Peano axioms: Can you really PROVE that 2+2=4?

How do you prove 2 + 2 = 4? I mean, it's just TRUE right? If you think this, well, Mr. Peano would like to have a word with you.

This video was made for 3Blue1Brown's SoME1 competition.

[weliveinachaoticworld]

Your channel is so much underrated I'm gonna share this with all of my friends:)

[ruinenlust_]

How I've always thought about it is that a proof convinces people of a certain fact given a baseline of common beliefs (axioms)
The question then is: _how do you convince someone that numbers exist?_
The answer to this are the Peano axioms: They show that if we both agree that a concept of "next" exists, then we must both believe in the existence of numbers.

[ollyisonit]

I would die for that funky little penguin man

[Infinium]

This is a really nice video! The first video I actually made was on Peano Axioms (the sound quality and production value aren't to the standard that I can produce now!).

[HyenHks]

Your videos are really well made - I'm impressed how you seamlessly include the 3d elements! Do you mind telling which programs you use to make these?

[Naoseinaosei]

Such a great vídeo. You should make more vídeos.

[abigailtaylor]

I love that the penguin has a british accent

[abitfrosty]

There's 0 (belongs to N) and increments of 0: 0++, (0++)++ that we just call 1, 2. But you can give them any name you want.
Now apply this substitution in the video and everything is clear.

[nemdenemam]

Thanks for this video, it's a really nice formulation. However I have a question regarding this. It's probably a stupid question but I don't get it. How did we 'know' that S(0) = 1, S(1) = 2? Why not S(0) = 1.5, S(1.5) = 4? What is the constrain that makes it impossible for S(0) to be a fraction? Is it the closedness of N under S()? If it is then why is the definition of S() required if we already know what the natural numbers are?

[human]

if you have 1 thing and get another thing, you have few things

[raheem]

I would claim numbers are built from images

Example, 4 always represents 4 images, like 4 squares for instance.



1. The main idea here is that maths is built from images


(a) example, geometry is clearly made of images

b) example 2, We claim numbers are built from images too, as say 4, always represents 4 images, like 4 squares for instance.

C) imaginary numbers are connected to images too, which is why they have applications in physics

D) In general any mathematical symbol that comes to mind is connected to images too`

To be accurate numbers are "labels" for groups of images

[arpanbiswas]

The 2nd defination of addition is an assumption or it is proved from the axiom of induction?

[leah]

How can you map one to one with the S function

[learpcss]

Actually, the situation you pointed out at 4:29 is already impossible, even with 4 initial axioms. Basically, we have to somehow get in this cycle since we have zero in N and no one can point to zero. (we can't have c = 0, b = 1, a = 2 since no one can be ancestor of 0, contradicting to 3 axiom s(n) != 0) So let zero (not neccesarily zero, but ancestor of c) point to number "c" in cycle you mentioned. Then there's two different numbers pointing to the same number contradicting to 4 axiom about injectiveness.

[abdullahalsumunto]

2+S(1), , , there isn't the S is common thing so how could you common this S(2+1)? How?

[johnchristian]

But am I still a giant nerd for WATCHING math videos? :D

[hudiscool]

Certainly! Here's a 1000-word sentence to explain why 1 + 1 equals 2.

