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What are the 'real numbers', really?

math.vanderbilt.edu

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Re: What are the 'real numbers', really?

#81

Post by thaumasiotes »

bjornsing wrote: post #73
thaumasiotes wrote: post #71

Earlier quoted context omitted.

Well, consider the ruler function[1], which is continuous on the irrationals and discontinuous on the rationals. The real numbers really are denser than the rationals; that's why something like the ruler function is possible (notably, a conceptual reverse, continuous on the rationals and discontinuous on the irrationals, cannot exist -- the rationals are too far apart). I'm pretty sure this is precisely the phenomeno…

Infinity is a pretty strange concept. :) I'm not sure arguing over it in this format is meaningful, but for the fun of it: Consider that the integral of the ruler function from 0 to 1 is 0 (as is stated in your reference 1). In layman's terms you could express this as "there are infinitely more irrational than rational numbers between 0 and 1". At the same time, "for every two rational numbers there are infinitely ma…

I don't understand the point you're trying to make?

The Cantor set shares the property that "for every two [points in the set] there are infinitely many [points in the set] in between", but no one would describe it as looking like a line. It's rather sparse.

Re: What are the 'real numbers', really?

#82

Post by GFK_of_xmaspast »

totemizer wrote: post #74
GFK_of_xmaspast wrote: post #58

Earlier quoted context omitted.

Guys like that in general have never seemed all that convincing to me.

I admit, some of his ideas are a bit.. well, I don't like when people talk about God seriously, and he sometimes mentions it, very rarely. But aside that, everything I can understand from what he says is true. It's a philosophical debate and if you are on the "real numbers" bandwagon (where most people are), you would lose integrity and your reputation might suffer even if you would speak to Wildberger about real num…

So I have a phd in math and I do tend to think less of other mathematicians who argue against infinite sets or uncountable sets and such, the argument's been over for a hundred years, you lost, deal with it. It's mathematical geocentrism.

Re: What are the 'real numbers', really?

#83

Post by GFK_of_xmaspast »

chowells wrote: post #6

What are "real numbers"? A horribly misnamed fiction. Nearly all of them cannot be represented with a finite amount of information. I strenuously object to naming an uncountable set "real" when only a countable subset (measure 0 of the full set) can be worked with in any way at all. We need to stop venerating the "real" numbers and start focusing on sets that are actually usable.

I don't see why we need to be shackled to the bounds of countability.

Re: What are the 'real numbers', really?

#84

Post by archgoon »

gizmo686 wrote: post #51
ColinWright wrote: post #42

Earlier quoted context omitted.

Indeed, and the constructable numbers are studied as a subset of the reals, as are the algebraics, and the computables. You can make a choice as to the domain of discourse. If you like, feel free to restrict it to the computables (or the constructables). Then apply the diagonal argument. Take the computable numbers between 0 and 1, including 0, not including 1. These are countable, so we can write them in a list, tak…

I can't find a flaw in your arguement, but it seems like it leads to a contradiction. Let a constructable number be one which can be unambiguously described in a finite string. Because we are working from a finite alphabet, we can trivially see that their is a bijection between the constructables and the integers (if we have n symbols, then each string can be read as an integer in base n, so the amount of constructab…

How do you know if a number is constructible? It is described by a computer program. Although the set of computer programs can be enumerated, determining if a program it will print out any digits is not something that cannot be determined. So yes, Halting Problem. :)

So you cannot list all constructables in a constructive manner because the list itself is not constructible.

A related concept

https://en.wikipedia.org/wiki/Chaitin%27s_constant

Re: What are the 'real numbers', really?

#85

Post by lmm »

ColinWright wrote: post #50
gweinberg wrote: post #46

The problem with "points on a number line" as a definition for real numbers is that it's not clear how you can tell if you have all of them. You can populate a number line as densely as you care to using just rational numbers, but that's not all of them, you're missing out on numbers like the square root of two. You can toss in the non-intergral powers of rational numbers, but you still won't have all of them, you're…

If you assume that there is no number bigger than zero but smaller than every positive number (basically the Archimedean property) then you can prove that "you've got them all." You use Dedekind cuts. Suppose there's a location on the line that's somehow missing - call it x. Let A be all the numbers less than x, let B be all the numbers greater than x, and that gives you your Dedekind cut. That Dedekind cut is, in a…

If you're using the Dedekind cut definition why use the line at all? Just say a real is any set of rationals bounded above, with arithmetic defined the obvious way; defining equality is slightly fiddly but it's fiddly with a number line too. What does the line visualization gain you?

