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The fallacy of Cantor (and his supporters such as Hilbert) lies in mixing the mathematical logic which is meaningful only for finite numbers, with the infinity

by zkmon 5y ago

The fallacy of Cantor (and his supporters such as Hilbert) lies in mixing the mathematical logic which is meaningful only for finite numbers, with the infinity as if it were a number that obeys comparison operations such as bigger, smaller etc. Almost anything can be proved/disproved or debated when one assumes such validity of the logical concepts to infinity. The acceptance of multiple infinities and some concepts around continuum (Dedekind cuts etc) are questionable due to this assumption.

mcguire 5y ago
"Almost anything can be proved/disproved or debated when one assumes such validity of the logical concepts to infinity." I strongly believe that statement to be false. Can you provide a contradiction that is provable using Cantor's definition of the cardinality of sets?
zkmon 5y ago
Many logical concepts which we take for granted, might not be meaningful in the realm of infinite. For example, something being equal to itself, some statement being true or false (no third state), things having unique identity or location, the concept or possibility of multiple things (and distinguishability), being able to compare one thing with another etc. All these make sense only in finite contexts but not in infinite or infinitesimal contexts. If we accept infinity as a participant in the mathematical logic, the infinitesimal also deserves a role as a counter-party.
- thaumasiotes 5y ago
  > If we accept infinity as a participant in the mathematical logic, the infinitesimal also deserves a role as a counter-party. Infinitesimals have a much bigger role in widely-known mathematics than infinities do; they are the basis for calculus.
  - qweqwweqwe-90i 5y ago
    Series play a big role in all of math...
- jyounker 5y ago
  Infinitesimals are just a different aspect of mathematical infinity. This is the magic that makes calculus work (from a formal standpoint).
  - im3w1l 5y ago
    While you can build a calculus using infinitesimals, the usual construction does not use them. https://en.wikipedia.org/wiki/Nonstandard_analysis https://en.wikipedia.org/wiki/Nonstandard_analysis
    - anikan_vader 5y ago
      The original construction did though, if you look at the writings of Newton/Leibniz. But their work was totally hand-wavey. Cauchy discovered limits in the 1800s, and it wasn’t until the 20th century that the original idea of infinitesimals was formalized. In some sense we have come full circle. I do think that infinitesimals are a much more intuitive way to understand calculus. It’s a shame it took so long to find positive numbers less than any real.
joppy 5y ago
Cardinalities of sets do have sensible operations like less than and greater than though - |A| <= |B| if there exists an injective function from A to B. One has to check that this inequality behaves in the ways that you expect, but it is a well-defined concept.
- IX-103 5y ago
  It is defined in the abstract, but I think there are a lot of implicit assumptions that get lost when dealing with cardinalities which may turn out to be important. Just because a bijective/injective/surjective function exists doesn't mean that such a transform is possible when you apply it to a specific situation. If sizes were sensible then for a set A composed of "every other integer" would be smaller than the a set B composed of "2 times every integer". If we were using calculus to take the limit of the ratio of sizes of the generating functions for A and B as the source set size goes to infinity then we find that A is in fact half the size of B. If we try to simply apply cardinality to A and B we find that the sets are exactly equal, since cardinality doesn't care about source sets or limits! This can a big deal if say, you're calculating probabilities across an infinite number of possible events/event configurations. There is a big difference between a 50% chance, a 0.000001% chance and a 100% chance. I'm not saying there's no use for cardinality and infinity classes, but they can easily be misapplied to allow you to be wrong, with confidence.
  - anikan_vader 5y ago
    > If sizes were sensible then… Sorry, but they simply don’t meet your (non-standard) definition of sensible. Both of the sets you mentioned can be interpreted as the set of even integers. This set is in bijection with itself (trivially), and thus is not considered to be strictly smaller than itself.
openasocket 5y ago
I don't think there's any confusion of the finite and infinite numbers in Cantor's diagonalization argument. Here's the most generic form of Cantor's argument: there is no surjective function from a set A to its power set P(A). We prove this by contradiction. Consider any function f: A -> P(A) , i.e. it takes elements of A and outputs subsets of A. Suppose this function is surjective: i.e. for all y in P(A) there is some x in A such that f(x) = y. But let q = { a in A | a is not in f(a) }. Clearly this is a valid set. And if f is surjective, there must be some x in A such that f(x) = q. Is x in q? If x were in q, then x would be in f(x), so that's a contradiction. If x were not in q, then by the definition of q x would be in q, which is also a contradiction. Thus we have a contradiction, so f cannot be surjective. As you can see, nowhere do we make any logical jumps that would only make sense in the case of finite sets. This proof is as straightforward as the proof that there is no set of all sets. We don't use the axiom of choice, the argument is even valid in constructive mathematics (though you have to make some adjustments). Now, we define one set X as being "less than" another set Y if and only if there is no surjective function from X to Y. You'll see that this definition corresponds exactly to the usual notions of size for finite sets, and makes intuitive sense (if for every y we have an x, there must be at least as many xs as ys). Now, just plug in the set of natural numbers into Cantor's theorem and you get: the power set of natural numbers is larger than the set of natural numbers.
- criloz2 5y ago
  If you are counting a set, you necessarily are creating a map between the item of the set you are counting and the natural numbers, this is basics, how come that you can say such statement about the sizes of power set of natural number and natural number?, from where the notion of the cardinality comes if not from counting, and counting implies a map to a natural number, so the set of natural number is bigger than its selves?
  - openasocket 5y ago
    Cardinality doesn't have to do with "counting" necessarily. Two sets X and Y are said to have the same cardinality if there is a function f : X -> Y where f is a bijection. By bijection we mean it has two properties: that for all a,b in X, if f(a) = f(b), then a = b (this is also called the injective property), and for all c in Y, there is some d in X such that f(d) = c (this is called the surjective property). Set X has cardinality less than Y if there is no such bijective function, but there is a function f: X -> Y that is injective. Conversely, X has cardinality larger than Y if there is no such bijective function, but there is a function f: X -> Y that is surjective. All you have to do to compare the size of two sets is to look at the functions mapping one to the other. No "counting" involved.
porb121 5y ago
> Almost anything can be proved/disproved or debated when one assumes such validity of the logical concepts to infinity. if this were true it would be trivial for you to give devastating examples

