Are natural numbers isomorphic to complex numbers?
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From: Arturo Magidin on 5 Jun 2010 14:42 On Jun 4, 11:57 pm, Akira Bergman <akiraberg...(a)gmail.com> wrote: > > > But what does that have to do with isomorphisms? "Isomorphic" means > > "has exactly the same structure". Even if we forgot the topological > > and cardinality differences between N and C, how can you think that > > they can have the same algebraic structure, given that C is is an > > algebraically closed field, and N is not even closed under > > subtraction? > > It means that N has all of the information that C has. You just have > to know how to look at it. That is not the meaning of "isomorphic". That's the problem here. If this is the kind of thing you are talking about, then you can say so. As to this, well, yes. Everything that is in C is "somehow" in N. How do I know? Easy: one can *construct* C from N using nothing but basic set theory. You can easily codify all the integers in terms of equivalence classes of pairs of natural numbers; you can easily codify all rationals in terms of integers (and hence in terms of natural numbers) as equivalence classes of pairs of integers. You can perhaps- not-so-easily codify all real numbers in terms of rationals (and hence in terms of integers, and hence in terms of natural numbers), either using things like Dedekind cuts or using things like Cauchy-sequences. And you can easily codify all complex numbers in terms of real numbers (and hence in terms of rationals, and hence in terms of integers, and hence in terms of natural numbers) as ordered pairs or as the plane, or as any number of other things. Everything in C is there, in potential, in N. More: N also contains, in the same sense, the potential for Hensel's p- adic numbers; it contains information about every single compact metric space (since every compact metric space is a continuous image of the Cantor set, which can be described in terms of real numbers, which are "in potential" inside of N). And much more. But, alas, that is not what "isomorphic" means. -- Arturo Magidin
From: porky_pig_jr on 5 Jun 2010 15:04 On Jun 5, 2:42 pm, Arturo Magidin <magi...(a)member.ams.org> wrote: > As to this, well, yes. Everything that is in C is "somehow" in N. How > do I know? Easy: one can *construct* C from N using nothing but basic > set theory. But why stop at N? By the same token, everything that is in C is "somehow" in empty set (Then construct N with, say, Von Neumann numerals, and then all way up to C). PPJ.
From: Arturo Magidin on 5 Jun 2010 15:07 On Jun 5, 2:04 pm, "porky_pig...(a)my-deja.com" <porky_pig...(a)my- deja.com> wrote: > On Jun 5, 2:42 pm, Arturo Magidin <magi...(a)member.ams.org> wrote: > > > As to this, well, yes. Everything that is in C is "somehow" in N. How > > do I know? Easy: one can *construct* C from N using nothing but basic > > set theory. > > But why stop at N? By the same token, everything that is in C is > "somehow" in empty set (Then construct N with, say, Von Neumann > numerals, and then all way up to C). Well, I stopped at N because that's what the OP is talking about. Now, you actually a lot of the axioms of Set Theory, not just the empty set (notice I said "basic set theory above"). Or you could go by way of Lawvere and get it out of Categories, for that matter. -- Arturo Magidin
From: Transfer Principle on 6 Jun 2010 01:12 On Jun 4, 10:19 pm, Ostap Bender <ostap_bender_1...(a)hotmail.com> wrote: > On Jun 4, 9:57 pm, Akira Bergman <akiraberg...(a)gmail.com> wrote: > > It means that N has all of the information that C has. You just have > > to know how to look at it. > It may be the case that a single atom contains all the information > about the entire Universe, but it is hard to see than an atom and the > Universe are isomorphic. A backhanded reference to AP and his Atom Totality theory, of course. Even though I am not an Atom Totalitarian, I see nothing wrong with considering AP's theories when they concern _mathematics_. As for Bergman, I'm not sure yet whether he's trying to introduce a new theory, discuss standard theory (in which case he's confused about what an algebraic isomorphic is), or neither.
From: Transfer Principle on 6 Jun 2010 01:19
On Jun 4, 11:39 pm, Ostap Bender <ostap_bender_1...(a)hotmail.com> wrote: > On Jun 4, 11:19 pm, Akira Bergman <akiraberg...(a)gmail.com> wrote: > > It is self explanatory from x^y = e^(y*log(x)). It has been on the net > > for years and still gets good traffic. It was agreed by sci.math crowd > > years ago. I don't know how it has been missed by so many math-heads > > for so many years. This is what I mean by the pitfalls of formalism > > heavy thinking. > > I don't have to advocate it. It advocates itself. > So, you agree with what was decided 180 years ago: that it is best to > define 0^0 as 1? Then what is your innovative contribution to this > question? > And what is this thing: > > 0^0 = {0,1} > What kind of a number is {0,1}? It is not an element of R nor of C, is > it? What multiplicaton group is it part of? I see nothing wrong with having a two-valued function, such as the sqrt function. But then this takes us into JSH territory, for wasn't it JSH who advocated that sqrt(4) should be {2,-2} rather than the principal sqrt, namely 2? As for 0^0, if we mean the cardinal (or ordinal) zero, then it does make sense to define 0^0 to be 1, since 0^0 is considered to equal {0}, the set whose lone element is the empty set (or empty function), and {0} evidently has a cardinality of one. Also, in tetration, it's often convenient to define 0^^2 to be 1, since lim_x->0 (x^^2) = lim_x->0 (x^x) = 1. |