There are more reals than naturals though, so which kind of number do you mean? — Pfhorrest
The supremum axiom of the reals refers to subsets of the reals and is therefore a second-order logical statement. It is not possible to characterize the reals with first-order logic alone: the Löwenheim–Skolem theorem implies that there exists a countable dense subset of the real numbers satisfying exactly the same sentences in first-order logic as the real numbers themselves. — Wikipedia insisting that the reals are a second-order theory
In mathematics, a real closed field is a field F that has the same first-order properties as the field of real numbers. Some examples are the field of real numbers, the field of real algebraic numbers, and the field of hyperreal numbers. — Wikipedia on real closed fields which are a first-order theory
The first-order theory of real closed fields is the first-order theory whose primitive operations are addition and multiplication, primitive predicates are = and ≤, and axioms are those of a real closed field. More precisely, a first-order theory is roughly speaking, a theory where quantifiers apply only to elements (not to sets of elements).
Alfred Tarski proved (c. 1931) that the first-order theory of real closed fields is complete, and decidable. This means that there exists a general procedure that takes as input an assertion expressed in this theory and decides which of the assertion and its negation is true (complete means that either the assertion or its negation is true). — Wikipedia: unlike natural-number logic, real-closed field logic is decidable
Then, they say something very interesting but really complicated about the generalized continuum hypothesis in conjunction with real closed fields. I think that this is the kind of things that could shed light on the true nature of infinite cardinality in real-number theory. — alcontali
In mathematics, a real closed field is a field F that has the same first-order properties as the field of real numbers. Some examples are the field of real numbers, the field of real algebraic numbers, and the field of hyperreal numbers. — Wikipedia on real closed fields which are a first-order theory
Still, the fact that an affinely extended real number system is possible, suggests that mathematical analysis may have exactly the same interpretation problem as Peano's arithmetic (PA), i.e. if one infinite cardinality satisfies the model, then all other upward infinite cardinalities also do. — alcontali
It means essentially that CH is equivalent to the fact that all models of the hyperreals are isomorphic. The idea is that the particular model of hyperreals you get depends on which nonprincipal ultrafilter you choose. If CH holds then all the models are isomorphic. — fishfry
The elements of any model of Peano arithmetic are linearly ordered and possess an initial segment isomorphic to the standard natural numbers. A non-standard model is one that has additional elements outside this initial segment. The construction of such models is due to Thoralf Skolem (1934). — Wikipedia on nonstandard models of arithmetic
There's a Mathoverflow thread about this, let me see if I can find it. Ah here it is. Good luck reading. MO as you know is a site for professional mathematicians so the best one can hope for is to understand a few of the words on the page. — fishfry
I don't know the answers to all the good questions you raise, but I can't help thinking that you're overthinking things and letting yourself get confused by Lowenheim-Skolem. — fishfry
The extra points don't participate in the field properties as I'm sure you know from calculus. — fishfry
This set together with the group operation of elliptic curves is an abelian group, with the point at infinity as an identity element. — Wikipedia on ECC
↪Frank Apisa That whole anthropological legend is apocryphal. — Banno
infinity is a number, but it has a characteristic that all real numbers do not possess. Namely, it is a number that is greater than any particular real number. — Michael Lee
Calculus was just school exam material for me consisting of endless symbol manipulation. I didn't particularly "care" about it. It is not that I have read anything about calculus ever since. It doesn't appear in computer-science subjects either. So, what am I supposed to do with it? — alcontali
Here is my solution to Zeno's paradox, things that happen in reality like motion do not exactly correspond to things in mathematics. — Michael Lee
When you say mathematics can correspond exactly to reality, what do you mean? — Qwex
Suppose there is one and only one thing in the Universe and absolutely nothing else. Then the only number that exists is one. — Michael Lee
I'm saying a number "exists" only if it can be instantiated by something in reality. — Michael Lee
since the Ultrafilter Lemma isn't constructively acceptable — sime
Examples of non-trivial ultrafilters are difficult (if not impossible) to give, as the only known proof of their existance relies on the Axiom of Choice. — The 'Art of Solving Problems' on giving examples for the ultrafilter concept
You invoked the extended real numbers and claimed it has something to do with L-S, which of course it does not. Unless I misunderstood your point. — fishfry
The supremum axiom of the reals refers to subsets of the reals and is therefore a second-order logical statement. It is not possible to characterize the reals with first-order logic alone: the Löwenheim–Skolem theorem implies that there exists a countable dense subset of the real numbers satisfying exactly the same sentences in first-order logic as the real numbers themselves. The set of hyperreal numbers satisfies the same first order sentences as R. Ordered fields that satisfy the same first-order sentences as R are called nonstandard models of R. This is what makes nonstandard analysis work; by proving a first-order statement in some nonstandard model (which may be easier than proving it in R), we know that the same statement must also be true of R. — Wikipedia on Löwenheim-Skolem in the context of real numbers
I had never read anything on model theory for real numbers. The materials I had run into were all about natural numbers. — alcontali
After all this discussion, I'm starting to reject my claim that Zeno's paradox can be solved by our inability to count and measure things. I'll think about this more at a later time. — Michael Lee
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