>Isn't induction somewhat a mathematical fallacy since you're proving something based on an assumption?
In some systems (peano arithmetic) induction is an axiom, in our modern mathematics it is a theorem.
Here is the theorem that we call "induction":
Let N be the set of natural numbers.
If the set M contains the first natural number (1 or 0 depending on your construction)
and if for every element of M, its successor is also in M then M=N
Then induction works like this. Suppose you want to prove: Every natural number is bigger than or equal than 0 (for simplicity).
Then let M be the set of all numbers that are bigger than or equal than 0.
Well, 0=0 so 0 must be in M
Then let x be any element of M. That means that x is larger than or equal than 0
then the successor of x is x+1 and x+1 is larger than x, and x is larger than or equal to 0, therefore x+1 is larger than or equal to 0.
Therefore every successor of an element of M, is in M.
Then M=N
So this is true for all natural numbers.
This is the justification. Non mathematicians are obviously allowed to use the babby unrigorous version of this because it is really equivalent so who cares.
