what's this i number, anyhow?

Isn't it also the case that Peano arithmetic can be used to define mathematical objects which satisfy the Peano axioms, but do not satisfy the properties of real numbers? If this is true, then how can the reals be uniquely defined without the above list?

(As noted elsewhere, these are properties and not axioms.)

Yes, on the subset of irrationals for which additional and multiplication are defined, they obey the rules above.

No, the fact that they sort-of obey these rules has nothing directly to do with them being a subset of the reals - they sort-of obey the rules for a completely ordered field because they are sort-of a completely ordered field. It is just that the reals are a nice example of a completely ordered field (and I think the only Archimedian completely ordered field).

I guess it all depends on how fundamental you want to go. If you want to work with the real numbers based on their properties without any thought to their construction, you can list those properties and call them "axioms" - statements that you /define/ to be true rather than proving them to be true. But since real numbers can be constructed from the Peano axioms, those axioms are more fundamental - and to my knowledge, the Peano axioms cannot themselves be derived from fewer or more fundamental axioms while still being powerful enough to make arithmetic work.

Yes, Peano arithmetic is used to define the natural numbers (0, 1, 2, ...). Other number systems, including the reals, are derived from them.

The reals can be defined in many ways - but any set that obeys the properties will be isomorphic to any others, and it doesn't matter which you pick.

satisfy

under multiplication because 2 is not an irrational.

Disproof by example: The square root of 2 (or 3) multiplied by itself is not irrational. The set of irrationals is NOT closed under multiplication.

QED

?-)

The confusion is over what we define to be the "axioms of arithmetic" as outlined elsewhere.

Wrong. The rational numbers are part of the set of irrational numbers.

The sets aren't necessarily exclusive.

Wrong. The rational numbers are part of the set of irrational numbers.

The sets aren't necessarily exclusive.

I had 5 apples, until you came along and stole them. Now I have a loss of 5 apples, and I'm so pissed I'm going to see if I can create -4 tires on your car! What's so abstract about a loss? :)

Now take one more tire from his car. That would be imaginary and abstract (until he buys more tires for his car and you take one).

OK. What is your experience of SQRT -1 ? Is this something I can put in my car trunk?

If you want to twist your car so the trunk goes sideways, yes. At least, that's what I feel about it.

In word problems (the few irritating ones with quadratics :) ), imaginary numbers show up as "no solution". Or more specifically, no solution in the reals. If you go and solve it anyway, the real part of the complex (conjugate pair) answer (or equivalently, the average of them) is usually the "closest to solving" it. So for example, the closest point of a line to a conic section (e.g., parabola, circle, etc.), given a line that doesn't intersect the given curve.

In the "spacially twisted" imaginary sense, you could kinda-sorta-imply that the plane is twisted by the imaginary component, giving two answers, one twisted to the left, one to the right; and by averaging them, you flatten out the plane again, but of course your number isn't the solution, because there can be no solution in the reals (the flat plane).

This might actually be a quantifiable method. One could contrive a means to compress or distort a plane curve as though it were a projection on a tilted or curled sheet of paper, and thus find the intersection with another curve; then through the process of flattening, the imaginary component is added back in, giving the correct result.

Hmm. But then, that might be easy enough for quadratics, but I don't think it would be independent of order; you may need a different method for cubics, and so on. And transcendental functions (i.e., of the exp(z) family), something altogether different, like the infinite spiral wormhole of the Ln(z) function.

Tim

Long time ago - yes, called by historians the "enlightenment." You could also call it the "endarkment." But you just had Newton, of course. And I was just consulting Will and Ariel Durant's "The Age of Voltaire." Euler has 2 pages in there - the book is not an intellectual-only history. He was a pupil of Johann Bernoulli. He existed as an academic leader in the various courts (benign dictators) of the day, primarily Russia and Germany.

Durant mentions he originated the concept of the function, as well as algebraic analysis.

His main student was LaGrange, who, like Euler, knew how to keep his mouth shut about politics, and was fortunate to survive, since he was close friends with Marie "Let them eat cake" Antoinette. He eventually became friends with Napoleon and designed the MKS system, one of the few positive results of a "planned" society.

LaGrange's work was called "Mecanique analytique." Thus began the "Valse Mecanique" of our modern systems of thought.

LaGrange was beset by depression, but at age 56, he married a 17 yo, and that evidently distracted him.

