notes now, but came across this piece that you might find helpful about my comments on set theory, Godel, and the natural numbers.

This is why it is so important to be precise enough in maths. The correct expression is not 'equivalence' but ' into ' or 'injective' or perhaps stronger 'into and onto' or 'one-to-one' or 'bijective' We also come back to the sine function I mentioned earlier. For 0>= x < 90 the set of all values of sin(x) has an infinite 'count' of values, all of which are finite. But worse, the count of this set is uncountable if x is taken from the real numbers. So your correspondence in this case is merely injective, but not bijective.

It is an axiom that if n exists then (n+1) exists and so on. so our sequence should be written 1, 2, 3, ...(n-2), (n-1), n, (n+1), (n+2),... This is consistent with n being finite, but never the largest element and yet the sequence does not terminate.

Every natural number is finite. It is just that the count of them does not terminate, since there is no last natural number. The set, N of all natural numbers is infinite because it has no limite to the count of its elements. There is a difference between the allowable size of a set and the allowable size of any element of that set. But this also means that the list does not terminate.

Some infinite sets do, some do not. For instance the set of all values of the function f(x) = exp(-x) for all x >= 0 have a greatest element exactly = to 1 This particular infinite set has no least element. A different function, eg the sine function has both a greatest and a least element. Sorry, no.

I am sorry you felt like that since nothing could be further from the truth. You are most definitely not an idiot since greater minds that yours or mine have been baffled by this question. I have noticed since that I mistakenly assumed your set S to be a set of number when in fact I see now that you stated clearly a set of sets. This brings us to the crux of greater minds since this is exactly the situation brought about by Russel's Paradox. That is when you try to apply Cantor - ZFC naive set theory to infinite sets that cannot be members of themselves. This is why Russell and Whitehead introduced the theory of types or classes, which is basically a reclassification of sets introducing a hierarchy of set types. This also paved the way for 'orders of logic' so ZFC is first order, infinite sets of sets is second order which is needed to correctly analyse infinite first order sets of numbers. In general you need a higher order of logic than the one you want to analyse and there are an unknown, perhaps indefinite or infinite, count of orders. This situation lead, in turn, to Godel's famous theorems about the subject. The best plain explanation of all this I have come across is put forwards in Hofstadter's award winning book Godel Escher Back

Propeller (impellor) type pumps work by generating a pressure difference between inlet and outlet. They are limited to the height of a standing column of the fluid when pumping against gravity (about 10 metres for water) A greater pressure difference would actually push the pumped material backwards. This is a lot less than 60m A positive displacement pump like a piston pump works by being one way. A piston pushes the pumped material in one direction (and can be pushed with as much force as necessary) and blocks the material from returning or going the other way. (Your gas meter is another positive displacement mechanism so that it can accurately record flow) On the other hand Nature achieves this displacement by sheer brute force, dumping large quantities of material with sufficient kinetic energy to create the swirling and upwelling. This could be melting ice in the right season or as I mentioned we have now discovered it could be submarine landslides down the continental slope. The sediment in the water is insoluble so in in suspension. That was why it was a sediment in the first place. I will grant there is the so called halocline, which is about solutes, but there is already no shortage of these at sea level since entropy works to try to homogenise bodies of solution.

I did actually answer your question since your list was an incomplete representation of the stated infinite set N. Of course N does not and can not appear on your list since N itself is not a natural number. To put it another way the problem is confusing a set, N which is not finite, with its elements (natural numbers) which are all finite.

Well the average continental shelf depth is just 60m The bottom of the continental slop varies between 3000 - 4000m This affect your previous descriptions and arguments. Both of these depths require somn sort of positive displacement mechanism. It's your thread and idea, not mine. But it does require more than guesswork and wishful thinking to make it into a realistic preposition. You have far to many 'mights' in your words sir.