A theoretical question about infinity
Hi Miki, greetings.
I understand that you are rejecting all sorts of arguments about the impossibility of a primeval world because it is not defined at all or an incorrect relation of such infinity {potential and concrete}. I have some thought experiment that is similar to these arguments, I would be happy if you would explain in what way it is not or how it is different from the other arguments.
So: there are two things that cannot exist at the same time. One is that space is infinite, and the other is that space grows over time. This is because if it is infinite it has nowhere left to grow. Like if space can grow to a certain size like 10 square meters, once it reaches a size of 10 meters it has nowhere left to grow, and the same thing {in my opinion} with infinity, meaning if it can grow to infinity and it is already there it has fulfilled its growth potential and therefore it has nowhere left to grow because it is already there.
I would appreciate your criticism and insights. Thank you very much. : )
First of all, I don’t understand your argument. What did you come to prove? If the world is ancient, meaning it has existed for an infinite amount of time, then it can’t grow? Do you mean grow in time? It can’t progress with the timeline? That’s not true, but regardless, I don’t understand what you came to prove.
As for your claim, you have a mathematical error. Unlike any finite number, infinity can certainly grow. For example, look at the set of natural numbers (=positive integers). How many of them are there? Infinity. And now look at the number of rational numbers (=a portion of natural numbers). How many of them are there? Also infinity. Although in Cantor’s cardinal theory this is the same infinity (there is a correspondence between the two groups), there are also infinities that are actually larger (like the real numbers, all the points on the real axis, whose number is greater than the natural numbers).
Another example. Consider the segment on the real axis (0,1). It has an infinite number of points (=representing real numbers). Now I will inflate the segment until it reaches length 2, meaning I am now on the segment (0,2). Both segments have the same infinite number of points, but the second is longer than the first.
Search Wikipedia for the entry “Hilbert’s Hotel” and you can read some amusing things about it.
First of all, thank you very much.
Secondly, my argument only talks about whether it is possible for the world to grow if space is infinite.
So what does this have to do with the pre-existence of the world? So it is ancient and has not grown.
There is a similarity between what I said and the antiquity of the world, but that is not the point. My point here is that even if the world is not ancient, infinite space still cannot grow. The point of my question is not whether the world is ancient or not, but the relationship to infinity.
Imagination is in the eye of the beholder. In any case, regardless of what this argument proves, it is wrong.
From what I remember, the two segments are A1, like the entire real line. And in general, according to Cantor's diagonal, it can be shown that it is larger than the set of natural numbers that are A, but still at least the same size as the real line. And in general, assuming that there are no intermediate sizes, it can be concluded that it is A1.
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