There are more reals than naturals though, so which kind of number do you mean? — Pfhorrest

Me? Any will do, depending on context I suppose.

ℵ₀ is a quantity that's not a real number, and ℵ₀ is the quantity of naturals/integers/rationals
ℵ₁ is a quantity that's not a real number, and ℵ₁ is the quantity of reals

It was really just a colloquial "definition", pointing out that ∞ ∉ R, |R| is ∞

Hence, mathematical analysis could suffer from the same fundamental interpretation problem surrounding infinity — alcontali

Classical real or complex analysis: very doubtful. Soft analysis: no telling where that is going. :cool:

It's just an opinion. — TheMadFool

Yep. And, it is wrong.

I don't see how inifinity is used in counting, nor is it a particular quantity. — Tim3003

Start with a poor definition and that's the soet of mess you get into,

See, instead, number.

nonsense.


It's not just that you are wrong, but that also you are not even not wrong.

That whole anthropological legend is apocryphal.

Because of this post, I started reading up on the axiomatization of real numbers, and of course, I have run into issues that I do not properly understand:

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

Real closed fields seem to be dramatically different from the standard real number system, but what is the actual difference?

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.

(if only I understood what they are saying ...)

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

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.

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.

https://mathoverflow.net/questions/136720/why-does-ch-imply-that-there-is-a-unique-ultrapower-of-mathbbn

Also see:
https://mathoverflow.net/questions/88292/non-zfc-set-theory-and-nonuniqueness-of-the-hyperreals-problem-solved

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.

I am confused myself over whether the completeness property is first or second order. I've seen explanations both ways. I believe it's second order. The hyperreals are a model of the first-order theory of the reals, but the hyperreals are not Cauchy-complete. That seems to imply that completeness must be second order.

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

Makes perfect sense. The algebraic numbers are not Cauchy-complete but they are a real closed field, just as the real numbers (which are Cauchy-complete) are.

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

This really isn't true, since the standard reals with Cauchy-completeness are second order. They provably have cardinality $2^{\aleph_0}$. This is the part where you're confusing yourself.

Also the extended reals of analysis with $\pm \infty$ have nothing to do with any of this. The extra points don't participate in the field properties as I'm sure you know from calculus.

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

This is incredible. I didn't know that this was possible. It is so unlike the models of PA:

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

Well, yeah, I wasn't aware of the fact they behave so differently ...

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

In my opinion, it is primarily a question of figuring out the foundational concepts embodied in the specialized vocabulary for the subject, i.e. real-numbers model theory. I had to do that for something a bit simpler, i.e. model theory and nonstandard models for PA. It looks like model theory related to real numbers is an entire subject in itself, even bigger than PA, with even more concepts to digest. It doesn't seem to have famous, celebrated theorems, though; unlike PA with Gödel's incompleteness theorems.

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

I wasn't sure if real-number theory is first order or second order. There is a disclaimer in the Löwenheim-Skolem page that it does NOT apply to second-order theories. In fact, I was aware of that disclaimer. Real-number theory turns out to be essentially second order. That was the main source of confusion.

To tell you the truth, this is the first time I have run into documentation about real-number model theory. It is totally new to me. As far as I am concerned, it is a completely new world. Very few concepts of natural-number model theory seem to transfer unchanged ...

The extra points don't participate in the field properties as I'm sure you know from calculus. — fishfry

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?

Field algebraic structures are more interesting to me. They reappear in cryptography. In elliptic-curve cryptography, the algebraic structure is specifically extended to contain $\infty$, which does effectively participate in the field. It is even the identity element for addition (without which the structure is not even a field):

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

What anthropological legend is apocryphal?

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

If infinity is a number greater than any real number, then by definition it is not a real number. So what kind of number do you propose that infinity is? If it is a real number greater than any real number, that is contradictory. If it is some other sort of number, how would we establish a relationship between this other number system, and the real numbers?

