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Re: Bijection Between Complex Numbers and Real Numbers?
Posted:
Feb 20, 2012 8:06 AM
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On Sun, 19 Feb 2012 01:50:02 -0700, Virgil <virgil@ligriv.com> wrote:
>In article
><607a757f-6f08-4e1e-a240-46eae3ba3d0d@p13g2000yqd.googlegroups.com>,
> Butch Malahide <fred.galvin@gmail.com> wrote:
>
>> On Feb 19, 1:25 am, Virgil <vir...@ligriv.com> wrote:
>> > In article <Pine.NEB.4.64.1202182143080.27...@panix3.panix.com>,
>> > William Elliot <ma...@panix.com> wrote:
>> >
>> > > On Sat, 18 Feb 2012, Michael Ejercito wrote:
>> >
>> > > > What bijective function exists such that every complex number maps
>> > > > to a unique real number, and likewise every real number maps to a
>> > > > unique complex number?
>> >
>> > There is certainly no such bijective function which preserves any of the
>> > arithmetical, order or topological properties between the two fields.
>>
>> That seems slightly overstated, in view of the fact that (assuming the
>> axiom of choice) the additive groups are isomorphic.
>
>But one need not assume the axiom of choice.
Doesn't matter. You made an assertion. AC implies that your assertion
is false. _Hence_ you cannot _prove_ your assertion, because
a proof of your assertion would then prove "not AC", and that's
impossible.
(Ok, it's impossible if ZF is consistent. When you prove ZF is
inconsistent let us know.)
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