Re: What are the 'real numbers', really?

#86

Post by ddebernardy »

Stal3r wrote: post #79

Can someone explain the setup of the 0=1 exercise? It's poorly worded. Is it saying find: (Y,1,+,1,×) or is it saying find what "1" has to be to make it a valid field?

Given the field (Y, 0, +, 1, *), you need to show that either 0 != 1, or 0 = 1 is the only element in the set.

I don't remember the precise proof, but if memory serves it derives from the existance of opposites and inverses, and 0 and 1 being unique in the set, due the commutative properties of abelean groups.

Re: What are the 'real numbers', really?

#87

Post by ddebernardy »

mherrmann wrote: post #48

This is a great article but unfortunately has one thing horribly wrong: Democracy far preceded the Age of Enlightenment. A form of democracy was already in place in ancient Greece at around 500 BC. Newton and the Age of Enlightenment were much later, at 1600+ AD. See Wikipedia: http://en.wikipedia.org/wiki/Democracy#History , http://en.wikipedia.org/wiki/Age_of_enlightenment , http://en.wikipedia.org/wiki/Isaac_Newto…

Hehe. I balked at that too...

It also states that you cannot order the field of complex numbers. Whereas I seem to recollect that there are ways to do so. For instance, z1 < z2 if x1 < x2 or x1 = x2 and y1 < y2.

Re: What are the 'real numbers', really?

#88

Post by pedrosorio »

DArcMattr wrote: post #18

I have a Master's in Applied Math. The comments about how "few students take [Real Analysis]" doesn't square with my experience and survey of an undergraduate mathematics education. Such a course is often called "Advanced Calculus", and is a required course for a Bachelors-level education in Math. I also understand in the European-style approach to teaching Math, students start off with a foundational approach to Cal…

"I also understand in the European-style approach to teaching Math, students start off with a foundational approach to Calculus through Real Analysis, and not the hand-wavy & computation-driven Calculus course."

Yes. Where I graduated, all engineering majors learn the axiomatic definition of the real numbers including the "supremum (least upper bound) axiom" at the beginning of the first calculus class.

Re: What are the 'real numbers', really?

#89

Post by ColinWright »

lmm wrote: post #85
ColinWright wrote: post #50

Earlier quoted context omitted.

If you assume that there is no number bigger than zero but smaller than every positive number (basically the Archimedean property) then you can prove that "you've got them all." You use Dedekind cuts. Suppose there's a location on the line that's somehow missing - call it x. Let A be all the numbers less than x, let B be all the numbers greater than x, and that gives you your Dedekind cut. That Dedekind cut is, in a…

If you're using the Dedekind cut definition why use the line at all? Just say a real is any set of rationals bounded above, with arithmetic defined the obvious way; defining equality is slightly fiddly but it's fiddly with a number line too. What does the line visualization gain you?

Because it was asked how we knew we "got them all", referring to points on the line. The reals are a way of modelling the line, the line is a way of visualising the reals. Each is complementary to the other.

And besides, the rationals are totally ordered, and their completion is totally ordered, so it makes sense to think of them as arranged in a line. The problem is that the reals are very, very strange in some ways, and people do get seduced into thinking they understand them, whereas usually it's just a case that they've got used to them.

Re: What are the 'real numbers', really?

#90

Post by claudius »

ddebernardy wrote: post #87
mherrmann wrote: post #48

This is a great article but unfortunately has one thing horribly wrong: Democracy far preceded the Age of Enlightenment. A form of democracy was already in place in ancient Greece at around 500 BC. Newton and the Age of Enlightenment were much later, at 1600+ AD. See Wikipedia: http://en.wikipedia.org/wiki/Democracy#History , http://en.wikipedia.org/wiki/Age_of_enlightenment , http://en.wikipedia.org/wiki/Isaac_Newto…

Hehe. I balked at that too... It also states that you cannot order the field of complex numbers. Whereas I seem to recollect that there are ways to do so. For instance, z1 < z2 if x1 < x2 or x1 = x2 and y1 < y2.

By your definition of 0 < p * q, which is not fulfilled by your < for p = i = q.
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