Prove 2n is always even etc

paulb203 · 2024-08-15 22:33:48

Having tried a few proof questions, I'm left wondering about the some of the assumptions made (axioms?) in the answers.

E.g,

Prove algebraically that the sum of any two consecutive integers is always an odd number

n+n+1
=2n+1

2n = even
+1 = odd

Therefore proven

*

Is it just accepted in maths that 2n=even, and that +1 makes it odd? Have those been proven? Are they classed as axioms? Self-evident?

Bob · 2024-08-16 00:37:04

In "Elementary Real and Complex Analysis" by Shilov, numbers of the form 2n are defined as even, and 2n+1 as odd. These definitions arise early in the axiom list, so you could certainly say they are axioms.

Your proof is therefore enough. I would add a little:

Let n be any integer. Then n + 1 is the next integer, so n and n+1 are consecutive.

Adding them gives n + n + 1 = 2n + 1 which is odd from the definition of an odd number.

Bob

paulb203 · 2024-08-17 02:00:09

Thanks, Bob.

“In mathematics or logic, an axiom is an unprovable rule or first principle accepted as true because it is self-evident or particularly useful. “Nothing can both be and not be at the same time and in the same respect” is an example of an axiom.”
Merriam-Webster
*
Unprovable. That’s interesting. I’m guessing that for any given axiom, if it was provable someone would have proved it (or would be attempting to prove it).
So we can’t prove 2n = Even. It is either ‘self-evident’, or, ‘particularly useful’? Or both?
It does seem self-evident. And I’m guessing it’s useful, for those who apply that kind of thing.
How do they know something is unprovable?
And what do you think of that Merriam-Webster definition?

Phrzby Phil · 2024-08-17 07:27:05

When mathematicians develop an axiom system for a subject, e.g., the axioms for plane geometry, or the axioms for set theory, two main criteria are:
1. the axioms must not contradict each other
2. the axioms must encompass what is considered to be the body of knowledge of the subject

Especially number 2 - this sort of seems like hand waving. But a most interesting example is the Zermelo-Fraenkel plus Axiom of Choice axioms for set theory (ZFC), developed and agreed to over years as encompassing what is considered to be a full description of set theory.

In 1900 Hilbert's first problem posed for the upcoming century is known as the continuum problem. In short, the problem is: is the size of the set of real numbers the very next transfinite size (i.e., aleph-1) after the size of the natural numbers (aleph-0).

BUT - In 1963 Paul Cohen proved - wait for it - you can have it either way. Either statement (Reals is aleph-1 or Reals is greater than aleph-1) is consistent with ZFC. So either statement is independent of ZFC.

You can find all of this in Wikipedia or any book on set theory.

Last edited by Phrzby Phil (2024-08-17 07:46:31)

Bob · 2024-08-17 07:33:21

Even and Odd are just words for something. So, having thought more about this, they are defined as having the properties ( 2n, and 2n+1) rather than axioms.

A major part of mathematics is concerned with building models to 'describe' something useful. Euclidean geometry, for example, enables us to work with triangles and circles and angles (and lots more) and leads to useful stuff like trigonometry. It's an idealised view because you cannot have a line with zero thickness, nor can you measure an angle with absolute accuracy. Nevertheless, it's very useful.

Number theory is built upon axioms such as x + y = y + x where x and y are real numbers. We can use the axioms to build theorems and then the theorems plus axioms to build new theorems. Eventually you end up with calculus and complex numbers.

The M-W definition looks ok to me.

If you can prove an axiom (presumably from some of the other axioms) then it isn't needed as an axiom, so you could develop your model with one less axiom.

For example. One axiom is "There exists a number (let's call it p for the moment) such that n + p = n. This number is called the additive identity. There is no axiom stating that p is unique because we can prove it.

Proof: suppose there are two identities p1 and p2. Consider the sum p1 + p2. It equals p1 because p2 is an identity, but it also equals p2 because p1 is an identity, thus p1 + p2 = p1 = p2 so the two identities are the same number.

Bob

Phrzby Phil · 2024-08-17 07:53:29

Bob adds another condition for axioms I should have mentioned - that of independence from the others.

It is considered elegant for the set of axioms for a theory to be as small as possible.

For centuries mathematicians tried to prove Euclid's parallel axiom from the others. It somehow seemed less elementary than the others. Some thought they had, but they had erred.

Starting in the 19th century, the creation of non-Euclidean geometries (i.e., models of geometry with other forms of the parallel axiom, yet with most or all of the others) proved (from outside of the axiom system) that the parallel axiom was in fact independent of the others.

paulb203 · 2024-08-19 23:24:03

Thanks, Bob, thanks, Phil

Agnishom · 2024-08-20 17:47:11

Bob wrote:

In "Elementary Real and Complex Analysis" by Shilov, numbers of the form 2n are defined as even, and 2n+1 as odd. These definitions arise early in the axiom list, so you could certainly say they are axioms.
Your proof is therefore enough. I would add a little:
Let n be any integer. Then n + 1 is the next integer, so n and n+1 are consecutive.
Adding them gives n + n + 1 = 2n + 1 which is odd from the definition of an odd number.
Bob

I agree with Bob here: the words are necessary; without the words, you have a soup of algebraic expressions

Math Is Fun Forum

#1 2024-08-15 22:33:48

Prove 2n is always even etc

#2 2024-08-16 00:37:04

Re: Prove 2n is always even etc

#3 2024-08-17 02:00:09

Re: Prove 2n is always even etc

#4 2024-08-17 07:27:05

Re: Prove 2n is always even etc

#5 2024-08-17 07:33:21

Re: Prove 2n is always even etc

#6 2024-08-17 07:53:29

Re: Prove 2n is always even etc

#7 2024-08-19 23:24:03

Re: Prove 2n is always even etc

#8 2024-08-20 17:47:11

Re: Prove 2n is always even etc

Board footer