Come again? How would the infinite of one line set the limit to the infinity of the other line?Bondi wrote:On the other hand, no matter how much I understand what Viktor is trying to convey, I do agree with you. Some words are philosophically very important, and are pretty much abused these days (they have been abused for centuries anyway). â€˜Infiniteâ€™ (or infinity) is probably the most abused word. Viktor wants to state it as an example, but he canâ€™t (or isnâ€™t able to) see the difference between his mathematical concept of an â€œinfininiteâ€ and the philosophical infinite. The mathematical concept is in no way should be called infinite. We can simply understand the underlying absurdity of that: if you call a line (as Viktor set it out) â€œinfiniteâ€ just because it is endless in both of its directions, then what do you say when you make up two or more lines endless in both their directions? Obviously, they canâ€™t be â€œinfiniteâ€ as two or more â€œinfinitiesâ€ would necessarily set limit to each other, therefore they canâ€™t be infinite.
BTW, this is where math comes in very handy. For example, it is mathematically provable (via diagonalization proof) that any countable infinity is as large as any other countable infinity -- e.g. one infinite line, of integers, is exactly the same size as two intersecting infinite lines, of real and imaginary integers. Similarly, the set of all integers is exactly as large as the set of all rational numbers (i.e. all fractions expressible as a ratio of two integers).
Think about it. If you start writing out a set of all rational fractions -- 1/2, 2/15, 89017985/3709817, etc -- this set will be exactly as large as the set of all integers. In fact, the set of all polynomials (with integer coefficients) is still as large as the set of all integers. On the other hand, the set of all numbers (rational and irrational) is larger, and thus is also larger than the set of all polynomials with integer coefficients.
Since Universe is quantized, it is in fact trivially demonstrable that the infinity of the Universe is exactly as large as the infinity of any countably infinite set such as integers. :)
No, one simply needs to understand more about transfinite mathematics.A line (or anything) is always limited by its own being. Viktor says that â€œblack holes are infiniteâ€â€”so then there are who knows how many â€œinfinitesâ€ around us in the Universe? One could hardly say any more absurd thing than that!
And at the end, one ends up with exactly the same thing they started with, but stated in a slightly different form. E.g. "all things are finite" means exactly, and nothing more than, "anything which is less than the totality, is less than the totality" -- which is a vacuous tautology (as opposed to contentful tautologies, such as mathematical proofs, which explicate the unobviously implicit).To cut it short, thatâ€™s exactly why I agree with you. Reasoning means, in part at least, that one is careful enough with their using of words.