Bach makes a basic error or assumption that’s widely rejected in math:
That there is any set of all sets. The notion is contradictory for more basic reasons like Russell’s paradox, so we use the “class of all sets” and define/construct sets so that there is no set of all sets. Proper classes are treated pretty differently from sets in many cases. Classes are collections of sets only. People don’t use the class of all sets to represent anything in the physical world, either, and I’d say that it probably can’t be used to represent anything physical, but that’s not a problem for infinities in general. There’s no class of all classes under standard set theory, since that would need to contain proper classes.
Even if we used the class of all sets to try to fix the argument, the power set operation has no natural extension to it in standard set theory. It would have to be the class of all subclasses of the class of all sets, which doesn’t exist under standard set theory because it would contain proper classes, but even if it did exist, that object would be different from the class of all sets, so there need not be any contradiction with them having different sizes. (I’d guess the class of all subclasses of the class of all sets would be strictly bigger by the same argument that the power set of a set is bigger than the set, under some set theory where that’s defined naturally and extends standard set theory.)
Sewell assumes subtraction with infinite cardinals should be well-defined like it is for finite numbers without (good) argument, but this is widely rejected. Also, there are ways to represent infinities so that the specific operations discussed are well-defined, e.g. representing the objects as sets and using set operations (unions, differences, partitioning) instead of arithmetic operations on numbers (addition, subtraction, division). N—N = 0 this way and N—N has no other value, where “-” means set difference and N is the set of natural numbers. Subtracting the even numbers (or odd numbers) from the natural numbers would be represented differently on the left-hand side, so that giving a different result isn’t a problem. EDIT: I think he quotes some similar arguments, but doesn’t really respond to them (or probably doesn’t respond well).
They seem to be arguing against strawmen. They don’t seem to understand the basics of standard axiomatic set theory well enough, and they wouldn’t make such bad arguments if they did. I would recommend you study axiomatic set theory if you’re still tempted to dismiss the logical possibility of infinity, or just accept that it’s likely to be logically possible by deferring to those who understand axiomatic set theory, because probably almost all of them accept its logical possibility.
(Again, I don’t intend to engage further, but I guess I’m bad at keeping that kind of promise.)
Understands the notion of infinity does not lead to contradictions in math. As you noted, infinity is one of the axioms of ZMC set theory, which is widely followed in math. So no wonder infinity is true (by definition) for most mathematicians!
Argues that math should be about the real world, so we should not be defining ad hoc rules which have no parallell in physical reality.
As an analogy (adapted from one used by William Craig), we can suppose I have 2 bags with infinite marbles. One contains marbles numbered with the even numbers, and the other marbles numbered with the odd numbers, so they have the same infinity of marbles. If I:
Give both bags to you, I will keep no bags, and therefore will have zero marbles. So inf—inf = 0.
Give 1 bag to you, I will keep 1 bag, and therefore will have infinite marbles. So inf—inf = inf.
This leads to 0 = inf, which is contradictory.
I appreciate one can say I have cheated by:
Using the same type of subtraction in both situations (indicated by “-”), whereas I should have used different symbols to describe the different types of subtractions.
Assuming I could perform the operations inf—inf, which is an indeterminate form.
However, as far as I can tell, reality only allows for one type of subtraction. If I have 3 apples in my hands (or x $ in a Swiss bank account ;)), and give you 2 apple, I will keep 1 apple. This is the motivation for 3 − 2 = 1.
In Sewell’s words:
“In classical mathematics the operation of subtraction on natural numbers yields definite answers, and so instances of subtraction can be grounded in real world examples of removal. The act of “removing” a subset of objects from a set of objects is just an instance of applying mathematical subtraction or division to physical collections in the real world”.
“There is nothing in transfinite mathematics implying that mathematical operations on infinite sets cannot be applied to logically possible infinite collections in the real world. So, if we are able to consistently subtract or divide infinite sets in transfinite mathematics, we should then without contradiction be able to carry out the removal of infinite subsets from infinite sets of real objects as well. Subtracting and dividing infinite sets should show what would happen in the real world if we could go about “removing” infinite subsets from infinite sets of physical objects. 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. Such a removal would then not be able to be performed in the real world, which does not permit logically contradictory states of affairs to occur. The application of inverse operations in transfinite mathematics to real world instances of removing infinite subsets then, is actually a test of the logical validity of infinite sets. If the math breaks down as we’ve seen, so does the logic of infinite sets 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.
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.