Well in my constructive understanding:

The 'Second-order' reals (as described via second-order logic) are also 'unique' from a constructionist perspective; for if the Axiom of Choice is rejected then second-order quantification over the sets of reals is strictly interpreted as quantifying over the constructable-sets of reals. Consequently, what we then have is a first-order countable model of the reals in 'second order' disguise. The reason why the real number field is unique in this interpretation is because we are actually still working within first-order logic; and since the Ultrafilter Lemma isn't constructively acceptable, the Löwenheim–Skolem theorem for first order-logic that depends upon it fails. Therefore constructive first-order models of the reals only possess models of countable cardinality. Consequently, there cannot exist models of constructive reals that are "non-standard" thanks to Tennenbaum's theorem that denies the existence of non-standard countable models that are recursive.

From this constructive perspective , the semantic intuition behind CH is trivially correct: There are no subsets of R whose size is greater than N but less than R, simply because the real numbers are encodings of natural number elements (via Godel numbering of the underlying computable total functions) and therefore they are of the same number. But alas there only exists an effective algorithm for deciding the provably total functions, i.e the provable real numbers, and hence there is no constructive proof that the number of provably constructive real numbers equals the number of constructable real numbers.

Infinity is not a real number because all real numbers have a definite magnitude or value whereas infinity does not. Infinity is always greater than any particular real number. Here's what I mean: the limit of y=1/x^2 as x "tends to infinity" is zero. I'm very unhappy with the "tends to infinity" part because it gives the feeling that x is moving to the right on the real number line with some kind of limited speed like a moving automobile. That is why Zeno's paradox arises because numbers can be divided infinitely and if that is the case, motion is impossible because the body must somehow traverse infinitely many numbers in a limited amount of time. That would mean movement amounts to saying some real thing pops out of existence at some point and then, over time, pops into existence at another point which is utterly absurd.

People frequently argue that problem has been solved by the concept of a limit. But that is not the case because unlike things in nature, there is no speed limit in mathematics. All operations in mathematics, counting 1, 2, 3, etc. happen instantaneously. x doesn't tend to infinity but is infinity. In the function above, x will never reach the x axis for any real value of x no matter how large it is, the moment you say x is some real number, essentially you have stopped "tending to infinity." Here is my solution to Zeno's paradox, things that happen in reality like motion do not exactly correspond to things in mathematics. Suppose you have a rod of a finite length. You cannot measure it with a ruler to determine its mathematical length exactly and mathematics always demands things be exact or it is just an estimate.

All the rules of mathematics for real numbers do not transfer over to arithmetic involving infinity. For example, for real numbers except zero (not all the rules in arithmetic apply to zero either for you cannot divide by it) you cannot perform an operation on one side of the equation only and expect to obtain a true result. Whereas (infinity = infinity) and (infinity + infinity) = infinity are both true.

non-finite

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

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.

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

If this were true, then mathematics could not give us truth. But it's not true, because mathematics can correspond exactly with reality. Consider that I have a table with some chairs. I can count the chairs and know that there is exactly six chairs there. If I want ten chairs, I can know that I need to get exactly four more. In some instances though, the mathematics is applied in a way which doesn't correspond exactly with reality, and this creates a problem like Zeno demonstrated. "Infinite division" does not correspond to reality, so this idea is itself a problem.

When you say mathematics can correspond exactly to reality, what do you mean? Numbers consist of 0-9 symbols. I'm sure the grass is nothing to do with those symbols. I would say, grass is a number, but it's not base 4.

There is infinity between the eyes and color.

Infinity is an eternity of time; in base 4 number, it is something, but that something is base 4, and not literal infinity.

When you say mathematics can correspond exactly to reality, what do you mean? — Qwex

As in my example, There are six chairs at the table. I need four more to have ten chairs at the table. If this is not an exact correspondence with the reality of that situation, what more is needed? What I want is ten chairs, that is the reality of the situation. Doesn't mathematics tell me exactly and precisely that I need four more chairs?

A 1 symbol on each chair is a direct correspondance?

I'd cut exactly and just put correspond.

Number can be any symbol, a C or an O, for example. It is also a 1 but that's only a property of C and O.

Yes, I could write 1 on one chair, 2 on the next, 3 on the next etc., to count them. Then I'd have a direct correspondence. After all the chairs are marked, I'd know that there is six chairs, and I could subtract six from ten to see that I still need four more to have the desired ten.

