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Well, I wouldn't say type theory is superior. Real numbers can actually be logically constructed as sets. So yes, Pi is a set. For example, you can define the

Well, I wouldn't say type theory is superior. Real numbers can actually be logically constructed as sets. So yes, Pi is a set. For example, you can define the set of real numbers as the set of equivalent classes of rational cauchy sequences. In order words, Pi is a set of rational cauchy sequences that are equivalent (this set is infinite). In simple (non formal) words: Pi is the set of sequences of rational numbers that get closer and closer to the numerical value of Pi. >>>> Is 3 an element of pi? If pi truly is a set Pi being a set doesn't mean that it is a set of real numbers. In fact, that would be illogical (recursive). If Pi is a set, it must be a set of things that are not real numbers. So it is not logical to ask if 3 is an element of Pi. You can ask if 3 is equal to Pi, in which case the answer is no.

cgmg 9y ago
> Real numbers can actually be logically constructed as sets. So yes, Pi is a set. You missed the point. The point is that while X (e.g. the real numbers) can be encoded as Y (e.g. Cauchy sequences), this doesn't mean that X is literally Y. That would entail that Cauchy sequences and Dedekind cuts, for example, are the exact same thing, which they are not. Rather, they are two different ways of encoding the same abstract structure (a Dedekind-complete ordered field). > Pi being a set doesn't mean that it is a set of real numbers. This is dodging the question. Is the set containing the empty set a member of pi? Such a question is meaningless without fixing a particular encoding, and it has nothing to do with the intrinsic properties of a real number.
- mathsandphysics 9y ago
  Thanks for your reply. >The point is that while X (e.g. the real numbers) can be encoded as Y (e.g. Cauchy sequences), this doesn't mean that X is literally Y. In my opinion, a mathematician does not know what the difference between "being literally" and "being encoded as" is - unless you assume that mathematics should aim to be a formal encoding of some pre-existing 'intrinsic reality', in which case what you're doing is physics and not math. In physics, the debate is more open: Is the wave function literally the electron? Or is the wave function just an encoding of the electron, which is the intrinsic real thing? Or is the electron the way we humans perceive the wave function, which is the intrinsic real thing? Anyway this is probably way too theoretical of a debate :)
  - cgmg 9y ago
    > In my opinion, a mathematician does not know what the difference between "being literally" and "being encoded as" is Wrong. The real numbers are literally a Dedekind-complete ordered field. The real numbers are encoded as Cauchy sequences, or as Dedekind cuts. Cauchy sequences are not Dedekind cuts: The former are equivalence classes of sequences. The latter are downward-closed sets without a greatest element. They are literally different objects. Regarding your electron comment, the answer is straightforward: the electron is an excitation of the electron spinor field, nothing more. Fields are more fundamental than particles.

cgmg 9y ago

> Real numbers can actually be logically constructed as sets. So yes, Pi is a set. You missed the point. The point is that while X (e.g. the real numbers) can be encoded as Y (e.g. Cauchy sequences), this doesn't mean that X is literally Y. That would entail that Cauchy sequences and Dedekind cuts, for example, are the exact same thing, which they are not. Rather, they are two different ways of encoding the same abstract structure (a Dedekind-complete ordered field). > Pi being a set doesn't mean that it is a set of real numbers. This is dodging the question. Is the set containing the empty set a member of pi? Such a question is meaningless without fixing a particular encoding, and it has nothing to do with the intrinsic properties of a real number.

mathsandphysics 9y ago
Thanks for your reply. >The point is that while X (e.g. the real numbers) can be encoded as Y (e.g. Cauchy sequences), this doesn't mean that X is literally Y. In my opinion, a mathematician does not know what the difference between "being literally" and "being encoded as" is - unless you assume that mathematics should aim to be a formal encoding of some pre-existing 'intrinsic reality', in which case what you're doing is physics and not math. In physics, the debate is more open: Is the wave function literally the electron? Or is the wave function just an encoding of the electron, which is the intrinsic real thing? Or is the electron the way we humans perceive the wave function, which is the intrinsic real thing? Anyway this is probably way too theoretical of a debate :)
- cgmg 9y ago
  > In my opinion, a mathematician does not know what the difference between "being literally" and "being encoded as" is Wrong. The real numbers are literally a Dedekind-complete ordered field. The real numbers are encoded as Cauchy sequences, or as Dedekind cuts. Cauchy sequences are not Dedekind cuts: The former are equivalence classes of sequences. The latter are downward-closed sets without a greatest element. They are literally different objects. Regarding your electron comment, the answer is straightforward: the electron is an excitation of the electron spinor field, nothing more. Fields are more fundamental than particles.