141–150 of 210 posts
Earlier quoted context omitted.
The Gaussian integers are well ordered as well. That '<' respects arithmetic operations isn't an aside; it's the core feature.
I did mention that part for a reason. And of course if one accepts the axiom of choice then every set can be well-ordered, but that would not force every ring to be a UFD.
Earlier quoted context omitted.
I think I wasn't clear enough. I never meant that axiomatic proofs in general are meaningless, only that axiomatic proofs of trivial arithmetic facts (1+1=2, 2+2=4...) are meaningless.
You have two choices here: 1. You assume "trivial arithmetic facts" as axioms. Result: You have an infinite number of axioms. (Whee!) The likelihood that you have snuck in non-trivial assumptions is pretty high, unless you are very strict about how you define "trivial" (which is probably as much work as just proving the trivial facts), and in that case, there's a high probability that some of your trivial facts are f…
The proof 2+2=4 is trivial given the peano axioms.
I suspect you know this was the point being made.
Why isn't 2+2==4 obvious? http://us.metamath.org/mpegif/mmset.html#trivia
Why is that chain of axiomatic proofs obvious? How is it obvious that one step follows the previous?
Definiton of Successor (S) and addition. It's trivial.
My favorite proofs of these basic number theory facts are in Apostols, Intro to Analytic Number Theory. At best the FTA using the proof there is only slightly less than obvious (of course obviousness probably varies widely based on mathematical background / maturity).
I know I'm going up against a brilliant mathematician and Fields medalist here, but I find this article to be unenlightening. It seems that Gowers has glossed over something about the integers that's built incredibly deeply into our intuition about them when he talks about Z[sqrt(-5)]: > These numbers have various properties in common with the integers: you can add them and multiply them, there are identities for bot…
Here's a stumper then: why do Z[sqrt(-1)] and Z[sqrt(-3)] have unique factorization but Z[sqrt(-5)] doesn't?
This might help.
Earlier quoted context omitted.
Actually, you're pretty much spot on. The word you want is "irreducible", rather than "indivisible". In general rings there are irreducible elements and prime elements, and they have different definitions. You're looking for a unique factorisation into irreducibles. https://en.wikipedia.org/wiki/Irreducible_element https://en.wikipedia.org/wiki/Prime_element
Thanks! Trying to read that is like trying to learn an entire new language by reading a sentence in it. Way too many words to already understand what it's even defining (commutative ring, irreducible polynomials, UFDs, principle ideal, nozero prime ideal... and I'm not following why p divides ab in R... or even exactly what that means). Searching youtube kahn academy and numberphile, but not turning up anything. exac…
Michael Artin's Algebra is a good, concrete book with tons of motivation and zero prerequisites. Wanna try it? ;)
https://www.amazon.co.uk/Wittgensteins-Lectures-Foundations-...
It's great fun to read.
All swans are white.
Now, there are various options. Some of them are:
- You can drop your claim that all swans are white.
- You state these aren’t swans because they aren’t white.
This is somewhat similar. Smart men have studied numbers for centuries, and have thought hard about what it means to be a number. They came up with a set of properties and laws that they thought was necessary and sufficient for a set of objects to behave like numbers. The fundamental theorem of arithmetic was not in that list, as it was thought one could derive it from simpler laws, or, possibly, nobody even thought of doing that, as they thought it to be obvious (I do not know enough of the history of mathematics to know which is true)
Suddenly, somebody comes up with a set of objects that abides by those laws, but for that set, the fundamental theorem of arithmetic does _not_ hold.
Now, you can claim that set of objects isn’t a set of numbers, but you can no longer claim that the fundamental theorem of arithmetic is obvious.