I'll be in quarantine for a couple of weeks soon. I shall try and get the field of real numbers with its order defined in that time. — fdrake
Oh are you or a loved one I'll? Sorry to hear. Be safe! — Kenosha Kid
Nah. Couldn't see my partner for a while, travelled to see her, quarantine restrictions got reimposed while I was here! — fdrake
This is a very ambitious thread. But probably no more than the first 900 pages of Penrose's The Road to Reality. I may not live long enough to see its completion, but you guys are younger, so there is hope. :worry: — jgill
What properties then characterise the ordering of naturals and fractions, and distinguish them from the ordering of province and country or the ordering of classifications of biological kinds? — fdrake
Dedekind, though the set of rationals be infinite in number and uncountable in density, they be still discontinuous in separation. Go to the cutting the place and bring me forth a continuous set of numbers that I might truly count the horrors in store for Man. — The Lord
This sort of thing? — Dedekind
What number didst thou cut at, and upon which side did that number fall? — The Lord
Be warned, for shit just got real. — The Lord
Christ, I think it'll take me longer to debug the mathjax than it did to write the comment. — Kenosha Kid
I thought he was defining the integers x and y as (equivalence classes of) ordered pairs of naturals (with the equivalence class part implied by saying that when the ordered pairs of naturals return the same value under subtraction then the “two” integers thus defined are the same integer). — Pfhorrest
A matter of notation and mathematical clarity. See definition 7.1
https://www.math.uchicago.edu/~may/VIGRE/VIGRE2011/REUPapers/Lian.pdf
Not a big deal. You are to be applauded for wading into this. — jgill
Let i be such that , then , , and finally .
And let show that the real 0 and this spirit i lie on the same scale, and that any multiplier x of i yield and move us along that scale, just as move us along the ordered real line.
And let it be seen that, whereas for any sum of two reals , there exists a number c such that , there is no pair of reals x & c such that , since cannot be real if x is real and vice versa, and thus the addition of i to x doth not move us along the real line, and the addition of x to i doth not move us alone the line to which i belongs.
And let us name this line the imaginary number line, and let i be known as the imaginary number, for only I the Lord can imagine it.
And let the sum of real and imaginary numbers be called complex, and be written , where a and b are real numbers that scale 1 and i respectively.
And thus for each pair of real numbers a and b, there exists a complex number , and this defines the set of complex numbers .
And since the real and imaginary parts are so orthogonal, let not this set be ordered, but only subsets of fixed a or b be ordered. — The Lord
And thus on that day the Lord created aman
a
m
a
n
. And he said, Let n=d, and thus He named him adam
a
d
a
m
. — Kenosha Kid
And thus on that day the Lord created aman — Kenosha Kid
Take the set of natural numbers , and count the ways that ye can fiddle with them, and note the result of your fiddling such that you can tellest me what thou did and what result it did have. — The Lord
Lord, I have fiddled all of the members of the set of natural numbers, and I have seen the following:
First, for some fiddles I performed, I ended up with an object unlike the set of natural numbers, such as to select two and make an ordered pair.
Second, for some fiddles I performed, I ended up with a subset of the original set, such as to add numbers > 0 or to multiply by numbers different to 1.
Third, for some fiddles I performed, I ended up with the same set, though the elements had been adulterated or moved.
Fourth, for some fiddles I performed, not even the elements were affected. — Morph
I have listened to your Morphisms and I am pleased. For you have discovered some Morphisms that leave each element in the set unaltered and thus the set as a whole unaltered, and we shall call this the identity morphism.
And you have discovered that some Morphisms alter the elements but leave the set as a whole unaltered, and we shall call these automorphisms.
And you have discovered that some Morphisms, such as the additions, alter the set but these alterings may be reversed by inverses called subtractions, and we shall call these isomorphisms.
And you have discovered that some Morphisms such as the multiplications act on sums of elements by acting on each element and summing, and we shall call these homomorphisms. And you have discovered further that some homomorphisms are not isomorphisms, such as multiplication by 0 which, in yielding unity, yields both an element and a subset of the set, but from which we cannot return to the set by any morphism bar the union with that set.
And finally you have discovered that some Morphisms do not yield objects of the same type you began with, such as the ordered pair you obtained from the set. — The Lord
Let the set of objects that you fiddle with be the domain of the morphism, and let the set of objects you end up with belong to the codomain, such that, in the additions and the subtractions, the codomain and the domain be as one, but the object yielded not be all that lies in the codomain but only that yielded by that morphism acting on that domain, and we shall call this subset of the codomain the image of the morphism. — The Lord
What? — Morph
My Lord, what do you call this small parcel of cephalic improvement?
As I understand it, we’re really saying “all objects with this structure have these properties”, but that’s technically true whether or not there “really” are any objects with that structure at all. — Pfhorrest
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