Originally posted by CZekeAh! That makes sense, thank you.
castlerook: Yeah, that's the concept of ordered fields (fields don't in general have order), which is quite interesting. I recently attended a talk about totally ordered fields and there are some surprising things we can say about them. There are basically only a few fields they can be: for instance, if a totally ordered field is complete (all Cauchy sequence ...[text shortened]... 0
So if 1 is 0, then so is everything else in the ring, and it can only be the zero ring.[/b]
Originally posted by Ramnedso do i. i was okay until i read this thread. and today has been quite a hardcore day so far (and its not even 11am!)-i've classified all groups up to order 15, started a program to calculate the ramsey number of (n,m), R(n,m), and i'm about to have a tutorial with a scary lecturer on, i believe, the Riemann integral.
I feel soooo -> 😛
i don't think i've ever looked at the natural numbers-they were always taken as said. it's the real numbers that are hard to define. dedekind cuts are baaaad!
These are questions to TheMaster37 or whoever can answer the questions I have.
I wasn't aware of the axioms of Zermelo and Frankel. But i know the axioms of Peano.
Didn't Peano cover it all? What does not Peano that Zermelo have and Frankel do have?
I short - what is the pros and cons to the axioms compared to eachother?
Originally posted by FabianFnasAren't they just describing different things? Peano's axioms describe the natural numbers. The Zermelo-Frankel axioms describe set theory.
These are questions to TheMaster37 or whoever can answer the questions I have.
I wasn't aware of the axioms of Zermelo and Frankel. But i know the axioms of Peano.
Didn't Peano cover it all? What does not Peano that Zermelo have and Frankel do have?
I short - what is the pros and cons to the axioms compared to eachother?
See, e.g., http://mathworld.wolfram.com/Zermelo-FraenkelAxioms.html
Originally posted by FabianFnasPeano describes only the natural numbers (without 0). Zermelo and Fraekel made axioms for set theory, wich includes an approach to numbers (ordinals). Z & F do have a 0 🙂
These are questions to TheMaster37 or whoever can answer the questions I have.
I wasn't aware of the axioms of Zermelo and Frankel. But i know the axioms of Peano.
Didn't Peano cover it all? What does not Peano that Zermelo have and Frankel do have?
I short - what is the pros and cons to the axioms compared to eachother?
Z&F go on to describe structures of sets and prove all kinds of relations between sets and ordinals. Quite interesting if you're into math 🙂
Originally posted by Ramnedi'm a sophomore too!
wow. I bet that I can name 10 - 15 things that you guys just wrote that I do not understand, as a sophomore in H.S.
You guys, if indeed you are correct, must have taken a hardcore college class.
we can be clueless together...
i always thought one factorial was one
so they lost me with that one....
as a H.S. sophomore, here's the best I can do...
using the Reflexive
Property of equality, we get:
x=x
let x=1
therefore, by the transitive property,
1=1
let us now force a zero on the right side of this equation
1=0
we get this zero by subtracting one whole number from the right hand side of the equation
simple algebra laws state that, with an equation, anything we do to one side, must be duplicaed on the opposite side in order to maintain equality...
this gives us that we must subtract one from he left side to get the balanced equation: 1-1 =0 or zero equals zero
if we do not subtract one from the left side, we get 1=0 or x=x-1
x=x-1 is not a balanced equation, therefore 1=0 is impossible