Suppose there is one and only one thing in the Universe and absolutely nothing else. Then the only number that exists is one. But in that case one is a meaningless predicate because everything is one. But if you divide that one thing into two pieces, then two exists and a half exists but one no longer exists. If we divide the one into three pieces, then three exists and a third exists, but both one and two no longer exists or any other number. If you object and say there are two (1/3) and one (1/3), I can say 2(1/3) is not the same as 2(1/2) that we used when we defined what two is and 1/3 is not the same as one when we defined what one is. I am certainly not saying mathematics is illogical or not useful. I'm saying things in reality are not numbers and I do not agree with Pythagoras that all is numbers. AFTER THOUGHT: okay, I confess that argument is lame, but all I'm trying to say is while it's easy to say there are six chairs at the table, no more no less, and if there are ten people attending the dinner, we need to get four more. However, it is possible, abet unlikely, we miscounted them. I'm almost certain E=mc^2 is exactly true, but it won't tell you how much matter is in a reactor that is converted to energy. That figure must be estimated by us and the equation will tell us exactly how much energy will be released based on that estimate. That is why I say it's hard to go from mathematical laws and concepts to reality and back again.

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

Wait, what? That's two things. The thing that exists and the number one. If a thing exists and there's no conscious entity around to comprehend it, there are no numbers. That would be my view. That numbers are an artifact of consciousness. There's a thing, but there's not the number one till someone experiences that thing; and moreover, evolves sufficient reason to count the thing. Counting's not an inherent part of the universe. It's something rational beings do. No experiencer, no numbers.

I'm saying a number "exists" only if it can be instantiated by something in reality. If many thing exist, then one cannot be instantiated by anything.

I'm saying a number "exists" only if it can be instantiated by something in reality. — Michael Lee

That seems unduly restrictive. By that criterion you would have rejected Riemann's non-Euclidean geometry in the 1840's because it was so obviously untrue about the world. Then when Einstein used Riemannian geometry to frame his general theory of relativity, you'd have had to change your mind. Except that by your logic, you'd have abandoned research into Riemann's work and Einstein would never have had the tool available.

Isn't it rather the job of math not to describe reality as we know it; but to provide concepts and tools that may be of use to future scientists? The existence of a mathematical object depends only on logical consistency and interestingness. Not on conformity to the limitations of contemporary knowledge of the world.

I'm not saying mathematics is illogical or wrong and thus theories about reality are illogical or wrong. Einstein's equation E=mc^2 correctly predicts how much energy will be produced from a certain amount of matter. What it cannot do is tell you how much matter is in a reactor say. The actual number must be estimated by us and then plugged into the equation to obtain the exact amount of energy based on that estimate. That is why I say reality is hard to fit into mathematics or mathematics into reality.

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

I see. ;-)

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

If I understood the explanations correctly, Löwenheim-Skolem applies to the theory of real closed fields (=first order theory) but not to to the theory of real numbers (=second order theory). That last bit wasn't immediately clear to me:

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.

I had never read anything on model theory for real numbers. The materials I had run into were all about natural numbers. — alcontali

To add to the Wiki quote, something I mentioned earlier: The hyperreals are not Cauchy-complete. No non-Archimedean field can be. Which leads to one of my little hobby horses. The constructive reals aren't complete because there are too few of them, only countably many. The hyperreals aren't complete because there are too many of them, the reals plus an uncountably infinite cloud of infinitesimals about each real. The standard reals are the Goldilocks model of the reals. Not too small and not too big to be Cauchy-complete. They're just right. And are therefore to be taken as the morally correct model of the reals.

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.

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

Zeno's paradox is best solved by observing how you would practically explain the paradox. To practically demonstrate the paradox requires one to repeatedly move an object along the same path, but ending the motion at the half-way point of the previously travelled distance and exclaiming "the object must have earlier travelled through this point".

In other words, a demonstration of Zeno's paradox can only explain what an object position is by destroying the object's motion. In other words, this demonstration shows that the construction of a position is incompatible with the construction of a motion, and hence is an intuitive demonstration of the Heisenberg Uncertainty Principle.

In my opinion, Zeno was close to discovering this principle characteristic of Quantum Mechanics, purely from ordinary phenomenological arguments.