One is free to propose different axioms that prove differently. — TonesInDeepFreeze
The answer to your problem is quite simple. In mathematics things are done by axiom. If you want to count to infinity and beyond, simply produce an axiom which allows you to do that, and bingo the infinite is countable, and you're ready to go beyond. Look closely at the following: — Metaphysician Undercover
But having a different concept and definition of infinitude doesn't thereby entail that there is a contradiction in set theory or mathematics. — TonesInDeepFreeze
Again, yes, there may be a contradiction between set theory and certain other formulations. But that does not entail that there is a contradiction within set theory. — TonesInDeepFreeze
set theory does not refer to an object named 'infinity' but rather to the property of being infinite, which is a crucial distinction. — TonesInDeepFreeze
or the angles in a triangle add up to 180 degrees — Philosopher19
where there are an infinite number of elements (including fractions and irrationals) between 0 and 1 — punos
You should see non-Euclidean geometry where the angles in a triangle can be more or less than 180 degrees. — Michael
My belief is that we can't just produce axioms. We can only recognise truths about Existence such as 1 add 1 equals 2 or the angles in a triangle add up to 180 degrees or one cannot count to infinity. — Philosopher19
1 add 1 equals 2 — Philosopher19
Imperfect triangles are imperfect by definition. I'm focused on absolutes. — Philosopher19
What do you mean by an "imperfect" triangle? — Michael
The angles in a true triangle add up to 180 degrees because that is the nature of Existence. — Philosopher19
What is this supposed to mean? — Michael
If there is no end to something, how can another thing with no end be twice as large as it? Don't they both have no ends? — Philosopher19
The angles in a true triangle add up to 180 degrees because that is the nature of Existence. It is not because someone said it or highlighted it. — Philosopher19
You cannot start counting 1,2,3,4,... ad infinitum and reach somewhere, anywhere. Infinity has neither a start or an end.If I count 1, 2, 3, 4 ad infinitum, will I reach infinity? One cannot count to infinity, and even if something like a number sequence goes on forever, it will not reach infinity. — Philosopher19
A set is a collection of objects (elements, members). I'm not sure if we can talk about an infinite set, although there are some theories about it (e.g. Zermelo–Fraenkel).To call {1,2,3,4,...} an infinite set is to imply that {1,2,3,4,...} consists of an infinite number of numbers. No doubt, even if 1, 2, 3, 4 goes on forever, an infinite number of numbers will never be reached. — Philosopher19
The angles in a true triangle add up to 180 degrees because that is the nature of Existence — Philosopher19
That's not Hilbert's paradox. — Michael
The statement that some infinities are bigger than others comes from set theory. The OP talks about infinities in the context of counting procedures. These are two different concepts of infinity. — DanCoimbra
First, imagine you have achieved immortality and are presented with two options: to receive $1 every day forever or $1 every year. Intuitively, you would choose $1 every day because, over the same infinite duration, you would accumulate more money. This illustrates that while both options extend to infinity in time, the rate at which you receive money differs, leading to a larger "size" of wealth in one scenario over the other. — punos
Now, let's consider a spatial analogy. Imagine two pipes, both of infinite length, but one has a diameter of 1 inch and the other has a diameter of 10 inches. Despite their lengths being equally infinite, the pipe with the larger diameter has a greater volume. This demonstrates that even with one dimension being infinite, other finite dimensions can contribute to a difference in "size" or capacity. — punos
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