by zkmon 5y ago

mcguire 5y ago

"Almost anything can be proved/disproved or debated when one assumes such validity of the logical concepts to infinity." I strongly believe that statement to be false. Can you provide a contradiction that is provable using Cantor's definition of the cardinality of sets?

zkmon 5y ago

Many logical concepts which we take for granted, might not be meaningful in the realm of infinite. For example, something being equal to itself, some statement being true or false (no third state), things having unique identity or location, the concept or possibility of multiple things (and distinguishability), being able to compare one thing with another etc. All these make sense only in finite contexts but not in infinite or infinitesimal contexts. If we accept infinity as a participant in the mathematical logic, the infinitesimal also deserves a role as a counter-party.

thaumasiotes 5y ago
> If we accept infinity as a participant in the mathematical logic, the infinitesimal also deserves a role as a counter-party. Infinitesimals have a much bigger role in widely-known mathematics than infinities do; they are the basis for calculus.
- qweqwweqwe-90i 5y ago
  Series play a big role in all of math...
jyounker 5y ago
Infinitesimals are just a different aspect of mathematical infinity. This is the magic that makes calculus work (from a formal standpoint).
- im3w1l 5y ago
  While you can build a calculus using infinitesimals, the usual construction does not use them. https://en.wikipedia.org/wiki/Nonstandard_analysis https://en.wikipedia.org/wiki/Nonstandard_analysis
  - anikan_vader 5y ago
    The original construction did though, if you look at the writings of Newton/Leibniz. But their work was totally hand-wavey. Cauchy discovered limits in the 1800s, and it wasn’t until the 20th century that the original idea of infinitesimals was formalized. In some sense we have come full circle. I do think that infinitesimals are a much more intuitive way to understand calculus. It’s a shame it took so long to find positive numbers less than any real.

joppy 5y ago

Cardinalities of sets do have sensible operations like less than and greater than though - |A| <= |B| if there exists an injective function from A to B. One has to check that this inequality behaves in the ways that you expect, but it is a well-defined concept.

