@robjohn I just answered an old question to rescue it from unanswered hell - should I have made it CW? (If so, can you do it?)

@OldJohn I don't think it needs to be CW.

@MikeSpivey OK - thanks

Can you prove the consistence of the peano axioms, just by using the peano axioms? Ie prove no contradiction can be derived based apon those axioms

@Ethan No - I think Gödel blew that one away

whose godel?

then wtf? how do we know anything we do is consistant?

But Henning is around, and if I say much more, he will correct me :)

???

Old jhon

then is this question pointless?

I mean if you cant prove the consistence of the peano axioms? then my question cant be solved either right?

@Ethan You really should read up on Gödel. He was one of the greatest mathematicians of the 20th century.

What formal system do most mathematicans use, as axioms for the real numbers/natural numbers etc

@Ethan this section explains the problem with consistency

@Ethan ZFC.

I don't understand zfs it sounds like a bunch of set theory stuff which I dont get

Im talking about axioms like

a+0=a

the additive identity

etc

@ethan - you need to read that section about consistency on wikipedia

which system

do most modern mathematicans use

for axioms on the reals/naturals

@Ethan the axioms you state are basically then theorems in ZFC.

I dont see them their

They don't use any axioms, they just know how to use them.

What?

There will always be a mathematician -Asaf for instance- that knows why.

@JonasTeuwen that is ethan's point I suppose

@Ethan you don't find those theorems there because they are non-trivial.

You will find them in a book though.

what?

If there not trivial, wouldn't it be important to show them?

@JayeshBadwaik So there is no point. Alrighty, off again.

@JonasTeuwen :P :P

@Ethan yes, but the space in the "margin" is too small.

I don't understand, I am looking for the most common system that modern mathematicians use as axioms, for the real numbers, natural numbers, and complex numbers

@Ethan okay, here is the deal

Axioms definging the operations on them, one with each other

read up construction of real numbers

@Ethan Best place to look is in books like Rudin's analysis books

I have the book

which book?

If one asks the question, it suggest to me you would need to study more elementary mathematics. Usually you should be able to figure out by yourself.

principles of mathematical analysis, rudin, second edition

Read the first chapter (the appendix of the first chapter consists of construction of real numbers)

OK - when you have read the relevant sections in Rudin, then you might want to ask here or on MSE about any bits that are not clear

I don't understand, it seems like I should have been taught these axioms, and shown weather or not they were consistent in say kinder garden? I mean how can people operate with quantiys and not know how or why the fundamental rules govern them

Those are not the axioms, but the constructions out of sets.

Yes i agree with Jonas

You cannot show the internal consistency of a consistent system.

He constructs the reals from rationals in the first chapter

but doesn't give axioms for rationals

See it as a dictionary, you have not enough available to define every word. Leading to the fact that you have to assume at least the meaning of some as 'granted'.

If a system is consistant, you can not show consistency? that sounds like a very deep statement, can you even prove that?

@Ethan Godel proved that some years ago.

link?

@Ethan wikipedia.

then if we want to abstract our system by say adding complex numbers? or some other opperators? how will we ever know were not contradicting some other result?

Actually, what I think he proved ws that you cannot prove the consistency of a system like that without using a system which is more powerful (in some sense) and hence also open to questions of consistency

@Ethan Here is a set theory based construction of natural numbers.

And here is the construction of rationals.

and here is the construction of integers.

@Ethan It will be fun if anyone tells you about the Banach-Tarski theorem :)

@OldJohn :P :P

Darn - I just did :(

@OldJohn no confetti for you!

@JayeshBadwaik :(

@OldJohn But that is about ... ultrafilters.

and the axiom of choice ...

@Ethan The Incompleteness Theorem

@OldJohn That way everything is!

You only need dependent choice I think, or actually the ultrafilter lemma.

Or Hahn-Banach!

Are we talking about Hahn-Banach or Banach-Tarski? I'm getting confused ... as usual

You need to have a non-measurable set for Banach-Tarski to work.

@JonasTeuwen oh yes

And that again is satisfied when Hahn-Banach holds.

OK

The Vitali construction only requires you to pick a countable set simultaneously.

i'm a big fan of constructive math, me

Has anybody studied functional equations? I can only find competeion based math books that talk about them, I would like to find a real book on them.

give an example pls

ok

find all functions that are only defined over the reals that satisfy

@Ethan I think there are very few books on them, because they are not really a branch of maths with many applications in other areas

$f(x+y)^2=f(x)^2+f(y)^2$

mhm

ok the key here is watch

Areas such as complex analysis have many more applications - and have many many good books

let x=0,y=0

f(0)^2=f(0)^2+f(0)^2

thus f(0)=0

then subsitute x=x, y=-x

f(0)^2=f(x)^2+f(-x)^2

0=f(x)^2+f(-x)^2

but sence they both have postive ranges

the sum of two postive numbers

can never be zero

thus the only solution

is f(x)=0

But what if we define the function only for positive integers?

${}{}$

the first 1, doesn't make a difference

i dont know about the second one lol

Why? $f(-x)$ is not defined. $x > 0$

so you cannot use your argument then.

Here I will give you an example of an aplication

You know ramanujans famous equality 2=sqrt(1+2sqrt(1+3sqrt(1+4sqrt(1+5sqrt(....

it might be a 2 or 3 or whatever

it simplifies to solving the functional equation

$f(x)^2=1+xf(x+1)$

Yes, duh.

which has a solution f(x)=(x+1)^2

How do you know if that will yield your solution?

How will it yield the solution?

in that context 'functional equations' just describe some set of functions and are therefore straight up a part of analysis

what do you mean by analysis

like less grubby calculus

it's what calculus is called in europe

like the Rudin book...

@JonasTeuwen Done.

@Ethan, also differential equations are a superset of 'algebraic functional equations' - since they also involve for example derivative operations - and that is a really popular subject

Damn - did Matt come and go while my back was turned?

maybe he is busy

Bye folks - back tomorrow

We're all gonna die.

Yep. Good eh.

@JonasTeuwen I mean. Eventually.

@JonasTeuwen How are you J?

I have a discus herniation. It kind of sucks.

I hate to the bone that people bring up limits when someone asks about infinity. I probably did it one or two times.

Yea life sucks. Must suck to be you.

@JonasTeuwen Darn. How did you get it?

I didn't have to do anything.

@JonasTeuwen I mean, how did the herniation happen?

There is no reason for it, just weak spot.

Please get your facts right

@JonasTeuwen Oh. And will you get operated on?

@BenjaLim What are you talking about?