Durant goes on to describe the advances of science such as electricity by Franklin. He then does a re-set back to 1730 with a chapter entitled "The Atheists." In a later volume, "Rousseau and Revolution," Durant amplifies on this change from feudal to nation-state.

What's missing to my mind from this effort is any history of the American Revolution. As well, there needs to be more examination of "inside revolutions" carried out by those who seek power first and wisdom second.

And today, you have two main camps in academia - the scientists and the Society- Shapers. The scientists since Newton have had all the fun - doing things which impacted the world and society. Galileo, Newton, and Darwin were what every academic dreams of - to be revolutionaries without any risk. It is attractive because you get attention and feel important. By contrast, the academic humanists have been the Capons in the hen house - writing poetry, posing, introspecting, creating "art," etc.

That's no fun, and a more exciting alternative has been the rise of the "society shaper." They are primarily leftists, and their goal is to erase your individuality by indoctrinating your children and making them conform. Control and power is more important than content. Today you cannot hope to succeed in academia unless you are leftist and share this vision. This is as real for your life as the road you drive on.

And I am tempted to see this social Marxism as the end result of the "enlightenment." I recently started a thread here called "woo woo science," to see what people thought about psychic influences. Many of the responders see themselves as machines. Their lives a "valse mecanique."

I plan to start another thread, "Is there a Science of Gawd?" To explore this "valse mecanique" further. Some of the connections to electronic design will become apparent. Besides those specific connections, in general, the "society shapers" wil probably not tolerate the rise of technology, but who knows? Both Steve Jobs and Bill Gates are "society shapers," so there may be full employment in any case.

You've got that one wrong. The rationals are defined as the set of p/q where p and q are integers, and q is non-zero (noting of course that any given rational can have infinitely many p/q representations). The irrationals are the set of real numbers that are not rational. By definition, they are exclusive.

See above.

the steep monotonic exponential function and oscillatory trigonometric f unctions is really very surprising. (Euler lived a long time ago.)

also call it the "endarkment."

You might - you are not only dim enough not to realise that it would have t o be "endarkenment" but clearly also much too dim to be unaware of what was going on at the time

iel Durant's "The Age of Voltaire." Euler has 2 pages in there - the book i s not an intellectual-only history. He was a pupil of Johann Bernoulli. He existed as an academic leader in the various courts (benign dictators) of the day, primarily Russia and Germany.

Try Jonathon Israel - his "Radical Enlightenment" lays the central emphasis on Spinoza, and treats Voltaire as a Johnny-come-lately populariser.

"Radical Enlightenment: Philosophy and the Making of Modernity, 1650-1750." 2001. ISBN 0-19-820608-9 HB; 0-199-25456-7 PB.

ociety- Shapers. The scientists since Newton have had all the fun - doing t hings which impacted the world and society. Galileo, Newton, and Darwin wer e what every academic dreams of - to be revolutionaries without any risk. I t is attractive because you get attention and feel important. By contrast, the academic humanists have been the Capons in the hen house - writing poet ry, posing, introspecting, creating "art," etc.

The academic humanists include lawyers, and other people who influence legi slators. Thinking that they have been "capons in the hen house" is simply i ll-informed.

"society shaper." They are primarily leftists,

Not true. The prototype was Herbert Spencer

and while he started off as a radical democrat, he became much more right-w ing as he got older, and his "Social Darwinism" is not in the least leftist .

This displays a distressing ignorance about the role that Spinoza and the e ncyclopedists played in getting people to thing about what society was abou t. Jonathon Israel makes the point the Spinoza had a lot of influence on th e French encyclopedists (who preceded Voltaire and did most of the heavy li fting).

One of their interests was in stopping religion from indoctrinating your ch ildren and persuading them to conform to the sillier delusions of organised religion. Their approach was to teach people to do critical thinking for t hemselves, from an early age. Haitic doesn't seem to have mastered that par ticular trick.

Haitic really doesn't have a clue what he's talking about.

You seem to have a problem distinguishing between that which exists independent of man and the artificial constructs used to describe it.

If you mean "real numbers", say "real numbers".

the set of real numbers inclusdes the rational numbers which helps id you want to, say, multiply sqrt(2) and sqrt(8)

Replace the i with a j and I'm sure you'll be happy, and there's always Wikipedia.