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Constructing $x+1$ creates a contradiction because the proof initially assumes that $x$ is the last natural number. However, by adding 1 to $x$, we get $x+1$, which is still a natural number and is strictly larger than $x$.

Conditions: The proof assumes there is a finite amount of $\mathbb{N}$.; $x$ is defined as the last natural number under this assumption.; The arithmetic property used is that if $x \in \mathbb{N}$, then $x+1 \in \mathbb{N}$ and $x+1 > x$.

Constructing $x+1$ creates a contradiction because the proof initially assumes that $x$ is the last natural number. However, by adding 1 to $x$, we get $x+1$, which is still a natural number and is strictly larger than $x$.

Conditions: The proof assumes there is a finite amount of $\mathbb{N}$.; $x$ is defined as the last natural number under this assumption.; The arithmetic property used is that if $x \in \mathbb{N}$, then $x+1 \in \mathbb{N}$ and $x+1 > x$.

The video proves that there are infinitely many natural numbers by using a proof by contradiction. It assumes the opposite—that there is a finite amount of natural numbers—and then shows that this assumption leads to a logical contradiction.

Conditions: The proof method is proof by contradiction (PBC).; The set under discussion is $\mathbb{N}$, the natural numbers.; The arithmetic fact used is that adding 1 to a natural number yields a larger natural number.

The video proves that there are infinitely many natural numbers by using a proof by contradiction. It assumes the opposite—that there is a finite amount of natural numbers—and then shows that this assumption leads to a logical contradiction.

Conditions: The proof method is proof by contradiction (PBC).; The set under discussion is $\mathbb{N}$, the natural numbers.; The arithmetic fact used is that adding 1 to a natural number yields a larger natural number.

In this video, PBC stands for "proof by contradiction." It is a mathematical method used to prove a statement by assuming the opposite of the desired statement and deriving an inconsistency or contradiction. The video explicitly abbreviates "proof by contradiction" as "Pbc." on the page before starting the first example.

Conditions: The context is a mathematical proof.; The method involves assuming the negation of the target statement.

In this video, PBC stands for "proof by contradiction." It is a mathematical method used to prove a statement by assuming the opposite of the desired statement and deriving an inconsistency or contradiction. The video explicitly abbreviates "proof by contradiction" as "Pbc." on the page before starting the first example.

Conditions: The context is a mathematical proof.; The method involves assuming the negation of the target statement.