I need some feedback on a proof from Peano axioms that $1a=a$

Question

I'm a high school student new to propositional logic. This is only my second attempt at writing a proof. I originally started with trying to prove the associativity of addition, but that was a bit complicated due to my minimal knowledge. I was advised to try and attempt to prove $1a = a$, that is $S(0)\cdot a = a$. Here is my attempt:

Prove from Peano axioms that for a natural number $a$, $1a = a$.

Assume that $a$ is a natural number. Assuming Peano's axioms apply to natural number $a$, we can say that if $a=0$, $a$ can also equal natural numbers $(1, 2, 3,...)$. So if $0$ multiplied by $1$ equals $0$, then $1a=a$ is true for every natural number.

Edit:

Lol, I know how to do proofs now:

Prove that for a natural number $a$, $1a = a$.

I'm not going to use Peano's axioms for this, I've been reading Spivak:

Taking the statement $ax = a$, I can perform the following manipulation:

$a^{-1} \cdot ax = a \cdot a^{-1}$

Which yields: $ x = 1$

(Or I can just divide both sides by $a$)

Proved it

Spivak FTW

Answer 1

You seem to be trying to use induction here. While that's a valid approach, what you have done does not work. Think about what your reasoning would allow you to extrapolate here: you are basically saying any property that holds for zero holds for all numbers. This can't be right.

To use induction to prove something is true of all natural numbers requires two steps. First you show it holds for zero (the base case). Second, you show that if it holds for an arbitrary number $n$ then it also holds for $n+1.$ The second step is missing here.

Here is an informal proof, based on the PA axioms on wikipedia.

The base case follows from the axiom $\forall x(x\cdot 0=0).$

For the induction step you need to prove $\forall x(1\cdot x = x \to1\cdot S(x)=S(x)).$ To show this, we can assume $1\cdot x=x$ and from this we must prove $1\cdot S(x)= S(x).$ We use the axiom $$x\cdot S(y) = x\cdot y+x.$$ Plussing in $1$ for $x$ and $x$ for $y$ gives $$ 1\cdot S(x) = 1\cdot x+1.$$ Now we can use our assumption and substitute $x$ for $1\cdot x.$ We will also remember that $1$ is just an abbreviation for $S(0)$ and sub that into the RHS. This gives $$ 1\cdot S(x) = x+S(0).$$

Now we use the axiom $x+S(y) = S(x+y).$ This axiom with $0$ subbed in for $y$ is $x+S(0) = S(x+0).$ We can use the axiom $x+0= x$ to reduce this to $$x+S(0) = S(x).$$ Finally, plugging this in to what we had before gives us $$1\cdot S(x) = S(x),$$ which completes our proof of the induction case.

(I should add, that in "reasoning by induction", I am actually invoking PA's induction axiom for the formula $1\cdot x = x$.)

Proving things from the axioms is tedious work. Furthermore, I haven't given a formal proof. Since you seem interested in logic, I would look for an introduction to formal logic, which will show you how to lay out fully formal proofs. Also a lot of courses have software for students to use for proof writing. It might be challenging at first, and you needn't let studying that stop you from understanding and working out proofs in the style I did above.

Answer 2

To show associativity, fix $m,k\in\mathbb{N}$.

Use induction on $n$ to show $\mathbb{N}=\{n\in\mathbb{N}:k+(m+n)=(k+m)+n\}$.

I clarify what our addition binary operation does. It is defined as follows.$$m+(0)=m\quad\text{and}\quad m+(n+1)=(m+(n))+1,$$where $(\cdot)+1$ is the successor function.

The proof you seek for $\mathbb{N}=\{n\in\mathbb{N}:1\times n=n\}$ is typically done by induction. First, I clarify what our multiplication binary operation does. It is defined as follows. $$m\times(0)=0\qquad\text{and}\qquad m(n+1)=(m\times(n))+m$$ A standard argument will begin with the Base Case. It is then followed by the Inductive Step.

Base Case. Observe $1\times 0=0$ by definition. The base case has been proven.

Inductive Step. Assume $1\times n= n$, where $n$ is any fixed natural number. Observe $$1\times(n+1)=(1\times(n))+1=(n)+1=n+1.$$ The inductive step has been proven.

Conclusion. By the principle of mathematical induction, for each natural number $n$, we have $1\times n=n$.

Answer 3

You'll have to do a proof by induction if you are using Peano postulates.

We need two things.

1) The recursive definition of multiplication:

$a*0 =0$

$a*(S(b)) = a + a*b$

2) the 9th Peano axiom (induction axiom):

If a statement about a natural number $a$, is true when $a = 0$ and if whenever it is true for $a = n$ for some non-specific $n$, it follows that the statement must be true for $a=S(n)$ (but not because of any specific property of the $n$ in particular; the $n$ is non-specific) then the statement is true for all natural numbers $a$.

.....

So the statement is: $1*a = a$.

(base step) It is true for $a = 0$: as $1*0 = 0$ by the definition of multiplication.

(induction step) If we know that for some $n$ that $1*n = n$. (Not all $n$; just this one particular $n$) the we know that:

$1*S(n) = 1 + 1*n = 1 + n = S(n)$.

Thus $1*S(n)=S(n)$

(conclusion) the statement is true for $a=0$; if it is true of $a=n$ it is true for $a=S(n)$, thus the two conditions for the 9th axiom are satisfied, and therefore we conclude the statement is true for all natural numbers.