The operative word here being "somewhere."Paradoxes indicate we have a wrong assumption somewhere, — Devans99
A 'true" continuity? What in the world is that? And while you're at it, please provide an epistemological ground for your "doubt." You seem to think it and other of your feelings are substantive with respect to any of your claims.So I doubt that true continuity is possible. — Devans99
The wrong assumption in this case is that the true continuity of space would require your hand to complete an actually infinite number of steps by passing through an actually infinite number of intermediate positions. As I have explained repeatedly now, the only individual positions that exist are whatever finite quantity of them we explicitly mark. If you still want to insist that real space is discrete, then make your case, but please stop pretending that this particular objection to its continuity is valid.Paradoxes indicate we have a wrong assumption somewhere, in this case, the assumption that it is possible to complete an actually infinite number of steps in a finite time is suspect. So I doubt that true continuity is possible. — Devans99
And while you're at it, please provide an epistemological ground for your "doubt." — tim wood
Every video and article I read about one-to-one-correspondence is garbage. They arbitrarily move infinity, place the first units together, send them off into infinity (without proving anything yet about uncountable vs countable) and exclaim "they are the same!". Nevermind ALL infinities are composed of units and you can do this trick with ALL infinities — Gregory
Why are you on about this? Everyone agrees with you and no one disagrees. As was said in my parochial school to the girls, do you want a medal or a chest to pin it on? — tim wood
I know you saw this, from @fishfry in the bijection thread:Suggesting the same information content. That is hard to swallow. — Devans99
(Thank you, fishfry.)(5) "Cantor's surprise." For any positive integer n, the n-dimensional Euclidean space Rn has the same cardinality as the real numbers.
Cantor originally thought that the real numbers had cardinality ℵ1; and the plane R2 had cardinality ℵ2, and Euclidean 3-space R3 had cardinality ℵ3, and so forth. [In math, n dimensional space just means the set of all n-tuples of real numbers, with pointwise addition and scalar multiplication by reals, just as with the usual x-y plane and x-y-z space....
He was surprised to realize that in fact all finite-dimensional Euclidean spaces have the same cardinality. Here's the proof. We'll show that the open unit interval and the open unit square have the same cardinality. That is, we'll show a bijection between the real numbers strictly between 0 and 1, and the set of ordered pairs in the x-y plane each of whose coordinates are strictly between 0 and 1.
Suppose (x,y) is a point in the open unit square with decimal representations x=.x1x2x3... and y=.y1y2y3... respectively. We map the pair (x,y) to a single real number by interleaving the digits to get .x1y1x2y2x3y3....
It's clear that you can reverse this process. Given any real number you can de-interleave its digits to get a pair of real numbers. We can extend the result from the unit interval to the entire real line via the tangent/arctangent. Of course this bijection is highly discontinuous, it has no nice properties at all.
You can clearly interleave n-digits this way, and that's the proof. When Cantor discovered this result he wrote to his friend Dedekind: "I see it but I don't believe it!" — fishfry
1. Axiom of infinity. It claims that the set of natural numbers exist. They exist in our minds where the impossible is possible, but there is nothing like it in reality, so maths should not claim 'they exist'. — Devans99
2. Axiom of choice. It claims it is possible to choose balls from an infinite number of bags. In reality, one cannot complete an infinite task, so it is impossible to make the infinite selection of balls. Hence maths should not claim it is possible. — Devans99
This indicates a confusion between existence in mathematics and actuality in metaphysics. They are not synonymous or equivalent. Everything that "exists" in mathematics is merely logically possible, not actual.Axiom of infinity. It claims that the set of natural numbers exist. They exist in our minds where the impossible is possible, but there is nothing like it in reality, so maths should not claim 'they exist'. — Devans99
This indicates a confusion between logical possibility and metaphysical possibility. Again, they are not synonymous or equivalent. It is logically possible to choose balls from an infinite number of bags, even though it is not metaphysically possible; i.e., it is actually impossible.Axiom of choice. It claims it is possible to choose balls from an infinite number of bags. In reality, one cannot complete an infinite task, so it is impossible to make the infinite selection of balls. Hence maths should not claim it is possible. — Devans99
I know you saw this, from fishfry in the bijection thread: — tim wood
This indicates a confusion between existence in mathematics and actuality in metaphysics. They are not synonymous or equivalent. Everything that "exists" in mathematics is merely logically possible, not actual. — aletheist
This indicates a confusion between logical possibility and metaphysical possibility. Again, they are not synonymous or equivalent. It is logically possible to choose balls from an infinite number of bags, even though it is not metaphysically possible; i.e., it is actually impossible. — aletheist
All of these are logically possible, just not metaphysically possible.I believe that the naturals and reals are purely mental constructs. They exist in our minds only (where the impossible is possible). They have the same status as talking trees and levitation - illogical/impossible things can exist in our minds but they cannot exist in reality. — Devans99
No one is claiming otherwise. When mathematicians state that the natural numbers "exist," they are not thereby calling them an actual infinity, only a potential infinity.An actual infinity of naturals (IE a set with a greater than any number of elements) is impossible. — Devans99
Incorrect--it is logically possible, just not metaphysically possible.It is not logically possible to complete a task that has no end. — Devans99
All of these are logically possible, just not metaphysically possible. — aletheist
No one is claiming otherwise. When mathematicians state that the natural numbers "exist," they are not thereby calling them an actual infinity, only a potential infinity. — aletheist
Incorrect--it is logically possible, just not metaphysically possible. — aletheist
Everything that "exists" in mathematics is merely logically possible, not actual. — aletheist
You simply refuse to acknowledge the definitions of terms that others are employing, and thus consistently (and persistently) attack straw men. Actual impossibility does not entail logical impossibility. Mathematical existence is not metaphysical actuality. The infinity of the natural numbers is potential, not actual. Continuity of space does not require an actual infinity of distinct positions. — aletheist
Everything that "exists" in mathematics is merely logically possible, not actual. — aletheist
Yes, given the standard mathematical definitions, the proposition that the number denoted by "5" possesses the character denoted by "prime" is true. Do you think that either of these terms denotes something actual?Do you think that "5 is prime" is true? — fishfry
No, it is defined as a potential infinity. One more time: mathematical existence does not entail metaphysical actuality. No one, except perhaps an extreme platonist, claims that there is an actual set containing all the natural numbers.The set of natural is defined in maths as an actual infinity
'The axiom of Zermelo-Fraenkel set theory which asserts the existence of a set containing all the natural numbers' — Devans99
No, it is defined as a potential infinity. One more time: mathematical existence does not entail metaphysical actuality. No one, except perhaps an extreme platonist, claims that there is an actual set containing all the natural numbers. — aletheist
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