/sci/ - Science & Math

[Anonymous No.7507181]

>using equivalence classes of Cauchy sequences instead of Dedekind cuts

[Anonymous No.7507233]

>giving a shit about the construction of the reals

[Anonymous No.7507243]

>the reals aren't actually real
Why do mathematicians do this shit

[Anonymous No.7507268]

>>7507181
I actually never saw the construction via Dedekind cuts but i´ve always thought that by constructing the reals as equivalence classes of Cauchy sequences you see very clearly that the critical property of the reals - and therefore the reason to expand the rationals - is their completeness. And also you get used to work with equivalence relations and classes, which is very usefull (think the construction of L_p spaces for example).

[Anonymous No.7507270]

>Hating to have trivial properties on + and ×

[Anonymous No.7507272]

>>7507268
Meant to ask: Why do you think Dedekind cuts are better?

[Anonymous No.7507278]

>>7507268
>equivalence relations and classes,
this completely sucks, yes.

dedekin cuts are directly usable in constructive mathematics, not the cauchy sequences.

[Anonymous No.7507282]

>>7507278
>constructive mathematics
Ok my bad, i should have known this would be just another Wilderburger shitpost.

[Anonymous No.7507286]

>>7507272
elegance I suppose. If you can do without quotienting, all the better. Plus I like how the Dedekind construction can be generalized to an arbitrary order

[Anonymous No.7507561]

Basically do you want your reals to be infinite sets of rationals or infinite sequences of rationals. Sets are clearly superior only analysisfags and physicists would disagree.

[Anonymous No.7507564]

>>7507268
>I actually never saw the construction via Dedekind cuts

It's like 2 pages of text. Google it you lazy fuck.

[Anonymous No.7507583]

>>7507268
It's brilliant, so long as you never actually have to do anything like add real numbers. Otherwise you'd best use p-adics or continued fractions, both of which actually have arithmetic that can be done by people, rather than the handwavy shit that pleases analysis faggots.

[Anonymous No.7507606]

>>7507268

This is the only correct answer

>>7507583

>implying that the construction of the reals has anything to do with calculation in the reals

try hard wilderburger

[Anonymous No.7507611]

>>7507606
>"construct" infinite sets
lel

[Anonymous No.7507648]

>>7507611
>but muh rational trigonometry

[Anonymous No.7507694]

>>7507648
>muh mystery school religion

[Anonymous No.7507817]

>>7507286
quotienting is nice algebraically tho. if you want to construct the hyperreals using an ultrapower, the filter you mod out by effectively acts like a cauchy

[Anonymous No.7507858]

I don't know I'm a math noob, but constructing the real numbers using equivalence classes of Cauchy sequences always seemed more intuitive to me than constructing them using Dedekind cuts. Actually though, today in my analysis class my professor gave a brief overview of the construction the real numbers using Dedekind cuts which was pretty elegant. I feel like I might change my mind on which approach is better once I learn more real analysis.

[Anonymous No.7507950]

>>7507858
The two construction depend on distinctly different properties of the reals. Cauchy sequences use distances between points and Dedekind cuts use the set's natural total order. Neither technique is more generally applicable than the other. The main difference is aesthetics and I prefer cuts because connected sets>sequences.

[Anonymous No.7507951]

>>7507858
The two construction depend on distinctly different properties of the reals. Cauchy sequences use distances between points and Dedekind cuts use the set's natural total order. Neither technique is more generally applicable than the other. The main difference is aesthetics and I prefer cuts because intervals>sequences.

[Anonymous No.7507954]

Hey mathfags. Are there any fundamentally algebraic constructions of the reals or is it strictly an analysis thing? It's been haunting me for a while.

[Anonymous No.7507955]

>>7507951
>Neither technique is more generally applicable than the other

Say what?

[Anonymous No.7507962]

>>7507955
Got a set without a natural total order? Then you won't want to use a dedikind cut construction.

Got a set without a natural metric? Then you won't to use a cauchy sequence construction.

On the other hand you can get instances of sets with natural total order and no natural metric and vice versa.

[Anonymous No.7507983]

>>7507954
It's purely analytical because the property of being complete involves properties of all sets of the real numbers (second-order properties), so there is no way of formalizing it nicely only with equations and algebraic tools in general.

At least that's what most people mean by algebraic methods.

[Anonymous No.7508025]

>>7507694
>completion of a uniform space
>mysterious in any way

Shove your finitism up you ass. :^)

>>7507954

Depends on what you mean by fundamentally algebraic. But I'll give it a go.

Suppose you consider the set X of all cauchy sequences in the rationals. They form a commutative unital ring under the usual operations on sequences, with 0 = (0,0,0, ... ) and 1 = (1,1,1,..). Consider the ideal U of sequences that converge to zero. The reals are the quotient ring X/U. You need to add order to the mix, but It's not that big of an issue.

Sure, this is just 'analysis' rephrased in the language of elementary algebra. But I tried.

>>7507983

Would you consider the property of being a PID an algebraic property?

>>7507962

My bad. I misinterpreted what you were saying.

[Anonymous No.7508035]

>>7507951
I probably prefer Cauchy sequences over Dedekind cuts (i.e. sequences over intervals) because I like algebra way more than analysis and I really dislike dealing with inequalities.

[Anonymous No.7508054]

>>7507951
I prefer sequences because metrics & equivalence relations > order relations.

>mfw had to use an order relation to state my disdain for order relations

[Anonymous No.7508070]

>>7508054

[Anonymous No.7508098]

[Anonymous No.7508173]

>>7508025
>The term "Mystery" derives from Latin mysterium, from Greek mysterion (usually as the plural mysteria μυστήρια), in this context meaning "secret rite or doctrine".

[Anonymous No.7508178]

>>750817
>completion of a uniform space
>sacramental rite
>unfounded doctrine