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You can't do this in general, and everywhere, so it's still not rigorous. Consider the embedding of the nats into the reals, where the reals are defined as a su
by jhanschoo 2y ago
You can't do this in general, and everywhere, so it's still not rigorous. Consider the embedding of the nats into the reals, where the reals are defined as a subset of the binary sequences (e.g. as the usual infinite binary expansion for (0,1] choosing those that do not terminate in the zeros sequence, but prefix with a Elias-gamma-coded, zigzag-encoded integer). But the usual definition of sequence is as a function from the nats; so are you again going to redefine your reals in terms of that?
In the end you still need to maintain a correspondence between the embedding and the original set as in the typical way where you do that and consider the subset notation as a shorthand that requires you to "lift" or "wrap" through the correspondence wherever necessary.
- whyever 2y agoI don't think you can define the reals as a mapping from natural numbers, reals are famously not countable.
- Tainnor 2y agoYou can't map the natural numbers to the set of real numbers, but you can map the natural numbers to the fractional part of a single real number.
- defrost 2y agoHow would this work when mapping them to the fractional part of the real number 2 ? Collapsing a countably infinite set to 0 doesn't seem useful or reversable.
- gizmo686 2y agoThe integer part is easy, since we already have the integers. Once you have D=[0,1), then you can define R=ZxD. That is to say, this definition of R seperates out the integral and fractional components of every real number.
- Tainnor 2y agofun n => 0 I'm not sure what you're trying to disagree with here.
- robinhouston 2y agoFortunately functions from the natural numbers are not countable either. This is Cantor's theorem.
- danwills 2y agoI'm not a set-theorist, but a set-theorist friend of mine once taught me that you can turn a countably-infinite set (such as the integers) into an uncountably-infinite one (like the reals) by applying the 'power set' operation (the set of all subsets). Not heaps sure what this really means with respect to whether 1 the integer is really completely related (as in, equal, or the-exact same-thing) to 1.0 the real though. Kinda seems like it might still need a bit more information to fully identify a real, even when it happens to be infinitely-close to an integer?
- Tainnor 2y agoMore generally, Cantor's diagonal argument shows that there is no surjective map of a set onto its powerset.
- housecarpenter 2y agoI think you misunderstood the parent comment. They're not talking about defining the reals by setting up a mapping where each real number corresponds to a unique natural number. They're talking about defining the reals by setting up a mapping where each real number corresponds to a unique mapping from the natural numbers to {0, 1} (i.e. a unique binary sequence). The set of all binary sequences is isomorphic to the power set of the natural numbers, which is uncountable.