Bach makes a basic error or assumption that’s widely rejected in math:
That there is any set of all sets. The notion is contradictory for more basic reasons like Russell’s paradox, so we use the “class of all sets” and define/construct sets so that there is no set of all sets. Proper classes are treated pretty differently from sets in many cases. Classes are collections of sets only. People don’t use the class of all sets to represent anything in the physical world, either, and I’d say that it probably can’t be used to represent anything physical, but that’s not a problem for infinities in general. There’s no class of all classes under standard set theory, since that would need to contain proper classes.
Even if we used the class of all sets to try to fix the argument, the power set operation has no natural extension to it in standard set theory. It would have to be the class of all subclasses of the class of all sets, which doesn’t exist under standard set theory because it would contain proper classes, but even if it did exist, that object would be different from the class of all sets, so there need not be any contradiction with them having different sizes. (I’d guess the class of all subclasses of the class of all sets would be strictly bigger by the same argument that the power set of a set is bigger than the set, under some set theory where that’s defined naturally and extends standard set theory.)
See this page for definitions and some discussion: https://en.wikipedia.org/wiki/Class_(set_theory)
Sewell assumes subtraction with infinite cardinals should be well-defined like it is for finite numbers without (good) argument, but this is widely rejected. Also, there are ways to represent infinities so that the specific operations discussed are well-defined, e.g. representing the objects as sets and using set operations (unions, differences, partitioning) instead of arithmetic operations on numbers (addition, subtraction, division). N—N = 0 this way and N—N has no other value, where “-” means set difference and N is the set of natural numbers. Subtracting the even numbers (or odd numbers) from the natural numbers would be represented differently on the left-hand side, so that giving a different result isn’t a problem. EDIT: I think he quotes some similar arguments, but doesn’t really respond to them (or probably doesn’t respond well).
They seem to be arguing against strawmen. They don’t seem to understand the basics of standard axiomatic set theory well enough, and they wouldn’t make such bad arguments if they did. I would recommend you study axiomatic set theory if you’re still tempted to dismiss the logical possibility of infinity, or just accept that it’s likely to be logically possible by deferring to those who understand axiomatic set theory, because probably almost all of them accept its logical possibility.
(Again, I don’t intend to engage further, but I guess I’m bad at keeping that kind of promise.)
Long story short, Sewell:
Understands the notion of infinity does not lead to contradictions in math. As you noted, infinity is one of the axioms of ZMC set theory, which is widely followed in math. So no wonder infinity is true (by definition) for most mathematicians!
Argues that math should be about the real world, so we should not be defining ad hoc rules which have no parallell in physical reality.
As an analogy (adapted from one used by William Craig), we can suppose I have 2 bags with infinite marbles. One contains marbles numbered with the even numbers, and the other marbles numbered with the odd numbers, so they have the same infinity of marbles. If I:
Give both bags to you, I will keep no bags, and therefore will have zero marbles. So inf—inf = 0.
Give 1 bag to you, I will keep 1 bag, and therefore will have infinite marbles. So inf—inf = inf.
This leads to 0 = inf, which is contradictory.
I appreciate one can say I have cheated by:
Using the same type of subtraction in both situations (indicated by “-”), whereas I should have used different symbols to describe the different types of subtractions.
Assuming I could perform the operations inf—inf, which is an indeterminate form.
However, as far as I can tell, reality only allows for one type of subtraction. If I have 3 apples in my hands (or x $ in a Swiss bank account ;)), and give you 2 apple, I will keep 1 apple. This is the motivation for 3 − 2 = 1.
In Sewell’s words:
“In classical mathematics the operation of subtraction on natural numbers yields definite answers, and so instances of subtraction can be grounded in real world examples of removal. The act of “removing” a subset of objects from a set of objects is just an instance of applying mathematical subtraction or division to physical collections in the real world”.
“There is nothing in transfinite mathematics implying that mathematical operations on infinite sets cannot be applied to logically possible infinite collections in the real world. So, if we are able to consistently subtract or divide infinite sets in transfinite mathematics, we should then without contradiction be able to carry out the removal of infinite subsets from infinite sets of real objects as well. Subtracting and dividing infinite sets should show what would happen in the real world if we could go about “removing” infinite subsets from infinite sets of physical objects. 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. Such a removal would then not be able to be performed in the real world, which does not permit logically contradictory states of affairs to occur. The application of inverse operations in transfinite mathematics to real world instances of removing infinite subsets then, is actually a test of the logical validity of infinite sets. If the math breaks down as we’ve seen, so does the logic of infinite sets in the real world”.
My reply here has some further context.
Sure, I trust your decisions regarding your time. Thanks for the discussion!
“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.