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Written By: Quarkoala

May 21, 2020

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If you haven't read "How to Prove that 1+1=2," I suggest checking it out before reading these comment responses.

There were two main confusions in the comments. We'll start with the confusion on the definition of 2. The reasoning was, in the proof, I use the fact that S(S(0))=2, but this fact relies on 1+1=2, so doesn't that make the proof circular?

Here's why not. It's because our precise definition of 2 is 2=S(S(0)). We could define 2 with 2=1+1, but this seems less natural when defining the set of natural numbers, and it wouldn't make a very fun story, since 1+1=2 would barely require a proof. But why does the definition 2=S(S(0)) make more sense?

Out of the recursive definition of natural numbers, using Peano's axioms, we get that we can list the natural numbers in the following way:

0

S(0)

S(S(0))

S(S(S(0)))

S(S(S(S(0))))

S(S(S(S(S(0)))))

...

Naturally, we want to line this up with the names we are used to, which have yet to be defined, so we define them as follows:

0=0

1=S(0)

2=S(S(0))

3=S(S(S(0)))

4=S(S(S(S(0))))

5=S(S(S(S(S(0)))))

...

Notice that the ellipsis means that it would take infinite definitions to define the symbols for all natural numbers, without formally defining a pattern. However, this turns out not to matter that much, since we can talk about individual natural numbers just fine, and the Peano axioms define the whole set of natural numbers.

Another confusion might be thinking that the definition of S is S(n)=n+1. While this statement is definitely a true statement, it can be proven to be true from the actual definition of S, which is in the Peano axioms. S(n)=n+1 is not a definition though, because addition is defined using S, so the reverse cannot be true. It is possible that one could define the natural numbers using addition instead of the function S, but S is simpler because it only has one input. If you used the definition of the natural numbers that relies only on addition, then yes, S(n)=n+1 would probably be your go-to definition.

I've been talking about the Peano axioms a lot, and multiple times, have been saying the equivalent of, "Don't worry if there seem to be all of these problems, the Peano axioms solve them." So, instead of leaving you to trust me, I'll just list them out. First, we have the axioms that define the set of natural numbers using the successor function, S (when n is said to be a natural number, it means that n is in the set of natural numbers):

S1) There exists a natural number, 0.

S2) For any natural numbers m and n, m=n iff S(m)=S(n). (iff means if and only if)

S3) There exists no natural number n where S(n)=0.

S4) For any set H where 0 is in H and (for any n) S(n) is an element of H if n is in H, all of the natural numbers are in H.

Notice that from these axioms, we already have all the natural numbers, and can thus give them names such as 1 and 2. From a certain point of view, we have defined the natural numbers by capturing the idea of counting using the successor function. There are also axioms that define addition on the set of natural numbers:

A1) For any natural number n, n+0=n.

A2) For any natural numbers m and n, m+S(n)=S(m+n).

Notice that the two axioms above are not needed to define the natural numbers. They are only needed to define addition, and can be used to show things such as S(0)+S(0)=S(S(0)), aka 1+1=2.

If this jumble of definitions is confusing, I'll give you another example of math where which statements are regarded as definitions and which statements are regarded as theormems is arbitrary. Recall that π is defined to be the circumference of any circle divided by its diameter. Using mathematical analysis (pure math part of calculus but more rigorous), it can be proven that π is the area of any circle divided by its radius squared. However, this could also be taken to be the definition of π, and then it would need to be and could be proven that π is the circumference of any circle divided by its diameter. It's arbitrary which one of these statements is the definition and which one is a theorem, just like it's arbitrary whether 2 should be defined as 2=1+1 or 2=S(S(0)).

I think that cleared up the first confusion. If I made any typos or errors, or you're still confuse, I encourage you to say so in the comments.

The second confusion is a lot easier to explain. It was about the first order logic proof, saying, isn't restating 2 in 9 not necessary? Well, actually, yes it is, and no it isn't. For a completely rigorous, no-short cut, no-shorthand, first order logic proof, it is necessary, even though it's obvious. This is because 2 and 9 are actually slightly different statements; 2 is 2=S(S(0)), and 9 is S(S(0))=2. Concluding 9 from 2 might seem obvious, but it is a step itself, and also relies on the symmetry property included in Id. If 10 was concluded from 8,*2,* and Id. instead of 8, *9,* and Id., then a step would be skipped. Actually, when I first wrote the proof, I did skip this step. Then, reading it over, I realized this mistake, and corrected it. I know that still, in order for the proof to be complete, it would need to mention universal instantiation, but I made it as complete and rigorous as I could without getting too complicated. Also, the proof would have been better if I implicitly didn't assume that everything (at least 1 and 2) was a natural number, but that would have introduced even more complications, and notation.

