Bah! Chaitin makes so much of his discovery that there is randomness in mathematics, but doesn't ever seem to have made the shift to thinking of structures as parametric over randomness.
To wit, generating random real numbers by simply flipping a countable number of coins and writing down the bits is trivial, and from that point of view:
1) An open set is just a set about which we can have finite information.
2) A closed set is just a set about which we can have complete information.
3) A compact set is just one we can quantify over and prove theorems about given only finite information.
4) A continuous function is one that has finite character: producing finite information about an element of the image only ever requires finite information about the argument.
These statements are a bit rough, but hey, if I ever have a gajillion dollars I can spend my newfound spare time merging the computational view on synthetic topology, Chaitin's algorithmic information theory, and the old "Age of Stochasticity" talk to get a view of mathematics that properly accounts for its finities and infinities.
http://www.stat.uchicago.edu/~lekheng/courses/191f09/mumford...
That guy is just a constructivist. Nothing new here.
It is no surprise that the set remains elusive, it doesn't exist.
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
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).
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.
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
But you can also assume ZF plus the negation of the axiom of choice, and get a system that is consistent if and only if ZF is consistent. It's not clear (to me, at least) that this other system will let you build the real numbers.
https://en.wikipedia.org/wiki/Ultrafinitism
...and "Set Theory, Should You Believe?"
https://web.archive.org/web/20110616020815/http://web.maths....
You can even write a set of them: {1, 2, 3, 4}
You cannot write the set of real numbers.
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).
Both real and natural numbers can be reasoned about despite their infinite size (and we even know that e.g. there must be "more" real numbers than natural numbers, even though both sets are infinite)
There are no bounds in the universe that can contain all the real or natural numbers.
and here is one of the natural numbers: ℕ
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).
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).
For those with some time to spare, you might enjoy:
Clearing Up Mysteries -- The Original Goal
http://bayes.wustl.edu/etj/articles/cmystery.pdf
"Put most briefly, Einstein held that the QM formalism is incomplete and that it is the job of theoretical physics to supply the missing parts; Bohr claimed that there are no missing parts. To most, their positions seemed diametrically opposed; however, if we can understand better what Bohr was trying to say, it is possible to reconcile their positions and believe them both. Each had an important truth to tell us. But Bohr and Einstein could never understand each other because they were thinking on different levels. When Einstein says QM is incomplete, he means it in the ontological sense; when Bohr says QM is complete, he means it in the epistemological sense. Recognizing this, their statements are no longer contradictory. Indeed, Bohr's vague, puzzling sentences -- always groping for the right word, never finding it -- emerge from the fog and we see their underlying sense, if we keep in mind that Bohr's thinking is never on the ontological level traditional in physics. Always he is discussing not Nature, but our information about Nature. But physics did not have the vocabulary for expressing ideas on that level, hence the groping."
>we understand "infinity" just fine.
I enjoyed "Meta Math" by Gregory Chaitin:
http://arxiv.org/abs/math/0404335
"So the set of real numbers, while natural—indeed, immediately given— geometrically, nevertheless remains quite elusive: Why should I believe in a real number if I can’t calculate it, if I can’t prove what its bits are, and if I can’t even refer to it? And each of these things happens with probability one! The real line from 0 to 1 looks more and more like a Swiss cheese, more and more like a stunningly black high-mountain sky studded with pin-pricks of light."