IX-103 5y ago
It is defined in the abstract, but I think there are a lot of implicit assumptions that get lost when dealing with cardinalities which may turn out to be important. Just because a bijective/injective/surjective function exists doesn't mean that such a transform is possible when you apply it to a specific situation. If sizes were sensible then for a set A composed of "every other integer" would be smaller than the a set B composed of "2 times every integer". If we were using calculus to take the limit of the ratio of sizes of the generating functions for A and B as the source set size goes to infinity then we find that A is in fact half the size of B. If we try to simply apply cardinality to A and B we find that the sets are exactly equal, since cardinality doesn't care about source sets or limits! This can a big deal if say, you're calculating probabilities across an infinite number of possible events/event configurations. There is a big difference between a 50% chance, a 0.000001% chance and a 100% chance. I'm not saying there's no use for cardinality and infinity classes, but they can easily be misapplied to allow you to be wrong, with confidence.
- anikan_vader 5y ago
  > If sizes were sensible then… Sorry, but they simply don’t meet your (non-standard) definition of sensible. Both of the sets you mentioned can be interpreted as the set of even integers. This set is in bijection with itself (trivially), and thus is not considered to be strictly smaller than itself.

openasocket 5y ago

I don't think there's any confusion of the finite and infinite numbers in Cantor's diagonalization argument. Here's the most generic form of Cantor's argument: there is no surjective function from a set A to its power set P(A). We prove this by contradiction. Consider any function f: A -> P(A) , i.e. it takes elements of A and outputs subsets of A. Suppose this function is surjective: i.e. for all y in P(A) there is some x in A such that f(x) = y. But let q = { a in A | a is not in f(a) }. Clearly this is a valid set. And if f is surjective, there must be some x in A such that f(x) = q. Is x in q? If x were in q, then x would be in f(x), so that's a contradiction. If x were not in q, then by the definition of q x would be in q, which is also a contradiction. Thus we have a contradiction, so f cannot be surjective. As you can see, nowhere do we make any logical jumps that would only make sense in the case of finite sets. This proof is as straightforward as the proof that there is no set of all sets. We don't use the axiom of choice, the argument is even valid in constructive mathematics (though you have to make some adjustments). Now, we define one set X as being "less than" another set Y if and only if there is no surjective function from X to Y. You'll see that this definition corresponds exactly to the usual notions of size for finite sets, and makes intuitive sense (if for every y we have an x, there must be at least as many xs as ys). Now, just plug in the set of natural numbers into Cantor's theorem and you get: the power set of natural numbers is larger than the set of natural numbers.

criloz2 5y ago
If you are counting a set, you necessarily are creating a map between the item of the set you are counting and the natural numbers, this is basics, how come that you can say such statement about the sizes of power set of natural number and natural number?, from where the notion of the cardinality comes if not from counting, and counting implies a map to a natural number, so the set of natural number is bigger than its selves?
- openasocket 5y ago
  Cardinality doesn't have to do with "counting" necessarily. Two sets X and Y are said to have the same cardinality if there is a function f : X -> Y where f is a bijection. By bijection we mean it has two properties: that for all a,b in X, if f(a) = f(b), then a = b (this is also called the injective property), and for all c in Y, there is some d in X such that f(d) = c (this is called the surjective property). Set X has cardinality less than Y if there is no such bijective function, but there is a function f: X -> Y that is injective. Conversely, X has cardinality larger than Y if there is no such bijective function, but there is a function f: X -> Y that is surjective. All you have to do to compare the size of two sets is to look at the functions mapping one to the other. No "counting" involved.

porb121 5y ago

> Almost anything can be proved/disproved or debated when one assumes such validity of the logical concepts to infinity. if this were true it would be trivial for you to give devastating examples