There were two main confusions in the comments. We'll start with the confusion on the definition of 2. The reasoning was, in the proof, I use the fact that S(S(0))=2, but this fact relies on 1+1=2, so doesn't that make the proof circular?

Here's why not. It's because our precise definition of 2 is 2=S(S(0)). We could define 2 with 2=1+1, but this seems less natural when defining the set of natural numbers, and it wouldn't make a very fun story, since 1+1=2 would barely require a proof. But why does the definition 2=S(S(0)) make more sense?

Out of the recursive definition of natural numbers, using Peano's axioms, we get that we can list the natural numbers in the following way:

0

S(0)

S(S(0))

S(S(S(0)))

S(S(S(S(0))))

S(S(S(S(S(0)))))

...

Naturally, we want to line this up with the names we are used to, which have yet to be defined, so we define them as follows:

0=0

1=S(0)

2=S(S(0))

3=S(S(S(0)))

4=S(S(S(S(0))))

5=S(S(S(S(S(0)))))

...

Notice that the ellipsis means that it would take infinite definitions to define the symbols for all natural numbers, without formally defining a pattern. However, this turns out not to matter that much, since we can talk about individual natural numbers just fine, and the Peano axioms define the whole set of natural numbers.

Another confusion might be thinking that the definition of S is S(n)=n+1. While this statement is definitely a true statement, it can be proven to be true from the actual definition of S, which is in the Peano axioms. S(n)=n+1 is not a definition though, because addition is defined using S, so the reverse cannot be true. It is possible that one could define the natural numbers using addition instead of the function S, but S is simpler because it only has one input. If you used the definition of the natural numbers that relies only on addition, then yes, S(n)=n+1 would probably be your go-to definition.

I've been talking about the Peano axioms a lot, and multiple times, have been saying the equivalent of, "Don't worry if there seem to be all of these problems, the Peano axioms solve them." So, instead of leaving you to trust me, I'll just list them out. First, we have the axioms that define the set of natural numbers using the successor function, S (when n is said to be a natural number, it means that n is in the set of natural numbers):

S1) There exists a natural number, 0.

S2) For any natural numbers m and n, m=n iff S(m)=S(n). (iff means if and only if)

S3) There exists no natural number n where S(n)=0.

S4) For any set H where 0 is in H and (for any n) S(n) is an element of H if n is in H, all of the natural numbers are in H.

Notice that from these axioms, we already have all the natural numbers, and can thus give them names such as 1 and 2. From a certain point of view, we have defined the natural numbers by capturing the idea of counting using the successor function. There are also axioms that define addition on the set of natural numbers:

A1) For any natural number n, n+0=n.

A2) For any natural numbers m and n, m+S(n)=S(m+n).

Notice that the two axioms above are not needed to define the natural numbers. They are only needed to define addition, and can be used to show things such as S(0)+S(0)=S(S(0)), aka 1+1=2.

If this jumble of definitions is confusing, I'll give you another example of math where which statements are regarded as definitions and which statements are regarded as theormems is arbitrary. Recall that π is defined to be the circumference of any circle divided by its diameter. Using mathematical analysis (pure math part of calculus but more rigorous), it can be proven that π is the area of any circle divided by its radius squared. However, this could also be taken to be the definition of π, and then it would need to be and could be proven that π is the circumference of any circle divided by its diameter. It's arbitrary which one of these statements is the definition and which one is a theorem, just like it's arbitrary whether 2 should be defined as 2=1+1 or 2=S(S(0)).

I think that cleared up the first confusion. If I made any typos or errors, or you're still confuse, I encourage you to say so in the comments.

The second confusion is a lot easier to explain. It was about the first order logic proof, saying, isn't restating 2 in 9 not necessary? Well, actually, yes it is, and no it isn't. For a completely rigorous, no-short cut, no-shorthand, first order logic proof, it is necessary, even though it's obvious. This is because 2 and 9 are actually slightly different statements; 2 is 2=S(S(0)), and 9 is S(S(0))=2. Concluding 9 from 2 might seem obvious, but it is a step itself, and also relies on the symmetry property included in Id. If 10 was concluded from 8,

Hope that cleared up some confusions. As usual, comment if you notice a mistake or typo, or are still confused.