3 ms·
"For all natural numbers, m = n if and only if S(m) = S(n)." Hmm, er, ah, ... That's an odd way to put it.
by mcguire 7y ago
"For all natural numbers, m = n if and only if S(m) = S(n)."
Hmm, er, ah, ... That's an odd way to put it.
- tialaramex 7y agoThe purpose of this is to make the successor function S an injection. That is, it assures us that all natural numbers are the successor to exactly one other natural number, except that zero is first and so it isn't the successor to anything (in the natural numbers). The number line you saw in primary school is fundamental to arithmetic. Without this rule we might worry that perhaps there can be other numbers between five and six, or that if we all count up from one some of us might never reach a hundred because we inadvertently divert and get to six million early. These concerns seem intuitively nonsensical, but Peano doesn't want to rely on intuition, he wants to use logic.
- mcguire 7y agoBut normally it would be written "For all natural numbers, S(m) = S(n) if and only if m = n." "if and only if" is symmetric, but English isn't, really.
- lidHanteyk 7y agoLucky 10000: "If and only if" [0] is a particular phrase in English with a particular meaning. It is symmetric, so that your version has the same logical content as the original. [0] https://en.wikipedia.org/wiki/If_and_only_if https://en.wikipedia.org/wiki/If_and_only_if
- mcguire 7y agoThe same logical content, yes, but... Do you write the Fibonacci function as "n * (n-1) = fib(n)"?
- Dylan16807 7y agoI don't know why you think it's backwards. It makes lots of sense to put "m=n" first. It's giving the definition of equality. "[term] is [definition]".