Just a few short remarks here:
Quote:
But the thing is, you can think of those sets as a function, that for each element tells you whether it is in the set. In that way you escape infinity altogether.
Do you really call this 'well defined'?
A function is a purely mathematical construct. Is this your way of escaping infinity, just by calling it a 'function'? In other words, it actually does not exist....
Now think about this next statement for some time:
IF there is an infinite set of some kind, it SHOULD contain everything there is. Otherwise, it by definition CAN NOT be infinite.
It is as simple as that. Infinite means it has no boundaries and consumes everything else. And no cheating here, it has to be a real collection, not some 'made up' one. Not something like 'I have this infinite collection of numbers here, because you don't. No, you have to actually make this infinite collection, and you may use any elementary particle you can find as 'marbles'. Then you need the ENTIRE universe to really make this collection, this set.
And if you take one particle out, it is not infinite anymore, it will become a finite collection. And you can't add a particle because an infinete set contains everything, so there is nothing outside of it to add to it.
Now think about the next statement for some time too:
If I create a set, and stop at any moment (because I'm finished) it is PER DEFINITION a finite set because in the process of making this set I cannot add an INFINITE number of items in a FINITE time span. Besides I COULD add some more. INFINITE sets require INFINITE time to make.
I see you raise your eyebrows again about my use of time here. But mathematicians think they are allowed to materialize anything they can come up with, out of thin air. They think they are math-magicians! No, you can't make up infinite sets, just because you have invented a symbol that means 'infinite'. "POOF, here's an infinite set, and POOF there's another one!" No, they don't exist! You don't have them|! Only in you imagination my friend! What's the use of thinking about pure abstract sets that can only exist within the rather limited symbols of math? Intellectual masturbation, that's what it is!
And even if I take math as my limitation, and consider 'spacetime' as something real, then you can't just say "here is something infinite in 3D space", it would mean you have to define it in 4D spacetime, which includes time as well. Now Cantor did not know that, did he?
In math nothing is well defined, certainly not in relation to the real world. Every equation is only comprehensible to those who know the 'meaning' of the symbols and the context the equation is used in and the restricitons that are obvious to the user. Take the simple equation 12 * 1/12 = 1. True, right?
No, not true for most objects in the world. Twelve pieces of cow don't add up to a living cow, just a pile of meat. Twelve pieces of a car don't add up to a car, just a pile of scrap metal.
We 'know' in advance it does not work that way, it is obvious to us, but it is not embedded in the language. So when does this previous mentioned equation count? Only in specific situations like amounts of 'stuff' like money.
If you take the symbol 'infinity' in math, but you then add that it might mean different things, I'm sorry but how can you state that it is well defined?
Zero is definitily NOT a natural number. In some cases (here we go again, NOT well defined) we can use it as such. But in essence it is the ABSENCE of a value. That is something else. To state there are zero cows in a field is another way of saying they are not there. But is usually refers to a situation in which it might be otherwise, or has been differently. Today there are zero cows in this field, yesterday there were 20 and tomorrow perhaps 15.
We will never refer to 'cows in space' (only pigs in space) because we won't find them there ever. We can only state 'There are NO cows in space'. To say there are zero cows in space is just nonsense. See? 'absence of'.
Even a simple equation 2 + 2 = 4 is only valid in certain situations. But not in the real world. You have to ADD the notion that it is only valid with 'same' objects. 2 + 2 carrots make 4 carrots, but 2 carrots plus 2 onions make 4... vegetables.
Ah, mister 'math', I hear you snorting... But wait, mathematicians make errors with this all the time! They forget the real world. That's how they have come up with 4 (5,6, 11...) dimensional space, just because they have found a notation that enables them to do calculations with it. And now, they really want us to believe these dimensions are curled up in tine portions of good 'ole 3D space just to make string theory work. But a 'dimension' is (at least in space) just a 'direction' not something you can curl up.
A natural number is per definition something that occurs natural, in nature. Otherwise it would NOT be well defined. Zero does NOT occur in nature. Show me where.
Why do we need a Cartesian coordinate system to solve problems in geometry and algebra? Because we need to define a reference point, the center of the grid. Just because IT IS NOT THERE IN NATURE. In nature there is no zero point, zero level or whatever. ANY point can be taken as zero, as a referencre point. Why is 0 Kelvin zero? Because there is the ABSENCE OF MOTION.
Oh, and what about 0 degrees celcius or fahrenheit? Now here's a hack of human ingenuïtiy. Ther are no absolute values in nature. But there ARE - under 'regular' conditions like gravity and pressure - certain observations to make, like the freezing of water. If we calibrate a device that will accurately show this threshold and add little marks to it that we call degrees, we have a way of defining temperature! And we call it... drum roll ZERO! So we can go up AND down on our scale. But is it real. No of course not! Change the gravity, change the pressure, change the purity of the water and you have other values, other thresholds. It only works in our little bubble we call earth's atmosphere.
