“On the other hand, if we would get mathematical nonsense by performing inverse operations in transfinite mathematics, then we would also get logical nonsense when trying to “remove” an infinite subset of real objects from an infinite set of them.”
This doesn’t follow and is false. The set difference operation is well-defined, so the result is not logical nonsense. The corresponding set cardinalities after a specific set difference will also be well-defined, since the cardinality function is also well-defined.
Plenty of apparently real things aren’t well-defined unless you specify them in enough detail, but that doesn’t make them nonsense. For example, the weight of a bag after removing an object whose weight is unknown. Or, the center of mass of two objects, knowing only their respective centers of mass (and distance between them).
There’s also no logical necessity for subtraction with infinite numbers to be well-defined, and it seems conceivable without logical contradiction that it’s not, even in the actual universe (e.g. if we model an infinite universe or the continuum using ZF(C) set theory for the infinities). It’s of course possible our universe has no infinities and arithmetic is always well-defined when representing any real objects in it, but there’s no decisive proof for either, and hence no decisive proof for the impossibility of infinity. It doesn’t follow by necessity from the finite case.
There’s also no logical necessity for subtraction with infinite numbers to be well-defined, and it seems conceivable without logical contradiction that it’s not, even in the actual universe (e.g. if we model an infinite universe or the continuum using ZF(C) set theory for the infinities).
In general, nothing can be proved to be logically true or false without assuming some claims are true. For instance, in order to show that a given mathematical hypothesis is true or false, one has to define some axioms. As an example, transitivity (if A is better than B, and B is better than C, then A is better than C) is usually assumed to be one of the axioms of rationality. Transitivity cannot be proved (without defining any axioms), it is true by definition, and I have no way to convince someone who argues that transitivity is false.
If the concept of infinity could be true, the whole would not always be the sum of its parts (e.g. inf/2 = inf). However, the whole always being the sum of its parts is axiomatically true to me, so I consider the concept of infinity to be false. Similarly to transitivity, I have no way to prove my axiom that the whole always is the sum of its parts.
For what is worth, I see expectational total hedonistic utilitarianism (ETHU) as the axiom of ethics/morality. On the one hand, it is impossible for anyone to prove it is true. For example, although I think the more likely a certain positive outcome is, the better, I have no way to prove one should maximise expected value. On the other hand, ETHU being true feels the same way to me as transitivity being true.
This doesn’t follow and is false. The set difference operation is well-defined, so the result is not logical nonsense. The corresponding set cardinalities after a specific set difference will also be well-defined, since the cardinality function is also well-defined.
To clarify the contradiction I mentioned above, if n denotes the cardinality operator, v the disjunction operator, ^ the conjunction operator, O the set of odd numbers, E the set of even numbers, ES the empty set, n(ES) = 0, and n(O) = n(E) = inf:
If I give both bags to you, I will keep no bags, and therefore will have zero marbles:
A1: n((O v E)\(O v E)) = n(O v E) - n((O v E) ^ (O v E)) = n(O v E) - n(O v E) = inf—inf.
B1: n((O v E)\(O v E)) = n(ES) = 0.
C1: A1^ B1 ⇒ inf—inf = 0.
If I give 1 bag to you, I will keep 1 bag, and therefore will have infinite marbles:
A2: n((O v E)\O) = n(O v E) - n((O v E) ^ O) = n(O v E) - n(O) = inf—inf.
B2: n((O v E)\O) = n((O v E)\E) = n(O) = inf.
C2: A2 ^ B2 ⇒ inf—inf = inf.
So there is a contradiction:
D: C1 ^ C2 ⇒ 0 = inf.
Since, 0 = inf is false, one of the following is false:
The relationship R ⇔ n(X\Y) = n(X) - n(X ^ Y), which I used above, exists in the real world.
Infinites exist in the real world.
I guess you would be inclined towards putting non-null weight into each one of these points being false. However, R essentially means the whole is the sum of its parts, which I cannot see being false in the real world. So I reject the existence of infinites in the real world.
“On the other hand, if we would get mathematical nonsense by performing inverse operations in transfinite mathematics, then we would also get logical nonsense when trying to “remove” an infinite subset of real objects from an infinite set of them.”
This doesn’t follow and is false. The set difference operation is well-defined, so the result is not logical nonsense. The corresponding set cardinalities after a specific set difference will also be well-defined, since the cardinality function is also well-defined.
Plenty of apparently real things aren’t well-defined unless you specify them in enough detail, but that doesn’t make them nonsense. For example, the weight of a bag after removing an object whose weight is unknown. Or, the center of mass of two objects, knowing only their respective centers of mass (and distance between them).
There’s also no logical necessity for subtraction with infinite numbers to be well-defined, and it seems conceivable without logical contradiction that it’s not, even in the actual universe (e.g. if we model an infinite universe or the continuum using ZF(C) set theory for the infinities). It’s of course possible our universe has no infinities and arithmetic is always well-defined when representing any real objects in it, but there’s no decisive proof for either, and hence no decisive proof for the impossibility of infinity. It doesn’t follow by necessity from the finite case.
In general, nothing can be proved to be logically true or false without assuming some claims are true. For instance, in order to show that a given mathematical hypothesis is true or false, one has to define some axioms. As an example, transitivity (if A is better than B, and B is better than C, then A is better than C) is usually assumed to be one of the axioms of rationality. Transitivity cannot be proved (without defining any axioms), it is true by definition, and I have no way to convince someone who argues that transitivity is false.
If the concept of infinity could be true, the whole would not always be the sum of its parts (e.g. inf/2 = inf). However, the whole always being the sum of its parts is axiomatically true to me, so I consider the concept of infinity to be false. Similarly to transitivity, I have no way to prove my axiom that the whole always is the sum of its parts.
For what is worth, I see expectational total hedonistic utilitarianism (ETHU) as the axiom of ethics/morality. On the one hand, it is impossible for anyone to prove it is true. For example, although I think the more likely a certain positive outcome is, the better, I have no way to prove one should maximise expected value. On the other hand, ETHU being true feels the same way to me as transitivity being true.
To clarify the contradiction I mentioned above, if n denotes the cardinality operator, v the disjunction operator, ^ the conjunction operator, O the set of odd numbers, E the set of even numbers, ES the empty set, n(ES) = 0, and n(O) = n(E) = inf:
If I give both bags to you, I will keep no bags, and therefore will have zero marbles:
A1: n((O v E)\(O v E)) = n(O v E) - n((O v E) ^ (O v E)) = n(O v E) - n(O v E) = inf—inf.
B1: n((O v E)\(O v E)) = n(ES) = 0.
C1: A1^ B1 ⇒ inf—inf = 0.
If I give 1 bag to you, I will keep 1 bag, and therefore will have infinite marbles:
A2: n((O v E)\O) = n(O v E) - n((O v E) ^ O) = n(O v E) - n(O) = inf—inf.
B2: n((O v E)\O) = n((O v E)\E) = n(O) = inf.
C2: A2 ^ B2 ⇒ inf—inf = inf.
So there is a contradiction:
D: C1 ^ C2 ⇒ 0 = inf.
Since, 0 = inf is false, one of the following is false:
The relationship R ⇔ n(X\Y) = n(X) - n(X ^ Y), which I used above, exists in the real world.
Infinites exist in the real world.
I guess you would be inclined towards putting non-null weight into each one of these points being false. However, R essentially means the whole is the sum of its parts, which I cannot see being false in the real world. So I reject the existence of infinites in the real world.