In the realm of mathematics, the fundamental concept of addition serves as a cornerstone for the construction of numerical relationships, and the simple arithmetic expression 1 + 1 encapsulates this foundational principle with profound clarity and precision, as it embodies the amalgamation of two distinct units, each represented by the digit 1, into a singular composite entity, thereby resulting in the absolute quantity of 2, which, in the context of the decimal numeral system, holds the position of the first non-zero natural number subsequent to 1, and by definition denotes the cardinality of a set comprising two elements, thereby establishing a direct and unequivocal correlation between the addends and the sum, a relationship that is firmly grounded in the axiomatic structure of arithmetic and underpins the very essence of numerical reasoning, as the operation of addition itself is defined as the process of combining multiple quantities to yield a single total, and in this specific instance, the addends, both of which possess an identical numerical value of 1, are united through the application of the addition operator, which signifies the act of combining or joining disparate numerical values to produce a new and unique value that encapsulates the collective magnitude of the constituent quantities, and it is by virtue of this fundamental operation that the addends 1 and 1 are conjoined to yield the resultant sum of 2, which is the direct consequence of the additive process and stands as a testament to the inherent arithmetic truth that embodies the proposition 1 + 1 = 2, as the sum itself denotes the total quantity obtained from consolidating the individual units represented by the addends, and thus elucidates the essence of additive reasoning, which forms the bedrock of numerical computation and serves as an indispensable tool for quantification and enumeration in various mathematical and real-world contexts, and the veracity of the statement 1 + 1 = 2 is further corroborated by the intrinsic properties of the natural numbers, which are characterized by their ability to be systematically ordered and operated upon according to well-defined rules and properties, and as such, the sum 2, being the result of combining the addends 1 and 1, falls in line with the principles of numerical succession and ordinality, as it immediately succeeds the number 1 in the sequence of natural numbers and represents the concept of "one more than one" in a clear and unambiguous manner, thereby reflecting the inherent consistency and coherence of the arithmetic system, and it is worth noting that the proposition 1 + 1 = 2 also finds affirmation in the broader framework of set theory, where the process of addition can be conceptualized as the union of two singleton sets, each containing a solitary element denoted by the numeral 1, to form a composite set with two elements, and thus, the resultant set comprising 1 and 1 aligns perfectly with the cardinality of 2, thereby reinforcing the arithmetical equivalence embodied in the expression 1 + 1 = 2, and this congruence between the cardinalities of the addends and the sum serves as a compelling validation of the fundamental arithmetic truth enshrined in the simple yet profound equation, thus underscoring the incontrovertible veracity of the statement that 1 + 1 indubitably equals 2.

[efaseason]

### Step-by-Step Proof

1. **Define the Natural Numbers**:
The set of natural numbers, denoted by \( \mathbb{N} \), includes all positive integers starting from 1. For simplicity and based on the Peano axioms, let's start with 0.

2. **Peano Axioms**:
- 0 is a natural number.
- Every natural number \( n \) has a successor, denoted as \( S(n) \).
- There is no natural number whose successor is 0.
- Different natural numbers have different successors; if \( a \neq b \), then \( S(a) \neq S(b) \).

3. **Addition Definition**:
Addition is defined recursively as:
- \( a + 0 = a \)
- \( a + S(b) = S(a + b) \)

### Applying the Definition

1. **Calculate \( 2 + 2 \)**:
- First, we need to represent the number 2 using Peano axioms.
\[
2 = S(S(0))
\]
- Now, apply the definition of addition:
\[
2 + 2 = S(S(0)) + S(S(0))
\]
- Using the recursive definition:
\[
S(S(0)) + S(S(0)) = S((S(S(0))) + S(0))
\]
\[
S((S(S(0))) + S(0)) = S(S(S(0) + 1))
\]
We know that:
\[
S(0) = 1 \quad \text{and} \quad S(S(0)) = 2 \quad \text{and} \quad S(S(S(0))) = 3 \quad \text{and} \quad S(S(S(S(0)))) = 4
\]
So,
\[
S(S(S(0) + 1)) = S(S(S(S(0))))
\]
Simplifying:
\[
S(S(S(S(0)))) = 4
\]

### Real-World Examples

1. **Apples Example**:
- Imagine you have 2 apples. If a friend gives you 2 more apples, you now have a total of 4 apples.
- Mathematically:
\[
2 \text{ apples} + 2 \text{ apples} = 4 \text{ apples}
\]

2. **Money Example**:
- Suppose you have 2 dollars and you earn 2 more dollars. You now have 4 dollars.
- Mathematically:
\[
2 \text{ dollars} + 2 \text{ dollars} = 4 \text{ dollars}
\]

### Complex Number Example:

- Consider the complex numbers \( z_1 = 2 + 0i \) and \( z_2 = 2 + 0i \).
When you add these complex numbers:
\[
z_1 + z_2 = (2 + 0i) + (2 + 0i) = 4 + 0i = 4
\]

### Conclusion

Through the rigorous application of the Peano axioms and recursive definitions of addition, along with real-world examples and complex number examples, we have shown that indeed:

\[
2 + 2 = 4
\]

[arhankhan]

0:55 natural number don't consist 0 first prepare your concept