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It doesn't appear that the set of rel numbers exists. If it did, someone should be able to point to it and say, "this is the set". No one can do that because th
by jsprogrammer 10y ago
It doesn't appear that the set of rel numbers exists. If it did, someone should be able to point to it and say, "this is the set". No one can do that because the available space to present the set appears finite, but producing the set would require an infinite amount of space (assuming you need some exclusive space to represent each number).
It is no surprise that the set remains elusive, it doesn't exist.
- chc 10y agoSets of numbers aren't physical things. Physical things can't be infinite, but I don't see why you think a set can't. By this thinking, not even the natural numbers can exist, since you can't point to them either.
- jsprogrammer 10y agoNatural numbers are easy to write: 1, 2, 3 You can even write a set of them: {1, 2, 3, 4} You cannot write the set of real numbers.
- chriswarbo 10y ago{1, 2, 3, 4} is also a set of real numbers. Why is it a valid existence proof of the naturals, but not the reals?
- Govannon 10y agoPossible linguistic pedantry; he used "a" set of natural numbers, but "the" set of real numbers. It's equally difficult to represent the set of natural numbers.
- jsprogrammer 10y agoIt's not pedantry, it's just what the words mean. To the question: I'm only saying that the "set of real numbers" and the "set of natural numbers" don't seem to exist. Despite numerous downvotes, no one has yet produced them here (or even a link to them).
- zAy0LfpBZLC8mAC 10y agohere is a representation of the real numbers: ℝ and here is one of the natural numbers: ℕ
- jsprogrammer 10y agoHow can I select an arbitrary, or even random, element from either?
- zAy0LfpBZLC8mAC 10y agoWho said that you could?
- jsprogrammer 10y agoZermelo? Axiom of choice says I should be able to select an element from the set of real numbers (assuming it exists; but, I believe the assumption to be counterfactual).
- DigitalPhysics 10y agoI agree. Even though the axiom of choice is independent of ZF, I don't think it is self-evident axiom. It actually seems pretty counter-intuitive if you believe in infinite-precision real numbers that have an infinite amount of information and can't be compressed. I have more to say on this in "Digital Physics" (the movie). -Khatchig
- jsprogrammer 10y agoAccording to Wikipedia, Zermelo formulated the axiom of choice. I think it makes sense though; if I (claim to) have a thing, I should be able to choose/pick/select/point-to it (seems to be almost[?] tautological).
- 10y ago
- GregBuchholz 10y agoYou may enjoy: https://en.wikipedia.org/wiki/Ultrafinitism https://en.wikipedia.org/wiki/Ultrafinitism ...and "Set Theory, Should You Believe?" https://web.archive.org/web/20110616020815/http://web.maths.unsw.edu.au/~norman/views2.htm https://web.archive.org/web/20110616020815/http://web.maths....
- witty_username 10y agoThe universe may be infinite.
- Retra 10y agoThe set of real numbers is as big as the set of functions from the naturals to themselves. In fact, that's a pretty convenient way of modeling the real numbers sometimes: as systematic names for functions on natural numbers.
- osoba 10y agoThe set of real numbers most certainly exists, it is defined as the unique such set (up to an isomorphism) that satisfies the following properties: It is at the same time a commutative additive group and a commutative multiplicative group such that the two group neutrals aren't the same. Furthermore the multiplication distributes over addition. There exists a total order relation such that the addition distributes over the order relation and multiplication agrees with the order relation (if (0 leq x) and (0 leq y) then (0 leq x*y)). Another axiom is required, which comes in many forms (all of which are equivallent) so take your pick, for example the supremum axiom https://en.wikipedia.org/wiki/Least-upper-bound_property https://en.wikipedia.org/wiki/Least-upper-bound_property You can read about any of this in any texbook of real analysis (for example Spivak or baby Rudin). If the axiomatic formulation of real numbers doesn't appeal to you, there is another way to define real numbers, namely to contruct them from natural numbers (and you seem convinced that natural numbers indeed exist). You can read about that in the appendix to chapter 1 of baby Rudin (Principles of Mathematical Analysis by Walter Rudin).
- soVeryTired 10y agoBy no means am I an expert in this stuff, but don't you need the axiom of choice (or maybe something just a little bit weaker) to construct the reals? I don't think it's fair to say the reals 'most certainly exist' without being misleading to a layman. They exist given some axioms that are used overwhelmingly often in mathematics, but you can still do some interesting stuff without those axioms, or with their negation.
- osoba 10y agoYou're correct. There is one object in all of mathematics that isn't defined (you've got to start from somewhere), it is just assumed that humans implicitly understand this concept, and it's the concept of a set (denoting a collection of objects). Along with it, certain properties of this object are assumed (among them the ability to choose an element of a non empty set - this is typically called the axiom of choice). For a full list you can take a look at http://mathworld.wolfram.com/Zermelo-FraenkelAxioms.html http://mathworld.wolfram.com/Zermelo-FraenkelAxioms.html