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Thank you very very very much! The most well-deserved (+1) I have ever given. You have done what I was beginning to think was impossible, you have resolved my issue completely (though I do not like the rule, I will definitely need to let it sink in more.).

Your issue is a non-issue. You don't make any sense. You never do, nor do you seem to care. Having an infinite element (in a particular sense that in the case of natural numbers is clear, and identifiable with a norm) or having infinitely many elements in a set (cardinality), or having a measure of a set are different things. You are --deliberately or not, I don't know-- confusing whether an element is finite (norm?) with how many elements there are in a set (cardinal?), or perhaps a measure (some concept of "extension" or "volume"). One way or another, several members are trying to help you grope towards these important concepts in mathematics, but you don't seem to care, and keep demanding them to address your silly "analogies." BTW, @wtf's last comments go in the direction of your pretence confusion.

You can bijectively map the natural numbers to your Boltzmann line (B-line) as follows: 0 <-> X 1 <-> 0 2 <-> 1 3 <-> 2 ... It's perfectly clear that you can do that, since your B-line and the naturals have the same cardinality. What you can NOT do is map them bijectively in an order-preserving manner. Why is that? Because they have a different order type. The natural numbers have no largest element, while the B-line does have a largest element, namely X. So: There is a bijection, but not an order-preserving bijection, between the points of the B-line and the points of the natural numbers. The two sets are cardinally equivalent, but not ordinally equivalent.

You've discovered the ordinal numbers! The first transfinite ordinal is called [math]\omega[/math], the lower-case Greek letter omega. It's a number that "comes after" all the finite natural numbers. In set theory it's exactly the same set as [math]\aleph_0[/math] but considered as an ordinal (representing order) rather than a cardinal (representing quantity). So the ordinal number line begins: 0, 1, 2, 3, 4, ... [math]\omega[/math], ... Now the point is, there is no "last" natural number [math]n[/math] that "reaches" or "is right before" [math]\omega[/math]. It doesn't work that way. If you are at [math]\omega[/math] and you take a step backwards, you will land on some finite natural number. But there are still infinitely many other natural numbers to the right of the one you landed on. You can jump back from [math]\omega[/math] to some finite natural number (which still has infinitely many natural numbers after it), but you can't jump forward a single step to get back to [math]\omega[/math]. That's just how it works. There's even a technical condition that lets us recognize why [math]\omega[/math] is special. A successor ordinal is an ordinal that has an immediate predecessor. All the finite natural numbers except 0 are successor ordinals. A limit ordinal is an ordinal that has no immediate predecessor. [math]\omega[/math] is a limit ordinal. That is, there is no other ordinal whose successor is [math]\omega[/math]. Note also that by this definition, 0 is also a limit ordinal. It's the only finite limit ordinal.

It's larvae. Photo is not sharp enough. It looks similar to biston betularia. https://www.google.com/search?q=biston+betularia https://www.google.com/search?q=peppered+moth+larvae

Sorry, I think I started with one thought and then veered off in a different direction. So the stuff that is mostly stable includes (IIRC) things like chlorogenic and ferulic acids, which tends to stick around. However other components including acetaldehydes, pentadiones actually increase, resulting in what some consider a rancid aroma. You can test that by measuring the pH over time. This is less pronounced if you start off with burned coffee beans in the first place (which often has less of fruity notes) but in certain (e.g. Colombian coffee) with citrus notes it becomes quite prominent after a little while. I think it is less about re-heating, but rather how long it oxidizes (and at which temperature). So warming up a fresh coffee should have little impact (unless heated excessively) but let it stand for a day or more more would (at least with decent beans). One can prolong the time by reducing oxygen and putting it in the fridge. Edit: actually I think I have read something taste profile changes after re-heating but I forgot if folks actually looked at the metabolites. I'll have to check. Edit2: I recall that there were taste tastes of coffee holding time on hot plates and I think the how long and how much heat it gets affected flavour rather rapidly. So likely it also depends on how you reheat it (and how hot). But again, that is rather murky memory territory now.

We know which will have the other for breakfast...

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.

I am thoroughly confused and tired. I have to think about this more.

The meter numbering is analogous to the set of natural numbers. There can't be an infinitely large natural number in the set of natural numbers.

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Its very confuse question.